Landau Problem Research Articles

The giant graviton expansion in $AdS_5 \times S^5$

The superconformal index of \frac1212-BPS states of \mathcal{N}=4𝒩=4U(N)U(N) super Yang-Mills theory has a known infinite qq-series expression with successive terms suppressed by q^NqN. We derive a holographic bulk interpretation of this series by evaluating the corresponding functional integral in the dual AdS_5 × S^55×S5. The integral localizes to a product of small fluctuations of the vacuum and of the collective modes of an arbitrary number of giant-gravitons wrapping an S^3S3 of maximal size inside the S^5S5. The quantum mechanics of the small fluctuations of one maximal giant is described by a supersymmetric version of the Landau problem. The quadratic fluctuation determinant reduces to a sum over the supersymmetric ground states, and precisely reproduces the first non-trivial term in the infinite series. Further, we show that the terms corresponding to multiple giants are obtained precisely by the matrix versions of the above super-quantum-mechanics.

Teleportation of a qubit using quasi-Bell states

In this paper, we study the exotic Landau problem at the classical level where two conserved quantities are derived. At the quantum level, the corresponding quantum operators of the conserved quantities provide two oscillator representations from which we derive two Boson Fock spaces. Using the normalized coherent states which are the minimum uncertainty states on noncommutative configuration space isomorphic to each of the boson Fock space, we form entangled coherent states which are Bell- like states labeled quasi-Bell states. The effect of non-maximality of a quasi-Bell state based quantum channel is investigated in the context of a teleportation of a qubit.

Landau problem in dynamical noncommutative space

This paper aims to investigate the Landau problem in the framework of a two-dimensional dynamical noncommutative (DNC) space. We study the deformed Landau problem using time-independent perturbation theory, wherein the energy shift notably depends on the DNC parameter τ. Using the accuracy of energy measurement, we put an upper bound on the parameter τ. Moreover, we study magnetoconductivity by employing the Kubo formula. This approach has allowed us to test the effects of noncommutative and DNC spaces on the behavior of magnetoconductivity. We show that dynamical noncommutativity of space has no effects on the x-component of the magnetoconductivity, but has a direct effect on its y-component.

Landau Levels versus Hydrogen Atom

The Landau problem and harmonic oscillator in the plane share a Hilbert space that carries the structure of Dirac’s remarkable so(2,3) representation. We show that the orthosymplectic algebra osp(1|4) is the spectrum generating algebra for the Landau problem and, hence, for the 2D isotropic harmonic oscillator. The 2D harmonic oscillator is in duality with the 2D quantum Coulomb–Kepler systems, with the osp(1|4) symmetry broken down to the conformal symmetry so(2,3). The even so(2,3) submodule (coined Rac) generated from the ground state of zero angular momentum is identified with the Hilbert space of a 2D hydrogen atom. An odd element of the superalgebra osp(1|4) creates a pseudo-vacuum with intrinsic angular momentum 1/2 from the vacuum. The odd so(2,3)-submodule (coined Di) built upon the pseudo-vacuum is the Hilbert space of a magnetized 2D hydrogen atom: a quantum system of a dyon and an electron. Thus, the Hilbert space of the Landau problem is a direct sum of two massless unitary so(2,3) representations, namely, the Di and Rac singletons introduced by Flato and Fronsdal.

The Landau Problem and non-Classicality in Symplectic Carroll Symmetry

Carroll’s group is shown as a group of transformations in a 5-dimensional space (𝒞) obtained from the embedding of the Euclidean space into a (4,1)-de Sitter space. Three of the five dimensions of 𝒞 are related to ℛ 3, and the other two to mass and time. A covariant formulation of Caroll’s group is established in phase space. The Landau problem was studied. Finally, the negative parameter of the Wigner function is calculated.

Dirac representation of the $$SO(3,2)$$ group and the Landau problem

Dirac representation of the $$SO(3,2)$$ group and the Landau problem

Weak-strong duality of the non-commutative Landau problem induced by a two-vortex permutation, and conformal bridge transformation

A correspondence is established between the dynamics of the two-vortex system and the non-commutative Landau problem (NCLP) in its sub- (non-chiral), super- (chiral) and critical phases. As a result, a trivial permutation symmetry of the point vortices induces a weak-strong coupling duality in the NCLP. We show that quantum two-vortex systems with non-zero total vorticity can be generated by applying conformal bridge transformation to a two-dimensional quantum free particle or to a quantum vortex-antivortex system of zero total vorticity. The sub- and super-critical phases of the quantum NCLP are generated in a similar way from the 2D quantum free particle in a commutative or non-commutative plane. The composition of the inverse and direct transformations of the conformal bridge also makes it possible to link the non-chiral and chiral phases in each of these two systems.

