Landau Hamiltonian Research Articles

HARDY INEQUALITIES AND IDENTITIES RELATED TO THE BAOUENDI-GRUSHIN VECTOR FIELDS AND LANDAU-HAMILTONIAN

In this paper, we present a weighted Hardy identity related to the Baouendi-Grushin vector fields and its applications in the context of differential inequalities. By selecting appropriate parameters, the Hardy identity related to the Baouendi-Grushin operator implies numerous sharp remainder formulae for Hardy type inequalities. In the commutative case, we obtain improved weighted Hardy inequalities in the setting of the Euclidean space. For example, in a special case, by dropping non-negative remainder terms, related to the Baouendi-Grushin operator, and choosing suitable parameters our identity allows us to derive an improved critical Hardy inequality for the radial derivative operator with a sharp constant that does not depend on the topological dimension. We employ the method of factorizing differential expressions, as used by Gesztesy and Littlejohn in [1]. In this paper, we demonstrate the application of the factorization method in the noncommutative Baouendi-Grushin setting. As an application of the Hardy identity related to the Baouendi-Grushin vector fields, we establish a Hardy inequality for the generalized Landau Hamiltonian (or the twisted Laplacian) with remainder terms.

On the spectrum of the Landau Hamiltonian perturbed by a periodic smooth electric potential

On the spectrum of the Landau Hamiltonian perturbed by a periodic smooth electric potential

Physical Aspects of 2014 Nobel Prize in Physiology or Medicine: 2. The First Principle and Universality Class for Grid Cells in the Brain

The main purpose of this review article is to use the fluctuation theory of phase transitions for studying the process of the emergence of hexagonal grid cells in the brain (2014 Nobel Prize in Physiology or Medicine). Particular attention is paid to the application of the Feynman’s classification of three stages of the study of natural phenomena for: 1) a brief description of the experimental stage of the discovery of the hexagonal structures of grid cells in human and animal brains; 2) the theoretical stage of research on the hexagon formation in the physical system of Benard cells, as well as the neurophysiological system of grid cells, discovered by Edward Mozer and May-Britt Mozer; 3) the most important stage, which allows one to formulate the first principle of the emergence of grid cells in the brain and, generally speaking, the first principle for the hexagon formation in different objects of inanimate and living nature. Our original theoretical findings are the following: (a) Polyakov’s conformal invariance hypothesis is violated for a system of grid cells in the brain; (b) the system of grid cells in the brain belongs to the universality class including the 3D Ising model in a magnetic field, as well as a real classical liquid-vapor system;(c) to formulate the first principle for a reliable theoretical justification of the emergence of hexagonal grid cells in the brain, it is necessary to use the fluctuating part of Gibbs thermodynamic potential (the Ginzburg–Landau Hamiltonian) for a system with chemical (biochemical) reactions.

Direct and inverse source problem for 2D Landau Hamiltonian operator

Abstract In the present paper, the unique solvability of the direct and inverse source problems for the pseudo-parabolic equation involving the bi-ordinal Hilfer fractional derivative and 2D Landau Hamiltonian operator is considered. Applying the Fourier analysis for the operator Landau Hamiltonian, the theorems of uniqueness and existence of solutions to direct and inverse source problems are proved. In the investigation of the inverse source problem, we have used the value of unknown function at the final time in order to find the right-hand side of the equation. It is also presented the stability result of the inverse problem.

Spectral asymptotics for the Landau Hamiltonian on cylindrical surfaces

Spectral asymptotics for the Landau Hamiltonian on cylindrical surfaces

On the spectrum of the Landau Hamiltonian perturbed by a periodic electric potential

We study the spectrum of the Landau Hamiltonian perturbed by a periodic electric potential $V\in L^2_{\mathrm{loc}}(\mathbb R^2;\mathbb R)$ assuming that the magnetic flux of the homogeneous magnetic field $B>0$ satisfies the condition $(2\pi)^{-1}Bv(K)=Q^{-1}$, $Q\in \mathbb N $, where $v(K)$ is the area of the unit cell $K$ of the period lattice of the potential $V$. For arbitrary periodic potentials $V\in L^2_{\mathrm {loc}}(\mathbb R^2;\mathbb R)$ with zero mean $V_0=0$ we show that the spectrum has no eigenvalues different from Landau levels. For periodic potentials $V\in L^2_{\mathrm{loc}}(\mathbb R^2;\mathbb R)\setminus C^{\infty}(\mathbb R^2;\mathbb R)$ we also show that the spectrum is absolutely continuous. Bibliography: 23 titles.

