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V. Jakubský

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Università degli Studi di Bergamo (University of Bergamo)

Adebayo D Agunbiade

Olabisi Onabanjo University, Ago-Iwoye, Nigeria

Indian Institute of Science Education and Research Mohali

Nikos Mantzakouras

National & Kapodistrian University of Athens

María Cecilia Tomasini

Universidad de Buenos Aires

University of California, Santa Cruz

Saleh Sultansoy

TOBB University of Economics and Technology

Yildiz Technical University

California Institute of Technology

Edward Bormashenko

Ariel University

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Papers by V. Jakubský

Lieb lattices and pseudospin-1 dynamics under barrier- and well-like electrostatic interactions

Physica E: Low-dimensional Systems and Nanostructures

This work considers the confining and scattering phenomena of electrons in a Lieb lattice subject... more This work considers the confining and scattering phenomena of electrons in a Lieb lattice subjected to the influence of a rectangular electrostatic barrier. In this setup, hopping amplitudes between nearest neighbors in orthogonal directions are considered different, and the next-nearest neighbor interaction describes spin-orbit coupling. This makes it possible to confine electrons and generate bound states, the exact number of which is exactly determined for null momentum parallel to the barrier. In such a case, it is proved that one even and one odd bound state is always generated, and the number of bound states increases for nonnull and increasing values of the parallel momentum. That is, current-carrying states are generated. In the scattering regime, the energies are determined so that resonant tunneling occurs. The existence of perfect tunneling energy in the form of super-Klein tunneling is proved to exist regardless of the bang gap opening. Finally, it is shown that perfect reflection appears when solutions are coupled to the intermediate flat-band solution.

Landau levels and snake states of pseudo-spin-1 Dirac-like electrons in gapped Lieb lattices

Journal of Physics: Condensed Matter

This work reports the three-band structure associated with a Lieb lattice with arbitrary nearest ... more This work reports the three-band structure associated with a Lieb lattice with arbitrary nearest and next-nearest neighbors hopping interactions. For specific configurations, the system admits a flat band located between two dispersion bands, where three inequivalent Dirac valleys are identified. Furthermore, quasi-particles are effectively described by a spin-1 Dirac-type equation. Under external homogeneous magnetic fields, the Landau levels are exactly determined as the third-order polynomial equation for the energy can be solved using Cardano’s formula. It is also shown that an external anti-symmetric field promotes the existence of current-carrying states, so-called snake states, confined at the interface where the external field changes its sign.

Coupled system of Dirac fermions with different Fermi velocities via composites of SUSY operators

Physics Letters A, 2022

We use the framework of supersymmetric transformations in the construction of coupled systems of ... more We use the framework of supersymmetric transformations in the construction of coupled systems of Dirac fermions. Its energy operator is a composite of the generators of the associated superalgebra, and the two coupled Dirac fermions acquire different Fermi velocities. We discuss in detail the peculiar spectral properties of the new system. The emergent phenomena like level crossing or generation of bound states in the continuum (BICs) are illustrated in two explicit examples.

Pseudo-Hermitian Operators in a Description of Physical Systems

Acta Polytechnica, 2007

We present some basic features of pseudo-hermitian quantum mechanics and illustrate the use of ps... more

Aharonov-Bohm Effect and the Supersymmetry of Identical Anyons

We briefly review the relation between the Aharonov-Bohm effect and the dynamical realization of ... more

The origin of the hidden supersymmetry

Physics Letters B, 2010

The hidden supersymmetry and related tri-supersymmetric structure of the free particle system, th... more The hidden supersymmetry and related tri-supersymmetric structure of the free particle system, the Dirac delta potential problem and the Aharonov-Bohm effect (planar, bound state, and tubule models) are explained by a special nonlocal unitary transformation, which for the usual N = 2 supercharges has a nature of Foldy-Wouthuysen transformation. We show that in general case, the bosonized supersymmetry of nonlocal, parity even systems emerges in the same construction, and explain the origin of the unusual N = 2 supersymmetry of electron in three-dimensional parity even magnetic field. The observation extends to include the hidden superconformal symmetry.

