Landau Problem Research Articles

Two-Dimensional Pauli Equation in Noncommutative Phase-Space

We study the Pauli equation in a two-dimensional noncommutative phase-space by considering a constant magnetic field perpendicular to the plane. The noncommutative problem is related to the equivalent commutative one through a set of two-dimensional Bopp-shift transformations. The energy spectrum and the wave function of the two-dimensional noncommutative Pauli equation are found, where the problem in question has been mapped to the Landau problem. In the classical limit, we have derived the noncommutative semiclassical partition function for one- and N- particle systems. The thermodynamic properties such as the Helmholtz free energy, mean energy, specific heat and entropy in noncommutative and commutative phasespaces are determined. The impact of the phase-space noncommutativity on the Pauli system is successfully examined.

Bloch waves and non-commutative tori of magnetic translations

We review the Landau problem of an electron in a constant uniform magnetic field. The magnetic translations are the invariant transformations of the free Hamiltonian. A Kähler polarization of the plane has been used for the geometric quantization. Under the assumption of quasi-periodicity of the wavefunction, the Zak’s magnetic translations in the Bravais lattice generate a non-commutative quantum torus. We concentrate on the case when the magnetic flux density is a rational number. The Bloch wavefunctions form a finite-dimensional module of the noncommutative torus of magnetic translations as well as of its commutant, which is the non-commutative torus of magnetic translations in the dual Bravais lattice. The bi-module structure of the Bloch waves is shown to be the connecting link between two Morita equivalent non-commutative tori. The main focus of our review is the Kähler structure on the Hilbert space of Bloch waves and its inherent quantum toric geometry. We reveal that the metaplectic group Mp(2,R) of the automorphisms of magnetic translation algebras is represented by the quantum optics squeezing operators.

Retraction Note to: Non-relativistic analytical solutions of the Kratzer potential for a perturbed system in a magnetic field

Non-relativistic analytical solutions of a quantum mechanical system, perturbed by an external magnetic field, are obtained for the case of Kratzer potential. Such problem is a kind of Landau problem named “Kratzer-Landau problem” due to the presence of the magnetic field. Paying attention to hyperspherical coordinates, an analytical formula for the Landau levels is achieved dealing with Schrodinger equation by using the perturbation method in the frame of the asymptotic iteration method. The analytical formula obtained is also tested by numerically cross-checking the energy eigenvalues with those of the literature in two dimensions (polar coordinates). Moreover, effects of the parameters, such as magnetic field strength and azimuthal angle (in three dimensions), on the eigenvalues are presented.

A note on broken dilatation symmetry in planar noncommutative theory

A riveting study of Ward-Takahashi identities is presented for a broken dilatation or scale symmetry in a generalized quantum Hall system. On using the “Peierls substitution” scheme, it is shown that noncommutativity between spatial coordinates emerges naturally at a large magnetic field limit. Thereafter, we derive a path-integral action for the corresponding noncommutative quantum system and discuss the equivalence between the considered noncommutative system and the generalized Landau problem thus rendering an effective commmutative description. We then derive an expression for the unintegrated scale or dilatation anomaly for the generalized Landau system using Fujikawa's method and is subsequently renormalised. In fact, we identify the Ward-Takahashi identities associated with broken dilatation symmetry to be anomalous which is a purely quantum effect induced from the noncommutative structure between spatial coordinates and therefore non-intuitive in nature.

Deformed quantum mechanics and the Landau problem

A deformation of the Landau problem based on a modification of Fock algebra is considered. Systems with the Hamiltonians [Formula: see text] where [Formula: see text] is the Landau Hamiltonian in the lowest level are discussed. The case [Formula: see text] is studied and it is shown that in this particular example, parameters of the problem can be fixed by using the quadratic Zeeman effect data and the Breit–Rabi formula.

Quantum walks simulating non-commutative geometry in the Landau problem

Non-Commutative Geometry (NCG) is considered in the context of a charged particle moving in a uniform magnetic field. The classical and quantum mechanical treatments are revisited, and a new marker of NCG is introduced. This marker is then used to investigate NCG in magnetic Quantum Walks (QWs). It is proven that these walks exhibit NCG at and near the continuum limit. For the purely discrete regime, two illustrative walks of different complexities are studied in full detail. The most complex walk does exhibit NCG, but the simplest, most degenerate one does not. Thus, NCG can be simulated by QWs, not only in the continuum limit but also in the purely discrete regime.