Exact renormalization group equation for lattice Ginzburg–Landau models adapted to the solution in the local potential approximation

The Wilson Green’s function approach and, alternatively, Feynman’s diffusion equation and the Hori representation have been used to derive an exact functional RG equation (EFRGE) that in the course of the RG flow interpolates between the interaction part of the lattice Ginzburg–Landau Hamiltonian and the logarithm of the generating functional of the S-matrix. Because the S-matrix vertices are the amputated correlation functions of the fluctuating field, it has been suggested that in the critical region the amputation of the long-range tails makes the S-matrix functional more localized and thus more amenable to the local potential approximation (LPA) than the renormalized free energy functional used in Wilson’s EFRGE. By means of a functional Legendre transform the S-matrix EFRGE has been converted into an EFRGE for the effective action (EA). It has been found that the field-dependent part of EA predicted by the equation is the same as calculated within the known EA EFRGE approaches but in addition it is accurately accounts for the field-independent terms. These are indispensable in calculation of such important quantities as the specific heat, the latent heat, etc. With the use of the derived EFRGE a closed expression for the renormalization counterterm has been obtained which when subtracted from the divergent solution of the Wetterich equation would lead to a finite exact expression for the EA thus making two approaches formally equivalent. The S-matrix equation has been found to be simply connected with a generalized functional Burgers’ equation which establishes a direct correspondence between the first order phase transitions and the shock wave solutions of the RG equation. The transparent semi-group structure of the S-matrix RG equation makes possible the use of different RG techniques at different stages of the RG flow in order to improve the LPA solution.

The Boltzmann distribution function for equilibrium fluctuations

A complex of issues related to the problem of proving the Boltzmann formula for the probability density (distribution) w(η) of the values of a certain set of parameters η of an equilibrium system is investigated. The classical Boltzmann formula expresses this distribution through the non-equilibrium free energy of the system F (η) , which depends on the mentioned parameters. This is possible because after the occurrence of fluctuations, the system finds itself in a non-equilibrium state, which evolves further to equilibrium. The specified parameters are chosen according to the problem under consideration. In the theory of phase transitions, they are called the order parameters. Questions under consideration include definition and construction of the free energy F (η) of a non-equilibrium system and distribution w(η ) for it in the microscopic theory. The approaches of Landau, Leontovich, and Peletminsky are discussed. It is proposed to investigate the results for states in the vicinity of equilibrium. The leading ideas of the research are considering the non-equilibrium state as being realized in the presence of an appropriate external field, using the Gibbs formula for the non-equilibrium system entropy, and applying the Boltzmann formula as a definition of the free energy of the non-equilibrium system. The article is a continuation of the authors' works and sets the task of clarifying some of their statements as well as simplifying and clarifying the calculations. Among other things, the following are discussed: the formula for the expansion of the free energy of the system in powers of the field, the simplification of the distribution w(η ) calculation, the normalization of the approximate expressions for the distribution w(η) , the possibilities of studying the free energy of the equilibrium system using our expression for the effective Landau Hamiltonian, the refinement of the calculation of the non-equilibrium free energy of a spatially inhomogeneous system, the investigation of new types of effective interactions with the Landau Hamiltonian.

Driving a pure spin current from nuclear-polarization gradients

A pure spin current is predicted to occur when an external magnetic field and a linearly inhomogeneous spin-only field are appropriately aligned. Under these conditions (such as originate from nuclear contact hyperfine fields that do not affect orbital motion) a linear, spin-dependent dispersion for free electrons emerges from the Landau Hamiltonian. The result is that spins of opposite orientation flow in opposite directions giving rise to a pure spin current. A classical model of the spin and charge dynamics reveals intuitive aspects of the full quantum mechanical solution. We propose optical orientation or electrical polarization experiments to demonstrate this outcome.

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.