Trigonometric identities, angular Schrödinger equations and a new family of solvable models

Physics Letters A, 2005

Angular parts of certain solvable models are studied. We find that an extension of this class may... more Angular parts of certain solvable models are studied. We find that an extension of this class may be based on suitable trigonometric identities. The new exactly solvable Hamiltonians are shown to describe interesting two-and three-particle systems of the generalized Calogero, Wolfes and Winternitz-Smorodinsky types.

Comment on: “Solution of the Dirac equation for the Woods–Saxon potential with spin and pseudospin symmetry” [Phys. Lett. A 338 (2005) 90]

Physics Letters A, 2006

The bound-state method employed by Guo and Sheng (loc. cit.) is shown inadequate since only one o... more The bound-state method employed by Guo and Sheng (loc. cit.) is shown inadequate since only one of their solutions remains compatible, in the spin-symmetric low-mass regime, with the physical boundary conditions. We clarify the problem and construct a new, correct solution in the pseudospin-symmetric regime.

Self-Isospectrality, Special Supersymmetry, and their Effect on the Band Structure

Physical Review Letters, 2008

We study a planar model of a non-relativistic electron in periodic magnetic and electric fields t... more We study a planar model of a non-relativistic electron in periodic magnetic and electric fields that produce a 1D crystal for two spin components separated by a half-period spacing. We fit the fields to create a self-isospectral pair of finite-gap associated Lamé equations shifted for a half-period, and show that the system obtained is characterized by a new type of supersymmetry. It is a special nonlinear supersymmetry generated by three commuting integrals of motion, related to the parityodd operator of the associated Lax pair, that coherently reflects the band structure and all its peculiarities. In the infinite period limit it provides an unusual picture of supersymmetry breaking.

Klein tunneling in carbon nanostructures: A free-particle dynamics in disguise

Physical Review D, 2011

The absence of backscattering in metallic nanotubes as well as perfect Klein tunneling in potenti... more The absence of backscattering in metallic nanotubes as well as perfect Klein tunneling in potential barriers in graphene are the prominent electronic characteristics of carbon nanostructures. We show that the phenomena can be explained by a peculiar supersymmetry generated by a first order Hamiltonian and zero-order supercharge operators. Like the supersymmetry associated with second order reflectionless finite-gap systems, it relates here the low-energy behavior of the charge carriers with the free-particle dynamics.

Supersymmetry in spherical molecules and fullerenes under perpendicular magnetic fields

Journal of Physics: Condensed Matter, 2013

Methods of supersymmetric quantum mechanics are used to obtain analytical solutions for massless ... more

Finite-gap systems, tri-supersymmetry and self-isospectrality

Journal of Physics A: Mathematical and Theoretical, 2008

We show that an n-gap periodic quantum system with parity-even smooth potential admits 2 n − 1 is... more We show that an n-gap periodic quantum system with parity-even smooth potential admits 2 n − 1 isospectral super-extensions. Each is described by a tri-supersymmetry that originates from a higher-order differential operator of the Lax pair and two-term nonsingular decompositions of it; its local part corresponds to a spontaneously partially broken centrally extended nonlinear N = 4 supersymmetry. We conjecture that any finite-gap system having antiperiodic singlet states admits a self-isospectral trisupersymmetric extension with the partner potential to be the original one translated for a half-period. Applying the theory to a broad class of finite-gap elliptic systems described by a two-parametric associated Lamé equation, our conjecture is supported by the explicit construction of the self-isospectral tri-supersymmetric pairs. We find that the spontaneously broken tri-supersymmetry of the self-isospectral periodic system is recovered in the infinite period limit.

Carbon nanotubes in an inhomogeneous transverse magnetic field: exactly solvable model

Journal of Physics A: Mathematical and Theoretical, 2014

A class of exactly solvable models describing carbon nanotubes in the presence of an external inh... more A class of exactly solvable models describing carbon nanotubes in the presence of an external inhomogeneous magnetic field is considered. The framework of the continuum approximation is employed, where the motion of the charge carriers is governed by the Dirac-Weyl equation. The explicit solution of a particular example is provided. It is shown that these models possess nontrivial integrals of motion that establish N = 2 nonlinear supersymmetry in case of metallic and maximally semiconducting nanotubes. Remarkable stability of energy levels with respect to small fluctuations of longitudinal momentum is demonstrated.