Dynamic and static properties of quantum Hall and harmonic oscillator systems on the non-commutative plane

We study two quantum mechanical systems on the noncommutative plane using a representation independent approach. First, in the context of the Landau problem, we obtain an explicit expression for the gauge transformation that connects the Landau and the symmetric gauge in noncommutative space. This lead us to conclude that the usual form of the symmetric gauge A⃗=−β2Ŷ,β2X̂, in which the constant β is interpreted as the magnetic field, is not true in noncommutative space. We also be able to establish a precise definition of β as a function of the magnetic field, for which the equivalence between the symmetric and Landau gauges is held in noncommutative plane. Using the symmetric gauge, we obtain results for the spectrum of the quantum Hall system, its transverse conductivity in the presence of an electric field, and other static observables. These results amend the literature on quantum Hall effect in the noncommutative plane in which the incorrect form of the symmetric gauge, in noncommutative space, is assumed. We also study the non-equilibrium dynamics of simple observables for this system. On the other hand, we study the dynamics of the harmonic oscillator in non-commutative space and show that, in general, it exhibits quasi-periodic behavior, in striking contrast with its commutative version. The study of dynamics reveals itself as a most powerful tool to characterize and understand the effects of non-commutativity.

Symmetry of Dirac two-oscillator system, gauge-invariance, and Landau problem

Role of gauge symmetry in the proton spin problem has intricate and unresolved aspects. One of the interesting approaches to gain physical insights is to explore the Landau problem in this context. A detailed study using the group theoretic method to understand the Landau problem establishes the significance of the gauge transformation intimately related with the space translation symmetry. An important implication of this result is that the E(2)-like Wigner’s little group for massless particles could throw more light on the question of gauge symmetry in QED and QCD. A generalized Landau-Zeeman Hamiltonian is proposed in which Dirac two-oscillator system and the symmetry of the group SO(3,2) become important. It is argued that nontrivial topology of pure gauge field holds promise to resolve the unsettled questions.

External gauge field-coupled quantum dynamics: Gauge choices, Heisenberg algebra representations and gauge invariance in general, and the Landau problem in particular

Even though its classical equations of motion are then left invariant, when an action is redefined by an additive total derivative or divergence term (in time, in the case of a mechanical system) such a transformation induces non-trivial consequences for the system’s canonical phase space formulation. This is even more true and then in more subtle ways for the canonically quantized dynamics, with, in particular, an induced transformation in the unitary configuration space representation of the Heisenberg algebra being used for the quantum system. When coupled to a background gauge field, such considerations become crucial for a proper understanding of the consequences for the system’s quantum dynamics of gauge transformations of that classical external background gauge field, while under such transformations, the system’s degrees of freedom, abstract quantum states and quantum dynamics are certainly strictly invariant. After a detailed analysis of these different points in a general context, these are then illustrated specifically in the case of the quantum Landau problem with its classical external background magnetic vector potential for which the most general possible parametrized gauge choice is implemented herein. The latter discussion aims as well to clarify some perplexing statements in the literature regarding the status of gauge choices to be made for the magnetic vector potential for that quantum system. The role of the global spacetime symmetries of the Landau problem and their gauge invariant Noether charges is then also emphasized.

On the two-dimensional time-dependent anisotropic harmonic oscillator in a magnetic field

A charged harmonic oscillator in a magnetic field, Landau problems, and an oscillator in a noncommutative space share the same mathematical structure in their Hamiltonians. We have considered a two-dimensional anisotropic harmonic oscillator with arbitrarily time-dependent parameters (effective mass and frequencies), placed in an arbitrarily time-dependent magnetic field. A class of quadratic invariant operators (in the sense of Lewis and Riesenfeld) have been constructed. The invariant operators (Î) have been reduced to a simplified representative form by a linear canonical transformation [the group Sp(4,R)]. An orthonormal basis of the Hilbert space consisting of the eigenvectors of Î is obtained. In order to obtain the solutions of the time-dependent Schrödinger equation corresponding to the system, both the geometric and dynamical phase-factors are constructed. A generalized Peres–Horodecki separability criterion (Simon’s criterion) for the ground state corresponding to our system has been demonstrated.