The noncommutative geometry of the Landau Hamiltonian: differential aspects

In this work we study the differential aspects of the noncommutative geometry for the magnetic C*-algebra which is a 2-cocycle deformation of the group C*-algebra of . This algebra is intimately related to the study of the quantum Hall effect in the continuous, and our results aim to provide a new geometric interpretation of the related Kubo’s formula. Taking inspiration from the ideas developed by Bellissard during the 80s, we build an appropriate Fredholm module for the magnetic C*-algebra based on the magnetic Dirac operator which is the square root (à la Dirac) of the quantum harmonic oscillator. Our main result consist of establishing an important piece of Bellissard’s theory, the so-called second Connes’ formula. In order to do so, we establish the equality of three cyclic 2-cocycles defined on a dense subalgebra of the magnetic C*-algebra. Two of these 2-cocycles are new in the literature and are defined by Connes’ quantized differential calculus, with the use of the Dixmier trace and the magnetic Dirac operator.

Classical Hamiltonian Systems with balanced loss and gain

Classical Hamiltonian systems with balanced loss and gain are considered in this review. A generic Hamiltonian formulation for systems with space-dependent balanced loss and gain is discussed. It is shown that the loss-gain terms may be removed completely through appropriate co-ordinate transformations with its effect manifested in modifying the strength of the velocity-mediated coupling. The effect of the Lorentz interaction in improving the stability of classical solutions as well as allowing a possibility of defining the corresponding quantum problem consistently on the real line, instead of within Stokes wedges, is also discussed. Several exactly solvable models based on translational and rotational symmetry are discussed which include coupled cubic oscillators, Landau Hamiltonian etc. The role of -symmetry on the existence of periodic solution in systems with balanced loss and gain is critically analyzed. A few non--symmetric Hamiltonian as well as non-Hamiltonian systems with balanced loss and gain, which include mechanical as well as extended system, are shown to admit periodic solutions. An example of Hamiltonian chaos within the framework of a non--symmetric system of coupled Duffing oscillator with balanced loss-gain and/or positional non-conservative forces is discussed. It is conjectured that a non--symmetric system with balanced loss-gain and without any velocity mediated interaction may admit periodic solution if the linear part of the equations of motion is necessarily symmetric —the nonlinear interaction may or may not be -symmetric. Further, systems with velocity mediated interaction need not be -symmetric at all in order to admit periodic solutions. Results related to nonlinear Schrödinger and Dirac equations with balanced loss and gain are mentioned briefly. A class of solvable models of oligomers with balanced loss and gain is presented for the first time along with the previously known results.

Perturbations of the Landau Hamiltonian: Asymptotics of Eigenvalue Clusters

We consider the asymptotic behavior of the spectrum of the Landau Hamiltonian plus a short-range continuous potential. The spectrum of the operator forms eigenvalue clusters. We obtain a Szegő limit theorem for the eigenvalues in the clusters as the cluster index and the field strength B tend to infinity with a fixed ratio \({\mathcal E}\). The answer involves the averages of the potential over circles of radius \(\sqrt{{\mathcal E}/2}\) (classical orbits). After rescaling, this becomes a semiclassical problem where the role of Planck’s constant is played by 2/B. We also discuss a related inverse spectral result.

Fractional quantum numbers, complex orbifolds and noncommutative geometry

This paper studies the conductance on the universal homology covering space Z of 2D orbifolds in a strong magnetic field, thereby removing the rationality constraint on the magnetic field in earlier works (Avron et al 1994 Phys. Rev. Lett. 73 3255–3257; Mathai and Wilkin 2019 Lett. Math. Phys. 109 2473–2484; Prieto 2006 Commun. Math. Phys. 265 373–396) in the literature. We consider a natural Landau Hamiltonian on Z and study its spectrum which we prove consists of a finite number of low-lying isolated points and calculate the von Neumann degree of the associated holomorphic spectral orbibundles when the magnetic field B is large, and obtain fractional quantum numbers as the conductance.

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.