Hidden superconformal symmetry of the spinless Aharonov–Bohm system

Journal of Physics A: Mathematical and Theoretical, 2010

A hidden supersymmetry is revealed in the spinless Aharonov-Bohm problem. The intrinsic supersymm... more A hidden supersymmetry is revealed in the spinless Aharonov-Bohm problem. The intrinsic supersymmetric structure is shown to be intimately related to the scale symmetry. As a result, a bosonized superconformal symmetry is identified in the system. Different self-adjoint extensions of the Aharonov-Bohm problem are studied in the light of this superconformal structure and interacting anyons. The scattering problem of the original Aharonov-Bohm model is discussed in the context of the revealed supersymmetry.

Solvability and -symmetry in a double-well model with point interactions

Journal of Physics A: Mathematical and General, 2005

The concept of point interactions offers one of the most suitable guides towards a quantitative a... more The concept of point interactions offers one of the most suitable guides towards a quantitative analysis of properties of certain specific non-Hermitian (so called PT −symmetric) quantum-mechanical systems. This is illustrated on a double-well model, an easy solvability of which is shown to lead to a clear picture of the mechanisms of the unavoided level crossing and of the spontaneous PT symmetry breaking at a certain strong-non-Hermiticity boundary. Below this limit the model is shown suitable for an explicit illustration of technicalities related to the standard probabilistic physical interpretation of bound states in PT −symmetric quantum mechanics.

$Research paper thumbnail of ${\mathcal{PT}}$-SYMMETRIC SUPERSYMMETRY IN A SOLVABLE SHORT-RANGE MODEL$

${\mathcal{PT}}$-SYMMETRIC SUPERSYMMETRY IN A SOLVABLE SHORT-RANGE MODEL

International Journal of Modern Physics A, 2006

The simplest purely imaginary and piecewise constant [Formula: see text]-symmetric potential loca... more The simplest purely imaginary and piecewise constant [Formula: see text]-symmetric potential located inside a larger box is studied. Unless its strength exceeds a certain critical value, all the spectrum of its bound states remains real and discrete. We interpret such a model as an initial element of the generalized non-Hermitian Witten's hierarchy of solvable Hamiltonians and construct its first supersymmetric partner in closed form.

Weakly non-Hermitian square well

Czechoslovak Journal of Physics, 2005

ABSTRACT Inside a box of size L we contemplate the simplest -symmetric piece-wise constant potent... more

Aharonov–Bohm effect on AdS2 and nonlinear supersymmetry of reflectionless Pöschl–Teller system

Annals of Physics, 2009

We explain the origin and the nature of a special nonlinear supersymmetry of a reflectionless Pös... more We explain the origin and the nature of a special nonlinear supersymmetry of a reflectionless Pöschl-Teller system by the Aharonov-Bohm effect for a nonrelativistic particle on the AdS 2. A key role in the supersymmetric structure appearing after reduction by a compact generator of the AdS 2 isometry is shown to be played by the discrete symmetries related to the space and time reflections in the ambient Minkowski space. We also observe that a correspondence between the two quantum non-relativistic systems is somewhat of the AdS/CFT holography nature.

Supersymmetries of the spin-1/2 particle in the field of magnetic vortex, and anyons

Annals of Physics, 2010

$Research paper thumbnail of Form-preserving Darboux transformations for $$4\times 4$$ Dirac equations$

Form-preserving Darboux transformations for $$4\times 4$$ Dirac equations

The European Physical Journal Plus

Darboux transformation is a powerful tool for the construction of new solvable models in quantum ... more Darboux transformation is a powerful tool for the construction of new solvable models in quantum mechanics. In this article, we discuss its use in the context of physical systems described by 4 × 4 Dirac Hamiltonians. The general framework provides limited control over the resulting energy operator, so that it can fail to have the required physical interpretation. We show that this problem can be circumvented with the reducible Darboux transformation that can preserve the required form of physical interactions by construction. To demonstrate it explicitly, we focus on distortion scattering and spin-orbit interaction of Dirac fermions in graphene. We use the reducible Darboux transformation to construct exactly solvable models of these systems where backscattering is absent, i.e. the models are reflectionless.

