A (2+1)-Dimensional Gauge Model with Electrically Charged Fermions

We consider a (2+1)-dimensional gauge theory with a nonzero fermion density and an initial Chern–Simons topological term, whose Lorentz invariance is spontaneously broken in a certain Lorentz reference frame by the generation of a constant homogenous magnetic field. We propose interpreting the number =±1, which characterizes the two nonequivalent representations of Dirac matrices in 2+1 dimensions, as a quantum number that explicitly describes the spin of the fermion. In particular, this interpretation allows determining the vacuum state of the model in a constant homogenous magnetic field as the state whose fermion and spin numbers are equal to zero.     

QED2+1 radiation effects in a strong magnetic field

We develop the eigenfunction method for the Dirac operator in a background magnetic field in the (2+1)-dimensional quantum electrodynamics (QED₂₊₁). In the eigenfunction repressentation, we find the exact solutions and the Green's functions of the Dirac equation in a strong constant homogeneous magnetic field in 2+1 dimensions. In the one-loop QED₂₊₁ approximation, we derive the effective Lagrangian, the density of vacuum fermions induced by the field, and the electron mass operator in a homogeneous background magnetic field. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 3, pp. 412–423, December, 1999.     

Magnetic Catalysis of the P-Parity-Breaking Phase Transition of the First Order and High-Temperature Superconductivity

In the framework of a (2+1)-dimensional P-even gauge theory based on the massive Gross–Neveu model, we show that an external magnetic field induces the P-parity-breaking phase transition of the first order. A dynamical generation of the Chern–Simons term and fractional particle spin and statistics occurs at the critical point. The results in the paper can be interesting in connection with recently discovered phase transitions in high-temperature superconductors.     

Zero level of a purely magnetic two-dimensional nonrelativistic Pauli operator for SPIN-1/2 particles

We study the manifold of complex Bloch-Floquet eigenfunctions for the zero level of a two-dimensional nonrelativistic Pauli operator describing the propagation of a charged particle in a periodic magnetic field with zero flux through the elementary cell and a zero electric field. We study this manifold in full detail for a wide class of algebraic-geometric operators. In the nonzero flux case, the Pauli operator ground state was found by Aharonov and Casher for fields rapidly decreasing at infinity and by Dubrovin and Novikov for periodic fields. Algebraic-geometric operators were not previously known for fields with nonzero flux because the complex continuation of “magnetic” Bloch-Floquet eigenfunctions behaves wildly at infinity. We construct several nonsingular algebraic-geometric periodic fields (with zero flux through the elementary cell) corresponding to complex Riemann surfaces of genus zero. For higher genera, we construct periodic operators with interesting magnetic fields and with the Aharonov-Bohm phenomenon. Algebraic-geometric solutions of genus zero also generate soliton-like nonsingular magnetic fields whose flux through a disc of radius R is proportional to R (and diverges slowly as R → ∞). In this case, we find the most interesting ground states in the Hilbert space L ₂ (ℝ ² ).     

Neutral Fermion with a Magnetic Moment in External Electromagnetic Fields

The interaction between a massive neutral fermion with a static (spin) magnetic dipole moment and an external electromagnetic field is described by the Dirac–Pauli equation. Exact solutions of this equation are obtained along with the corresponding energy spectrum for an axially symmetric external magnetic field and for some centrally symmetric electric fields. It is shown that the spin–orbital interaction of a neutral fermion with a magnetic moment determines both the characteristic properties of the quantum states and the fermion energy spectrum. It is found that (1) the discrete energy spectrum of a neutral fermion depends on the projection of the fermion spin on a certain quantization axis, (2) the ground energy level of a fermion in these electric fields as well as the energy levels of all bound states with a fixed value of the quantum number characterizing the projection of the fermion spin in the electric field E = er is degenerate and the degeneration order is countably infinite, and (3) the energy spectra of neutral fermions and antifermions with spin magnetic moments are symmetric in centrally symmetric fields. Bound states of a neutral fermion with a magnetic moment in an external electric field do exist even if the Dirac–Pauli equation does not explicitly contain the term with the fermion mass. In addition, in centrally symmetric electric fields, there exist a countably infinite set of pairs of isolated charge-conjugate zero-energy solutions of the Dirac–Pauli equation.     