Beyond Diophantine Wannier diagrams: Gap labelling for Bloch–Landau Hamiltonians

It is well known that, given a 2d purely magnetic Landau Hamiltonian with a constant magnetic field b which generates a magnetic flux \varphi per unit area, then any spectral island \sigma_b consisting of M infinitely degenerate Landau levels carries an integrated density of states \mathcal{I}_b=M \varphi . Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any 2d Bloch–Landau operator H_b which also has a bounded \mathbb{Z}^2 -periodic electric potential. Assume that H_b has a spectral island \sigma_b which remains isolated from the rest of the spectrum as long as \varphi lies in a compact interval [\varphi_1,\varphi_2] . Then \mathcal{I}_b=c_0+c_1\varphi on such intervals, where the constant c_0\in\mathbb{Q} while c_1\in \mathbb{Z} . The integer c_1 is the Chern marker of the spectral projection onto the spectral island \sigma_b . This result also implies that the Fermi projection on \sigma_b , albeit continuous in b in the strong topology, is nowhere continuous in the norm topology if either c_1\ne0 or c_1=0 and \varphi is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields.

Gaplessness of Landau Hamiltonians on Hyperbolic Half-planes via Coarse Geometry

We use coarse index methods to prove that the Landau Hamiltonian on the hyperbolic half-plane, and even on much more general imperfect half-spaces, has no spectral gaps. Thus the edge states of hyperbolic quantum Hall Hamiltonians completely fill up the gaps between Landau levels, just like those of the Euclidean counterparts.

The Noncommutative Geometry of the Landau Hamiltonian: Metric Aspects

This work provides a first step towards the construction of a noncommutative geometry for the quantum Hall effect in the continuum. Taking inspiration from the ideas developed by Bellissard during the 80's we build a spectral triple for the C∗-algebra of continuous magnetic operators based on a Dirac operator with compact resolvent. The metric aspects of this spectral triple are studied, and an important piece of Bellissard's theory (the so-called first Connes' formula) is proved.

Asymptotic Growth of the Local Ground-State Entropy of the Ideal Fermi Gas in a Constant Magnetic Field

We consider the ideal Fermi gas of indistinguishable particles without spin but with electric charge, confined to a Euclidean plane {{mathbb {R}}}^2 perpendicular to an external constant magnetic field of strength B>0. We assume this (infinite) quantum gas to be in thermal equilibrium at zero temperature, that is, in its ground state with chemical potential mu ge B (in suitable physical units). For this (pure) state we define its local entropy S(Lambda ) associated with a bounded (sub)region Lambda subset {{mathbb {R}}}^2 as the von Neumann entropy of the (mixed) local substate obtained by reducing the infinite-area ground state to this region Lambda of finite area |Lambda |. In this setting we prove that the leading asymptotic growth of S(LLambda ), as the dimensionless scaling parameter L>0 tends to infinity, has the form Lsqrt{B}|partial Lambda | up to a precisely given (positive multiplicative) coefficient which is independent of Lambda and dependent on B and mu only through the integer part of (mu /B-1)/2. Here we have assumed the boundary curve partial Lambda of Lambda to be sufficiently smooth which, in particular, ensures that its arc length |partial Lambda | is well-defined. This result is in agreement with a so-called area-law scaling (for two spatial dimensions). It contrasts the zero-field case B=0, where an additional logarithmic factor ln (L) is known to be present. We also have a similar result, with a slightly more explicit coefficient, for the simpler situation where the underlying single-particle Hamiltonian, known as the Landau Hamiltonian, is restricted from its natural Hilbert space text{ L}^2({{mathbb {R}}}^2) to the eigenspace of a single but arbitrary Landau level. Both results extend to the whole one-parameter family of quantum Rényi entropies. As opposed to the case B=0, the corresponding asymptotic coefficients depend on the Rényi index in a non-trivial way.

The geometry of (non-Abelian) Landau levels

The purpose of this paper is threefold: First of all the topological aspects of the Landau Hamiltonian are reviewed in the light of (and with the jargon of) the theory of topological insulators. In particular it is shown that the Landau Hamiltonian has a generalized even time-reversal symmetry (TRS). Secondly, a new tool for the computation of the topological numbers associated with each Landau level is introduced by combining the Dixmier trace and the (resolvent of the) harmonic oscillator. Finally, these results are extended to models with non-Abelian magnetic fields. Two models are investigated in details: the Jaynes–Cummings model and the “Quaternionic” model.