Lieb lattices and pseudospin-1 dynamics under barrier- and well-like electrostatic interactions

Physica E: Low-dimensional Systems and Nanostructures

This work considers the confining and scattering phenomena of electrons in a Lieb lattice subject... more This work considers the confining and scattering phenomena of electrons in a Lieb lattice subjected to the influence of a rectangular electrostatic barrier. In this setup, hopping amplitudes between nearest neighbors in orthogonal directions are considered different, and the next-nearest neighbor interaction describes spin-orbit coupling. This makes it possible to confine electrons and generate bound states, the exact number of which is exactly determined for null momentum parallel to the barrier. In such a case, it is proved that one even and one odd bound state is always generated, and the number of bound states increases for nonnull and increasing values of the parallel momentum. That is, current-carrying states are generated. In the scattering regime, the energies are determined so that resonant tunneling occurs. The existence of perfect tunneling energy in the form of super-Klein tunneling is proved to exist regardless of the bang gap opening. Finally, it is shown that perfect reflection appears when solutions are coupled to the intermediate flat-band solution.

Landau levels and snake states of pseudo-spin-1 Dirac-like electrons in gapped Lieb lattices

Journal of Physics: Condensed Matter

This work reports the three-band structure associated with a Lieb lattice with arbitrary nearest ... more This work reports the three-band structure associated with a Lieb lattice with arbitrary nearest and next-nearest neighbors hopping interactions. For specific configurations, the system admits a flat band located between two dispersion bands, where three inequivalent Dirac valleys are identified. Furthermore, quasi-particles are effectively described by a spin-1 Dirac-type equation. Under external homogeneous magnetic fields, the Landau levels are exactly determined as the third-order polynomial equation for the energy can be solved using Cardano’s formula. It is also shown that an external anti-symmetric field promotes the existence of current-carrying states, so-called snake states, confined at the interface where the external field changes its sign.

Coupled system of Dirac fermions with different Fermi velocities via composites of SUSY operators

Physics Letters A, 2022

We use the framework of supersymmetric transformations in the construction of coupled systems of ... more We use the framework of supersymmetric transformations in the construction of coupled systems of Dirac fermions. Its energy operator is a composite of the generators of the associated superalgebra, and the two coupled Dirac fermions acquire different Fermi velocities. We discuss in detail the peculiar spectral properties of the new system. The emergent phenomena like level crossing or generation of bound states in the continuum (BICs) are illustrated in two explicit examples.

Pseudo-Hermitian Operators in a Description of Physical Systems

Acta Polytechnica, 2007

We present some basic features of pseudo-hermitian quantum mechanics and illustrate the use of ps... more

Aharonov-Bohm Effect and the Supersymmetry of Identical Anyons

We briefly review the relation between the Aharonov-Bohm effect and the dynamical realization of ... more

The origin of the hidden supersymmetry

Physics Letters B, 2010

The hidden supersymmetry and related tri-supersymmetric structure of the free particle system, th... more The hidden supersymmetry and related tri-supersymmetric structure of the free particle system, the Dirac delta potential problem and the Aharonov-Bohm effect (planar, bound state, and tubule models) are explained by a special nonlocal unitary transformation, which for the usual N = 2 supercharges has a nature of Foldy-Wouthuysen transformation. We show that in general case, the bosonized supersymmetry of nonlocal, parity even systems emerges in the same construction, and explain the origin of the unusual N = 2 supersymmetry of electron in three-dimensional parity even magnetic field. The observation extends to include the hidden superconformal symmetry.

Trigonometric identities, angular Schrödinger equations and a new family of solvable models

Physics Letters A, 2005

Angular parts of certain solvable models are studied. We find that an extension of this class may... more Angular parts of certain solvable models are studied. We find that an extension of this class may be based on suitable trigonometric identities. The new exactly solvable Hamiltonians are shown to describe interesting two-and three-particle systems of the generalized Calogero, Wolfes and Winternitz-Smorodinsky types.

