Landau Problem Research Articles

The squashed fuzzy sphere, fuzzy strings and the Landau problem

We discuss the squashed fuzzy sphere, which is a projection of the fuzzy sphere onto the equatorial plane, and use it to illustrate the stringy aspects of noncommutative field theory. We elaborate explicitly how strings linking its two coincident sheets arise in terms of fuzzy spherical harmonics. In the large N limit, the matrix-model Laplacian is shown to correctly reproduce the semi-classical dynamics of these charged strings, as given by the Landau problem.

Landau instability at liquid film evaporation

The Landau problem on the evaporation front stability is generalized to the case of finite thickness of the evaporating liquid layer. The analysis of the influence of additional factors, the impermeability condition of solid wall, and resulting pulsations of mass velocity is carried out.

The Noncommutative Landau Problem in Podolsky's Generalized Electrodynamics**Supported by the National Natural Science Foundation of China under Grant No 11464005, the Guizhou Province Science and Technology Agency Fund under Grant No 20132255, and the Science Foundation of Guizhou Province Masters under Grant No (2012)61.

The Landau problem in Podolsky's generalized electrodynamics is studied by the method of diagonalization in noncommutative phase space and we find that the different noncommutative effects for a certain system led by the nonuniqueness of generalized Bopp shift can be avoided. The exact energy eigenvalues are found, and the result shows that the energy spectra are generically non-degenerate. Furthermore, we obtain the special energy spectra of noncommutative space and commutative space.

Constructing the quantum Hall system on the Grassmannians Gr2(CN)

In this talk, we give the formulation of Quantum Hall Effects (QHEs) on the complex Grassmann manifolds Gr2(CN). We set up the Landau problem in Gr2(CN), solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the SU(N) Wigner D-functions for charged particles on Gr2(CN) under the influence of abelian and non-abelian background magnetic monopoles or a combination of these thereof. For the simplest case of Gr2(C4) we provide explicit constructions of the single and many- particle wavefunctions by introducing the Plucker coordinates and show by calculating the two-point correlation function that the lowest Landau level (LLL) at filling factor v = 1 forms an incompressible fluid. Finally, we heuristically identify a relation between the U(1) Hall effect on Gr2(C4) and the Hall effect on the odd sphere S5, which is yet to be investigated in detail, by appealing to the already known analogous relations between the Hall effects on CP3 and CP7 and those on the spheres S4 and S8, respectively. The talk is given by S. Kürkçüoğlu at the Group 30 meeting at Ghent University, Ghent, Belgium in July 2014 and based on the article by F.Ballı, A.Behtash, S. Kürkçüoğlu, G.Ünal [1].

Rigorous Computation of a Radially Symmetric Localized Solution in a Ginzburg--Landau Problem

We present a rigorous numerical method for proving the existence of a localized radially symmetric solution for a Ginzburg--Landau-type equation. This has a direct application to the problem of finding spots in the Swift--Hohenberg equation. The method is more generally applicable to finding radially symmetric solutions of stationary PDEs on the entire space. One can rewrite such a problem in the form of a singular ODE. We transform this ODE into a finite domain and use a Green's function approach to formulate an appropriate integral equation. We then construct a mapping whose fixed points coincide with solutions to the ODE, and we show via computer-aided analytic estimates that the mapping is contracting on a small neighborhood of a numerically determined approximate solution.

The supersymmetric Wong equations and the non-abelian Landau problem

A Lagrangian formulation is given extending to supersymmetry, or the motion of a charged point particle with spin in a non-abelian external field. The classical formulation is constructed for any external static non-abelian SU(N) gauge potential. As an illustration, a specific gauge is fixed, enabling canonical quantization and the study of the supersymmetric non-abelian Landau problem. The spectrum of the quantum Hamiltonian operator follows in accordance with the supersymmetric structure.