Gauge invariant energy spectra in 2-dimensional noncommutative quantum mechanics

In this paper, we consider an electron moving on a 2D noncommutative plane immersed in a constant magnetic field under the influence of a polynomial potential. We obtain gauge invariant energy spectra of this system using 2-parameter family of unitarily equivalent irreducible representations of the nilpotent Lie group GNC that were worked out in detail in Chowdhury (2017). We work out the cases of anisotropic harmonic potential and the Hall potential as physical applications of the proposed method. We also show subsequently that straightforward generalization of the Landau problem and the quantum Hall effect in the noncommutative setting using minimal coupling prescription as is done naively on many occasions in the literature violate gauge invariance of energy spectra.

Conformal generation of an exotic rotationally invariant harmonic oscillator

An exotic rotationally invariant harmonic oscillator (ERIHO) is constructed by applying a non-unitary isotropic conformal bridge transformation (CBT) to a free planar particle. It is described by the isotropic harmonic oscillator Hamiltonian supplemented by a Zeeman type term with a real coupling constant $g$. The model reveals the Euclidean ($|g|<1$) and Minkowskian ($|g|>1$) phases separated by the phases $g=+1$ and $g=-1$ of the Landau problem in the symmetric gauge with opposite orientation of the magnetic field. A hidden symmetry emerges in the system at rational values of $g$. Its generators, together with the Hamiltonian and angular momentum produce non-linearly deformed $\mathfrak{u}(2)$ and $\mathfrak{gl}(2,{\mathbb R})$ algebras in the cases of $0<|g|<1$ and $\infty>|g|>1$, which transmute one into another under the inversion $g\rightarrow -1/g$. Similarly, the true, $\mathfrak{u}(2)$, and extended conformal, $\mathfrak{gl}(2,{\mathbb R})$, symmetries of the isotropic Euclidean oscillator ($g=0$) interchange their roles in the isotropic Minkowskian oscillator ($|g|=\infty$), while two copies of the $\mathfrak{gl}(2,{\mathbb R})$ algebra of analogous symmetries mutually transmute in Landau phases. We show that the ERIHO system is transformed by a peculiar unitary transformation into the anisotropic harmonic oscillator generated, in turn, by anisotropic CBT. The relationship between the ERIHO and the subcritical phases of the harmonically extended Landau problem, as well as with a plane isotropic harmonic oscillator in a uniformly rotating reference frame, is established.

Emergent entropy of exotic oscillators and squeezing in three-wave mixing process

We demonstrate the existence of entanglement in a system of harmonic oscillators placed in the noncommutative plane (“exotic oscillators”) by computing the entanglement entropy as measured by the von Neumann entropy of the reduced density matrix. It is explicitly verified that the entanglement arises from the noncommutativity, which controls the coupling strength between the spatial modes. This can easily be generalized to the case where the momentum components also satisfy noncommutative relations, so that the entire phase space becomes noncommutative. It is noted that the underlying mathematical structure is reminiscent of the Unruh effect and using the Landau problem in the presence of a harmonic interaction, a concrete physical realization has been illustrated. Finally, we show that phase-space noncommutativity can give rise to the non-classical effect of squeezing, which generally results from the non-linearity of a medium in a three-wave mixing process.

Relativistic Landau and oscillator levels in a symmetric gauge field in a non-commutative complex space

In this work, we obtain the exact solution for the relativistic Landau problem plus oscillator potential in a complex symmetric gauge field in a non-commutative complex space, using the algebraic techniques of creation and annihilation operators. It is shown that the relativistic Landau problem in a complex symmetric gauge field in a non-commutative complex space has a similar behavior to the Pauli equation in the presence of a symmetric gauge field in a commutative ordinary space. We derive the exact non-commutative Landau and oscillator energy levels, while the non-relativistic limit of the energy spectrum is obtained. We show that the energy is not degenerate and is split into two levels as in the Zeeman effect.