Comment on: “Solution of the Dirac equation for the Woods–Saxon potential with spin and pseudospin symmetry” [Phys. Lett. A 338 (2005) 90]

Physics Letters A, 2006

The bound-state method employed by Guo and Sheng (loc. cit.) is shown inadequate since only one o... more The bound-state method employed by Guo and Sheng (loc. cit.) is shown inadequate since only one of their solutions remains compatible, in the spin-symmetric low-mass regime, with the physical boundary conditions. We clarify the problem and construct a new, correct solution in the pseudospin-symmetric regime.

Self-Isospectrality, Special Supersymmetry, and their Effect on the Band Structure

Physical Review Letters, 2008

We study a planar model of a non-relativistic electron in periodic magnetic and electric fields t... more We study a planar model of a non-relativistic electron in periodic magnetic and electric fields that produce a 1D crystal for two spin components separated by a half-period spacing. We fit the fields to create a self-isospectral pair of finite-gap associated Lamé equations shifted for a half-period, and show that the system obtained is characterized by a new type of supersymmetry. It is a special nonlinear supersymmetry generated by three commuting integrals of motion, related to the parityodd operator of the associated Lax pair, that coherently reflects the band structure and all its peculiarities. In the infinite period limit it provides an unusual picture of supersymmetry breaking.

Klein tunneling in carbon nanostructures: A free-particle dynamics in disguise

Physical Review D, 2011

The absence of backscattering in metallic nanotubes as well as perfect Klein tunneling in potenti... more The absence of backscattering in metallic nanotubes as well as perfect Klein tunneling in potential barriers in graphene are the prominent electronic characteristics of carbon nanostructures. We show that the phenomena can be explained by a peculiar supersymmetry generated by a first order Hamiltonian and zero-order supercharge operators. Like the supersymmetry associated with second order reflectionless finite-gap systems, it relates here the low-energy behavior of the charge carriers with the free-particle dynamics.

Supersymmetry in spherical molecules and fullerenes under perpendicular magnetic fields

Journal of Physics: Condensed Matter, 2013

Methods of supersymmetric quantum mechanics are used to obtain analytical solutions for massless ... more

Finite-gap systems, tri-supersymmetry and self-isospectrality

Journal of Physics A: Mathematical and Theoretical, 2008

We show that an n-gap periodic quantum system with parity-even smooth potential admits 2 n − 1 is... more We show that an n-gap periodic quantum system with parity-even smooth potential admits 2 n − 1 isospectral super-extensions. Each is described by a tri-supersymmetry that originates from a higher-order differential operator of the Lax pair and two-term nonsingular decompositions of it; its local part corresponds to a spontaneously partially broken centrally extended nonlinear N = 4 supersymmetry. We conjecture that any finite-gap system having antiperiodic singlet states admits a self-isospectral trisupersymmetric extension with the partner potential to be the original one translated for a half-period. Applying the theory to a broad class of finite-gap elliptic systems described by a two-parametric associated Lamé equation, our conjecture is supported by the explicit construction of the self-isospectral tri-supersymmetric pairs. We find that the spontaneously broken tri-supersymmetry of the self-isospectral periodic system is recovered in the infinite period limit.

Carbon nanotubes in an inhomogeneous transverse magnetic field: exactly solvable model

Journal of Physics A: Mathematical and Theoretical, 2014

A class of exactly solvable models describing carbon nanotubes in the presence of an external inh... more A class of exactly solvable models describing carbon nanotubes in the presence of an external inhomogeneous magnetic field is considered. The framework of the continuum approximation is employed, where the motion of the charge carriers is governed by the Dirac-Weyl equation. The explicit solution of a particular example is provided. It is shown that these models possess nontrivial integrals of motion that establish N = 2 nonlinear supersymmetry in case of metallic and maximally semiconducting nanotubes. Remarkable stability of energy levels with respect to small fluctuations of longitudinal momentum is demonstrated.