Landau's problem in the noncommutative quantum mechanics and linearized gravitational waves: DeWitt's approach

We present a systematic study of the noncommutative interaction of gravitational waves with matter at the quantum mechanical level in the long wavelength and small velocity limit. In particular, the noncommutative quantum dynamics of the Landau's problem is employed to measure the transmission amplitudes in terms of the Riemann's tensor elements by a gravitational wave in linearized approximation. It turns out that the Hamiltonian of the charged particles derived based on DeWitt's approach is valid by the deformed electromagnetic fields.

Landau problem on the ellipsoid, hyperboloid and paraboloid of revolution

We define the Landau problem on two-dimensional ellipsoid, hyperboloid and paraboloid of revolution. Starting from the two-center McIntosh–Cisneros–Zwanziger (MICZ)–Kepler system and making the reduction into these two-dimensional surfaces, we obtain the Hamiltonians of the charged particle moving on the corresponding surface of revolution in the magnetic field conserving the symmetry of the two-dimensional surface (Landau problem). For each case we figure out the values of parameter for which the qualitative character of the motion coincides with that of a free particle moving on the same two-dimensional surface. For the case of finite trajectories we construct the action-angle variables.

Quantum Hall effect on the GrassmanniansGr2(CN)

Quantum Hall effects on the complex Grassmann manifolds ${\mathbf{Gr}}_{2}({\mathbb{C}}^{N})$ are formulated. We set up the Landau problem in ${\mathbf{Gr}}_{2}({\mathbb{C}}^{N})$ and solve it using group theoretical techniques and provide the energy spectrum and the eigenstates in terms of the $SU(N)$ Wigner $\mathcal{D}$ functions for charged particles on ${\mathbf{Gr}}_{2}({\mathbb{C}}^{N})$ under the influence of Abelian and non-Abelian background magnetic monopoles or a combination of these. In particular, for the simplest case of ${\mathbf{Gr}}_{2}({\mathbb{C}}^{4})$, we explicitly write down the $U(1)$ background gauge field as well as the single- and many-particle eigenstates by introducing the Pl\"ucker coordinates and show by calculating the two-point correlation function that the lowest Landau level at filling factor $\ensuremath{\nu}=1$ forms an incompressible fluid. Our results are in agreement with the previous results in the literature for the quantum Hall effect on $\mathbb{C}{P}^{N}$ and generalize them to all ${\mathbf{Gr}}_{2}({\mathbb{C}}^{N})$ in a suitable manner. Finally, we heuristically identify a relation between the $U(1)$ Hall effect on ${\mathbf{Gr}}_{2}({\mathbb{C}}^{4})$ and the Hall effect on the odd sphere ${S}^{5}$, which is yet to be investigated in detail, by appealing to the already-known analogous relations between the Hall effects on $\mathbb{C}{P}^{3}$ and $\mathbb{C}{P}^{7}$ and those on the spheres ${S}^{4}$ and ${S}^{8}$, respectively.

Magnetic Flux Quantization of the Landau Problem

Landau problem has a very important application in modern physics, in which two-dimensional electron gas system and quantum Hall effect are outstanding. In this paper, first we review the solution of the Pauli equation, then using the single electron wave function, we calculate moving area expectations of the ideal 2-dimensional electron gas system and the per unit area’s degeneracy of the electron gas system. As a result, how to calculate the magnetic flux of the electron gas system is given. It shows that the magnetic flux of 2-dimensional electron gas system in magnetic field is quantized, and magnetic flux quantization results from the quantization of the moving area expectations of electron gas system.

Erratum to: A geometric Ginzburg–Landau problem

For surfaces embedded in a three-dimensional Euclidean space, consider a functional consisting of two terms: a version of the Willmore energy and an anisotropic area penalising the first component of the normal vector, the latter weighted with the factor TeX . The asymptotic behaviour of such functionals as TeX tends to 0 is studied in this paper. The results include a lower and an upper bound on the minimal energy subject to suitable constraints. Moreover, for embedded spheres, a compactness result is obtained under appropriate energy bounds.