Non-commutative phase space Landau problem in the presence of a minimal length

The deformed Landau problem under a electromagnetic field is studied, where the Heisenberg algebra is constructed in detail in non-commutative phase space in the presence of a minimal length. We show that, in the presence of a minimal length, the momentum space is more practical to solve any problem of eigenvalues. From the Nikiforov-Uvarov method, the energy eigenvalues are obtained and the corresponding wave functions are expressed in terms of hypergeometric functions. The fortuitous degeneration observed in the spectrum shows that the formulation of the minimal length complements that of the non-commutative phase space. Изучается деформированная задача Ландау в электромагнитном поле, в которой алгебра Гейзенберга подробно строится в некоммутативном фазовом пространстве при наличии минимальной длины. Мы показываем, что при наличии минимальной длины импульсное пространство более практично для решения любой проблемы собственных значений. С помощью метода Никифорова-Уварова получаются собственные значения энергии, а соответствующие волновые функции выражаются через гипергеометрические функции. Случайное вырождение, наблюдаемое в спектре, показывает, что формулировка минимальной длины дополняет формулировку некоммутативного фазового пространства.

The Landau problem and nonclassicality

Exploring the concept of the extended Galilei group [Formula: see text]. Representations for field theories in a symplectic manifold have been derived in association with the method of the Wigner function. The representation is written in the light-cone of a de Sitter space–time in five dimensions. A Hilbert space is constructed, endowed with a symplectic structure, which is used as a representation space for the Lie algebra of [Formula: see text]. This representation gives rise to the spin-0 Schrödinger (Klein–Gordon-like) equation for the wave functions in phase space, such that the dependent variables have the content of position and linear momentum. This is a particular example of a conformal theory, such that the wave functions are associated with the Wigner function through the Moyal product. We construct the Pauli–Schrödinger (Dirac-like) equation in phase space in its explicitly covariant form. In addition, we analyze the gauge symmetry for spin-1/2 particles in phase space and show how implement the minimal coupling in this case. We applied to the problem of an electron in an external field, and we recovered the nonrelativistic Landau levels. Finally, we study the parameter of negativity associated with the nonclassicality of the system.

Near-square primes conjecture

In 1912, Edmund Landau listed four basic problems about prime numbers in the International Congress of Mathematicians. These problems are now known as Landau's problems. Landau's fourth problem asked whether there are infinitely many primes which are of the form $n^{2}+1$ for some integer $n$. This problem remains open. We prove this conjecture is indeed true.

Resolvent-based optimal estimation of transitional and turbulent flows

Abstract

Role of guiding center in Landau level system and mechanical and pseudo orbital angular momenta

There is an interesting but not-so-popular quantity called pseudo orbital angular momentum (OAM) in the Landau-level system, besides the well-known canonical and mechanical OAMs. The pseudo OAM can be regarded as a gauge-invariant extension of the canonical OAM, which is formally gauge invariant and reduces to the canonical OAM in a certain gauge. Since both of the pseudo OAM and the mechanical OAM are gauge invariant, it is impossible to judge which of those is superior to the other solely from the gauge principle. However, these two OAMs have totally different physical meanings. The mechanical OAM shows manifest observability and clear correspondence with the classical OAM of the cyclotron motion. On the other hand, we demonstrate that the standard canonical OAM as well as the pseudo OAM in the Landau problem is the concept which crucially depend on the choice of the origin of the coordinate system. We try to reveal the relation between the pseudo OAM and the mechanical OAM as well as their observability by paying special attention to the role of guiding-center operator in the Landau problem.

Conformal bridge between asymptotic freedom and confinement

We construct a nonunitary transformation that relates a given "asymptotically free" conformal quantum mechanical system $H_f$ with its confined, harmonically trapped version $H_c$. In our construction, Jordan states corresponding to the zero eigenvalue of $H_f$, as well as its eigenstates and Gaussian packets are mapped into the eigenstates, coherent states and squeezed states of $H_c$, respectively. The transformation is an automorphism of the conformal $\mathfrak{sl}(2,{\mathbb R})$ algebra of the nature of the fourth-order root of the identity transformation, to which a complex canonical transformation corresponds on the classical level being the fourth-order root of the spatial reflection. We investigate the one- and two-dimensional examples that reveal, in particular, a curious relation between the two-dimensional free particle and the Landau problem.