Verification of the Landau Equation and Hardy’s Inequality

We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interaction exponent (2), a weighted Poincaré inequality is a natural consequence of the traditional weighted Hardy inequality, which in turn implies that the norms of solutions propagate in the L1 space. Now, the L estimate is based on the work of De Giorgi, Nash, and Moser, as well as a few weighted Sobolev inequalities.

Conformal bridge transformation, mathcal{PT} - and supersymmetry

Supersymmetric extensions of the 1D and 2D Swanson models are investigated by applying the conformal bridge transformation (CBT) to the first order Berry-Keating Hamiltonian multiplied by i and its conformally neutral enlargements. The CBT plays the role of the Dyson map that transforms the models into supersymmetric generalizations of the 1D and 2D harmonic oscillator systems, allowing us to define pseudo-Hermitian conjugation and a suitable inner product. In the 1D case, we construct a mathcal{PT} -invariant supersymmetric model with N subsystems by using the conformal generators of supersymmetric free particle, and identify its complete set of the true bosonic and fermionic integrals of motion. We also investigate an exotic N = 2 supersymmetric generalization, in which the higher order supercharges generate nonlinear superalgebras. We generalize the construction for the 2D case to obtain the mathcal{PT} -invariant supersymmetric systems that transform into the spin-1/2 Landau problem with and without an additional Aharonov-Bohm flux, where in the latter case, the well-defined integrals of motion appear only when the flux is quantized. We also build a 2D supersymmetric Hamiltonian related to the “exotic rotational invariant harmonic oscillator” system governed by a dynamical parameter γ. The bosonic and fermionic hidden symmetries for this model are shown to exist for rational values of γ.

Physical symmetries and gauge choices in the Landau problem

Due to a special nature of the Landau problem, in which the magnetic field is uniformly spreading over the whole two-dimensional plane, there necessarily exist three conserved quantities, i.e. two conserved momenta and one conserved orbital angular momentum for the electron, independently of the choice of the gauge potential. Accordingly, the quantum eigen-functions of the Landau problem can be obtained by diagonalizing the Landau Hamiltonian together with one of the above three conserved operators with the result that the quantum mechanical eigen-functions of the Landau problem can be written down for arbitrary gauge potential. The purpose of the present paper is to clarify the meaning of gauge choice in the Landau problem based on this gauge-potential-independent formulation, with a particular intention of unraveling the physical significance of the concept of gauge-invariant-extension of the canonical orbital angular momentum advocated in recent literature on the nucleon spin decomposition problem. At the end, our analysis is shown to disclose a physically vacuous side face of the gauge symmetry.

Hyperons and Θs+ in holographic QCD

We revisit the holographic description of strange baryons in the context of the Sakai-Sugimoto construction, by considering the strange quark mass as heavy. Hyperons are described by a massive $(K,{K}^{*})$ multiplet, bound to a light-flavor instanton in bulk, much in the spirit of the Callan-Klebanov construction. The modular Hamiltonian maps onto the Landau problem, a charged particle in a two-dimensional external magnetic field, induced by the bulk Chern-Simons interaction, plus spin-orbit coupling. The ensuing holographic hyperon spectrum compares fairly with the empirical one. The holographic strange pentaquark baryon ${\mathrm{\ensuremath{\Theta}}}_{s}^{+}$ is shown to be unbound.

SO(5) Landau model and 4D quantum Hall effect in the SO(4) monopole background

We investigate the $SO(5)$ Landau problem in the $SO(4)$ monopole gauge field background by applying the techniques of the non-linear realization of quantum field theory. The $SO(4)$ monopole carries two topological invariants, the second Chern number and a generalized Euler number, specified by the $SU(2)$ monopole and anti-monopole indices, $I_+$ and $I_-$. The energy levels of the $SO(5)$ Landau problem are grouped into $\text{Min}(I_+, I_-) +1$ sectors, each of which holds Landau levels. In the $n$-sector, $N$th Landau level eigenstates constitute the $SO(5)$ irreducible representation with $(p,q)_5=(N+I_+ + I_--n, N+n)_5$ whose function form is obtained from the $SO(5)$ non-linear realization matrix. In the $n=0$ sector, the emergent quantum geometry of the lowest Landau level is identified as the fuzzy four-sphere with radius being proportional to the difference between $I_+$ and $I_-$. The Laughlin-like wavefunction is constructed by imposing the $SO(5)$ lowest Landau level projection to the many-body wavefunction made of the Slater determinant. We also analyze the relativistic version of the $SO(5)$ Landau model to demonstrate the Atiyah-Singer index theorem in the $SO(4)$ gauge field configuration.