Fractional quantum Hall physics in topological flat bands

We present a pedagogical review of the physics of fractional Chern insulators with a particular focus on the connection to the fractional quantum Hall effect. While the latter conventionally arises in semiconductor heterostructures at low temperatures and in high magnetic fields, interacting Chern insulators at fractional band filling may host phases with the same topological properties, but stabilized at the lattice scale, potentially leading to high-temperature topological order. We discuss the construction of topological flat band models, provide a survey of numerical results, and establish the connection between the Chern band and the continuum Landau problem. We then briefly summarize various aspects of Chern band physics that have no natural continuum analogs, before turning to a discussion of possible experimental realizations. We close with a survey of future directions and open problems, as well as a discussion of extensions of these ideas to higher dimensions and to other topological phases. Nous présentons une revue didactique sur la physique des isolants de Chern, qui se concentre plus particulièrement sur lʼeffet Hall quantique fractionnaire. Habituellement, cet effet apparaît typiquement dans des hétérostructures semiconductrices à basse température et sous champ magnétique fort. En revanche, les isolants de Chern peuvent abriter des phases topologiques aux propriétés similaires, mais stabilisées à lʼéchelle du paramètre de réseau, ce qui peut conduire un ordre topologique à haute température. Nous décrivons la construction des modèles avec bande(s) plate(s), passons en revue les résultats numériques et établissons une comparaison entre les isolants de Chern sur réseau et le problème de Landau défini dans le continuum. Nous discutons alors brièvement les aspects de la physique des isolants de Chern qui nont pas dʼanalogues dans le continuum, avant de passer aux possibles réalisations expérimentales. Nous concluons par une liste de perspectives et de problèmes encore ouverts dans ce domaine, ainsi que par une discussion des extensions de ces idées à des dimensions supérieures et à dʼautres phases topologiques.

Exact solutions of the (2+1) dimensional Dirac equation in a constant magnetic field in the presence of a minimal length

We study the ($2+1$)-dimensional Dirac equation in a homogeneous magnetic field (relativistic Landau problem) within a minimal length or generalized uncertainty principle scenario. We derive exact solutions for a given explicit representation of the generalized uncertainty principle and provide expressions of the wave functions in the momentum representation. We find that in the minimal length case, the degeneracy of the states is modified, and that there are states that do not exist in the ordinary quantum mechanics limit ($\ensuremath{\beta}\ensuremath{\rightarrow}0$). We also discuss the massless case, which may find application in describing the behavior of charged fermions in new materials like graphene.

SOLUTIONS FOR THE LANDAU PROBLEM USING SYMPLECTIC REPRESENTATIONS OF THE GALILEI GROUP

Symplectic unitary representations for the Galilei group are studied. The formalism is based on the noncommutative structure of the star-product, and using group theory approach as a guide, a consistent physical theory in phase space is constructed. The state of a quantum mechanics system is described by a quasi-probability amplitude that is in association with the Wigner function. As a result, the Schrödinger and Pauli–Schrödinger equations are derived in phase space. As an application, the Landau problem in phase space is studied. This shows how this method of quantum mechanics in phase space is to be brought to the realm of spatial noncommutative theories.

Particle–antiparticle asymmetry from magnetogenesis through the Landau mechanism

Motivated by string theory an extension of the Landau problem to quantum field theory is considered. We show that the commutator between momenta of the fields violates Lorentz and CPT invariance leading to an alternative method of understanding the question of particle–antiparticle asymmetry. The presence of magnetic field at very early moments of the universe would then suggest that the particle–antiparticle asymmetry can be understood as a consequence of magnetogenesis.