Quantum mechanics in magnetic backgrounds with manifest symmetry and locality

The usual methods for formulating and solving the quantum mechanics of a particle moving in a magnetic field respect neither locality nor any global symmetries which happen to be present. For example, Landau’s solution for a particle moving in a uniform magnetic field in the plane involves choosing a gauge in which neither translation nor rotation invariance are manifest. We show that locality can be made manifest by passing to a redundant description in which the particle moves on a U(1)-principal bundle over the original configuration space and that symmetry can be made manifest by passing to a corresponding central extension of the original symmetry group by U(1). With the symmetry manifest, one can attempt to solve the problem by using harmonic analysis and we provide a number of examples where this succeeds. One is a solution of the Landau problem in an arbitrary gauge (with either translation invariance or the full Euclidean group manifest). Another example is the motion of a fermionic rigid body, which can be formulated and solved in a manifestly local and symmetric way via a flat connection on the non-trivial U(1)-central extension of the configuration space SO(3) given by U(2).

Symplectic Schrödinger equation for many-body systems

Using elements of symmetry, as gauge invariance, many aspects of a Schrödinger equation in phase space are analyzed. The number (Fock space) representation is constructed in phase space and the Green function, directly associated with the Wigner function, is introduced as a basic element of perturbative procedure. This phase space representation is applied to the Landau problem and the Liouville potential.

Rising and lowering operator approach to the problem of a charged particle in a uniform magnetic field

In this paper, we analyse the quantization of cyclotron orbits for a 2D non-relativistic and spinless charged particle under the influence of a static and uniform magnetic field, the Landau problem. We follow the algebraic approach of rising and lowering operators. The Hamiltonian is expressed in terms of the usual rising and lowering operators of an isotropic quantum oscillator. Through a Bogoliubov transformation, we show that the Hamiltonian may be reduced to the one of a quantum oscillator that accounts for the formation of infinitely degenerate Landau levels. The explicit form of the position-space wave functions of Fock–Darwin states for a vanishing parabolic confinement potential is worked out by applying the techniques of ladder operators and the induction method. To explore the internal dynamics of the electron states, we compute the Fock–Darwin states’ density currents expressing them in terms of the ladder operators. In the supplementary material, we provide the reader with Mathematica algorithm that can help perform some of the calculations exposed in the paper. The charged particle in a magnetic field is used to explore some of the fundamental concepts of quantum mechanics as commuting operators, simultaneous eigenstates, degeneracy, rising and lowering operators, etc. This approach can be used as supplementary material in undergraduate final-year courses like quantum mechanics or solid state physics.

Symmetries of deformed supersymmetric mechanics on Kähler manifolds

Based on the systematic Hamiltonian and superfield approaches we construct the deformed $\mathcal{N}=4,8$ supersymmetric mechanics on K\"ahler manifolds interacting with constant magnetic field, and study their symmetries. At first we construct the deformed $\mathcal{N}=4,8$ supersymmetric Landau problem via minimal coupling of standard (undeformed) $\mathcal{N}=4,8$ supersymmetric free particle systems on K\"ahler manifold with constant magnetic field. We show that the initial "flat" supersymmetries are necessarily deformed to $SU(2|1)$ and $SU(4|1)$ supersymmetries, with the magnetic field playing the role of deformation parameter, and that the resulting systems inherit all the kinematical symmetries of the initial ones. Then we construct $SU(2|1)$ supersymmetric K\"ahler oscillators and find that they include, as particular cases, the harmonic oscillator models on complex Euclidian and complex projective spaces, as well as superintegrable deformations thereof, viz. $\mathbb{C}^N$-Smorodinsky-Winternitz and $\mathbb{CP}^N$-Rosochatius systems. We show that the supersymmetric extensions proposed inherit all the kinematical symmetries of the initial bosonic models. They also inherit, at least in the case of $\mathbb{C}^N$ systems, hidden (non-kinematical) symmetries. The superfield formulation of these supersymmetric systems is presented, based on the worldline $SU(2|1)$ and $SU(4|1)$ superspace formalisms.