Hidden superconformal symmetry of the spinless Aharonov–Bohm system

Journal of Physics A: Mathematical and Theoretical, 2010

A hidden supersymmetry is revealed in the spinless Aharonov-Bohm problem. The intrinsic supersymm... more A hidden supersymmetry is revealed in the spinless Aharonov-Bohm problem. The intrinsic supersymmetric structure is shown to be intimately related to the scale symmetry. As a result, a bosonized superconformal symmetry is identified in the system. Different self-adjoint extensions of the Aharonov-Bohm problem are studied in the light of this superconformal structure and interacting anyons. The scattering problem of the original Aharonov-Bohm model is discussed in the context of the revealed supersymmetry.

Solvability and -symmetry in a double-well model with point interactions

Journal of Physics A: Mathematical and General, 2005

The concept of point interactions offers one of the most suitable guides towards a quantitative a... more The concept of point interactions offers one of the most suitable guides towards a quantitative analysis of properties of certain specific non-Hermitian (so called PT −symmetric) quantum-mechanical systems. This is illustrated on a double-well model, an easy solvability of which is shown to lead to a clear picture of the mechanisms of the unavoided level crossing and of the spontaneous PT symmetry breaking at a certain strong-non-Hermiticity boundary. Below this limit the model is shown suitable for an explicit illustration of technicalities related to the standard probabilistic physical interpretation of bound states in PT −symmetric quantum mechanics.

$Research paper thumbnail of ${\mathcal{PT}}$-SYMMETRIC SUPERSYMMETRY IN A SOLVABLE SHORT-RANGE MODEL$

${\mathcal{PT}}$-SYMMETRIC SUPERSYMMETRY IN A SOLVABLE SHORT-RANGE MODEL

International Journal of Modern Physics A, 2006

The simplest purely imaginary and piecewise constant [Formula: see text]-symmetric potential loca... more The simplest purely imaginary and piecewise constant [Formula: see text]-symmetric potential located inside a larger box is studied. Unless its strength exceeds a certain critical value, all the spectrum of its bound states remains real and discrete. We interpret such a model as an initial element of the generalized non-Hermitian Witten's hierarchy of solvable Hamiltonians and construct its first supersymmetric partner in closed form.

Weakly non-Hermitian square well

Czechoslovak Journal of Physics, 2005

ABSTRACT Inside a box of size L we contemplate the simplest -symmetric piece-wise constant potent... more

Aharonov–Bohm effect on AdS2 and nonlinear supersymmetry of reflectionless Pöschl–Teller system

Annals of Physics, 2009

We explain the origin and the nature of a special nonlinear supersymmetry of a reflectionless Pös... more We explain the origin and the nature of a special nonlinear supersymmetry of a reflectionless Pöschl-Teller system by the Aharonov-Bohm effect for a nonrelativistic particle on the AdS 2. A key role in the supersymmetric structure appearing after reduction by a compact generator of the AdS 2 isometry is shown to be played by the discrete symmetries related to the space and time reflections in the ambient Minkowski space. We also observe that a correspondence between the two quantum non-relativistic systems is somewhat of the AdS/CFT holography nature.

Supersymmetries of the spin-1/2 particle in the field of magnetic vortex, and anyons

Annals of Physics, 2010

$Research paper thumbnail of Form-preserving Darboux transformations for $$4\times 4$$ Dirac equations$

Form-preserving Darboux transformations for $$4\times 4$$ Dirac equations

The European Physical Journal Plus

Darboux transformation is a powerful tool for the construction of new solvable models in quantum ... more Darboux transformation is a powerful tool for the construction of new solvable models in quantum mechanics. In this article, we discuss its use in the context of physical systems described by 4 × 4 Dirac Hamiltonians. The general framework provides limited control over the resulting energy operator, so that it can fail to have the required physical interpretation. We show that this problem can be circumvented with the reducible Darboux transformation that can preserve the required form of physical interactions by construction. To demonstrate it explicitly, we focus on distortion scattering and spin-orbit interaction of Dirac fermions in graphene. We use the reducible Darboux transformation to construct exactly solvable models of these systems where backscattering is absent, i.e. the models are reflectionless.