Dynamics of a Dirac fermion in the presence of spin noncommutativity

In this work, we gain further insight in the physical aspects of the spin noncommutativity, which mix spacetime and spin degrees of freedom in a noncommutative scenario, in a Lorentz invariant way. Due to the combination of noncommutativity and Lorentz invariance, time also becomes nonlocal so the modified Dirac equation in general contains an infinite tower of time derivatives. Nevertheless, we prove the existence of a conserved probability current in any order in the noncommutativity parameter and for any background field Aμ. Finally, we study the Landau problem in the presence of spin noncommutativity. In this case, we are able to derive a simple Hermitean noncommutative correction to the Hamiltonian operator. We show that the degeneracy of the excited states is lifted by the noncommutativity at the second order of perturbation theory. This is to be contrasted to the case of canonical noncommutativity, where these corrections are of the first order, imposing much stronger restrictions to the noncommutativity.

Near-horizon dynamics of a particle in extreme Reissner–Nordström and Clément–Gal’tsov black hole backgrounds: action-angle variables

We analyze the periodic motion in the conformal mechanics describing particles moving near the horizon of extreme Reissner–Nordström and axion–dilaton (Clément–Gal’tsov) black holes. For this purpose, we extract the (two-dimensional) compact (angular) parts of these systems and construct their action-angle variables. In the first case, we obtain the well-known spherical Landau problem, which possesses hidden so(3) symmetry, while in the latter case the system does not have a hidden constant of motion. In both the cases, we indicate the existence of ‘critical points’, separating the regions of periodic motions with qualitatively different properties.

Noncommutativity in (2+1)-dimensions and the Lorentz group

In this article we considered models of particles living in a three-dimensional space-time with a nonstandard noncommutativity induced by shifting canonical coordinates and momenta with generators of a unitary irreducible representation of the Lorentz group. The Hilbert space gets the structure of a direct product with the representation space, where we are able to construct operators which realize the algebra of Lorentz transformations. We study the modified Landau problem for both Schr\"odinger and Dirac particles, whose Hamiltonians are obtained through a kind of non-Abelian Bopp's shift of the dynamical variables from the ones of the usual problem in the normal space. The spectrum of these models are considered in perturbation theory, both for small and large noncommutativity parameters. We find no constraint between the parameters referring to noncommutativity in coordinates and momenta but they rather play similar roles. Since the representation space of the unitary irreducible representations $SL(2,\mathbb{R})$ can be realized in terms of spaces of square-integrable functions, we conclude that these models are equivalent to quantum mechanical models of particles living in a space with an additional compact dimension.

An optimal linear solver for the Jacobian system of the extreme type-II Ginzburg–Landau problem

This paper considers the extreme type-II Ginzburg–Landau equations, a nonlinear PDE model for describing the states of a wide range of superconductors. Based on properties of the Jacobian operator and an AMG strategy, a preconditioned Newton–Krylov method is constructed. After a finite-volume-type discretization, numerical experiments are done for representative two- and three-dimensional domains. Strong numerical evidence is provided that the number of Krylov iterations is independent of the dimension n of the solution space, yielding an overall solver complexity of O(n).

Landau electron in a rotating environment: A general factorization of time evolution

For the Landau problem with a rotating magnetic field and a confining potential in the (changing) direction of the field, we derive a general factorization of the time evolution operator that includes the adiabatic factorization as a special case. The confining potential is assumed to be of a general form and it can correspond to nonlinear Heisenberg equations of motion. The rotation operator associated with the solid angle Berry phase is used to transform the problem to a rotating reference frame. In the rotating reference frame, we derive a natural factorization of the time evolution operator by recognizing the crucial role played by a gauge transformation. The major complexity of the problem arises from the coupling between motion in the direction of the magnetic field and motion perpendicular to the field. In the factorization, this complexity is consolidated into a single operator which approaches the identity operator when the potential confines the particle sufficiently close to a rotating plane perpendicular to the magnetic field. The structure of this operator is clarified by deriving an expression for its generating Hamiltonian. The adiabatic limit and non-adiabatic effects follow as consequences of the general factorization which are clarified using the magnetic translation concept.