A remarkable dynamical symmetry of the Landau problem

We show that the dynamical group of an electron in a constant magnetic feld is the group of symplectomorphisms Sp(4, R). It is generated by the spinorial realization of the conformal algebra so(2,3) considered in Dirac’s seminal paper ”A Remarkable Representation of the 3 + 2 de Sitter Group”. The symplectic group Sp(4,R) is the double covering of the conformal group SO(2,3) of 2+1 dimensional Minkowski spacetime which is in turn the dynamical group of a hydrogen atom in 2 space dimensions. The Newton-Hooke duality between the 2D hydrogen atom and the Landau problem is explained via the Tits-Kantor-Koecher construction of the conformal symmetries of the Jordan algebra of real symmetric 2 × 2 matrices. The connection between the Landau problem and the 3D hydrogen atom is elucidated by the reduction of a Dirac spinor to a Majorana one in the Kustaanheimo-Stiefel spinorial regularization.

Relativistic Topological Insulator Model

A relativistic topological insulator model in three spatial dimensions which is a non trivial extension of the non-abelian Landau problem is proposed. The model is exactly soluble and energy levels have both a discrete and a continuous degeneracy. The chromomagnetic field is strong and the fermions are confined in a plane and the physical effects that appear reflects the ${\bm Z}_2$ symmetry.

Dynamics, symmetries, anomaly and vortices in a rotating cosmic string background

Non-relativistic conformally invariant systems in a rotating cosmic string (conical) spacetime are analyzed at the classical and quantum levels by means of the gravitoelectromagnetic interpretation of the background. Solutions of the equations of motion are found by employing a local canonical transformation, that leads to their natural interpretation in terms of Riemann surfaces. The cone parameter α and the angular velocity Ω of the background determine the existence of hidden symmetries. Globally defined higher order integrals associated with perihelion of geodesic orbits appear at rational values of α. For the harmonic oscillator system with frequency ω, the integrals responsible for the trajectory closure arise only for rational values of α and |γ| = |Ω/ω|, with |γ| = 1 corresponding to the Landau problem. We face a quantum anomaly problem since the hidden symmetry operators can only be constructed when α is integer. Such operators are non-local in the case of the free particle. For the harmonic oscillator, the symmetry generators are obtained with the help of the conformal bridge transformation. We also study a multi-particle version of the harmonic oscillator system with |γ| = 1 using the mean-field theory and find that the emerging vortex structure respects a singular point of the background.

Revisiting the compatibility problem between the gauge principle and the observability of the canonical orbital angular momentum in the Landau problem

As is widely-known, the eigen-functions of the Landau problem in the symmetric gauge are specified by two quantum numbers. The first is the familiar Landau quantum number n, whereas the second is the magnetic quantum number m, which is the eigen-value of the canonical orbital angular momentum (OAM) operator of the electron. The eigen-energies of the system depend only on the first quantum number n, and the second quantum number m does not correspond to any direct observables. This seems natural since the canonical OAM is generally believed to be a gauge-variant quantity, and observation of a gauge-variant quantity would contradict a fundamental principle of physics called the gauge principle. In recent researches, however, Bliokh et al. analyzed the motion of helical electron beam along the direction of a uniform magnetic field, which was mostly neglected in past analyses of the Landau states. Their analyses revealed highly non-trivial m-dependent rotational dynamics of the Landau electron, but the problem is that their papers give an impression that the quantum number m in the Landau eigen-states corresponds to a genuine observable. This compatibility problem between the gauge principle and the observability of the quantum number m in the Landau eigen-states was attacked in our previous letter paper. In the present paper, we try to give more convincing answer to this delicate problem of physics, especially by paying attention not only to the particle-like aspect but also to the wave-like aspect of the Landau electron.