A phenomenological description of temperature dependence of magnetoresistance in La2/3Ca1/3MnO3-δ thin films

The electrical resistance in zero magnetic field and magnetoresistance in different external magnetic fields have been measured in a temperature range of 77—300 K. It is found that the temperature dependence of magnetoresistance can be well described by a phenomenofogical formula of

Uniqueness of selfdual periodic Chern–Simons vortices of topological-type

In analogy with the abelian Maxwell–Higgs model (cf. Jaffe and Taubes in Vortices and monopoles, 1980) we prove that periodic topological-type selfdual vortex-solutions for the Chern–Simons model of Jackiw–Weinberg [Phys Rev Lett 64:2334–2337, 1990] and Hong et al. Phys Rev Lett 64:2230–2233, 1990 are uniquely determined by the location of their vortex points, when the Chern–Simons coupling parameter is sufficiently small. This result follows by a uniqueness and uniform invertibility property established for a related elliptic problem (see Theorem 3.6 and 3.7). Research supported by M.I.U.R. project: Variational Methods and Nonlinear Differential Equations.     

The Chern-Simons Invariants of Hyperbolic Manifolds Via Covering Spaces

One important invariant of a closed Riemannian 3-manifold isthe Chern–Simons invariant [1]. The concept was generalizedto hyperbolic 3-manifolds with cusps in [11], and to geometric(spherical, euclidean or hyperbolic) 3-orbifolds, as particularcases of geometric cone-manifolds, in [7]. In this paper, westudy the behaviour of this generalized invariant under changeof orientation, and we give a method to compute it for hyperbolic3-manifolds using virtually regular coverings [10]. We confineourselves to virtually regular coverings because a coveringof a geometric orbifold is a geometric manifold if and onlyif the covering is a virtually regular covering of the underlyingspace of the orbifold, branched over the singular locus. Thereforeour work is the most general for the applications in mind; namely,computing volumes and Chern–Simons invariants of hyperbolicmanifolds, using the computations for cone-manifolds for whicha convenient Schläfli formula holds (see [7]). Among otherresults, we prove that every hyperbolic manifold obtained asa virtually regular covering of a figure-eight knot hyperbolicorbifold has rational Chern–Simons invariant. We giveexplicit examples with computations of volumes and Chern–Simonsinvariants for some hyperbolic 3-manifolds, to show the efficiencyof our method. We also give examples of different hyperbolicmanifolds with the same volume, whose Chern–Simons invariants(mod ) differ by a rational number, as well as pairs of differenthyperbolic manifolds with the same volume and the same Chern–Simonsinvariant (mod ). (Examples of this type were also obtainedin [12] and [9], but using mutation and surgery techniques,respectively, instead of coverings as we do here.) 1991 MathematicsSubject Classification 57M50, 51M10, 51M25.     

Second Order Semiclassics with Self-Generated Magnetic Fields

We consider the semiclassical asymptotics of the sum of negative eigenvalues of the three-dimensional Pauli operator with an external potential and a self-generated magnetic field B. We also add the field energy bòB²{\beta \int B^{2}} and we minimize over all magnetic fields. The parameter β effectively determines the strength of the field. We consider the weak field regime with β h ² ≥ const > 0, where h is the semiclassical parameter. For smooth potentials we prove that the semiclassical asymptotics of the total energy is given by the non-magnetic Weyl term to leading order with an error bound that is smaller by a factor h^1+e{h^{1+\varepsilon}} , i.e. the subleading term vanishes. However for potentials with a Coulomb singularity, the subleading term does not vanish due to the non-semiclassical effect of the singularity. Combined with a multiscale technique, this refined estimate is used in the companion paper (Erdős et al. in Scott correction for large molecules with a self-generated magnetic field, Preprint, 2011) to prove the second order Scott correction to the ground state energy of large atoms and molecules.     

An Answer to S. Simons’ Question on the Maximal Monotonicity of the Sum of a Maximal Monotone Linear Operator and a Normal Cone Operator

The question whether or not the sum of two maximal monotone operators is maximal monotone under Rockafellar’s constraint qualification—that is, whether or not “the sum theorem” is true—is the most famous open problem in Monotone Operator Theory. In his 2008 monograph “From Hahn-Banach to Monotonicity”, Stephen Simons asked whether or not the sum theorem holds for the special case of a maximal monotone linear operator and a normal cone operator of a closed convex set provided that the interior of the set makes a nonempty intersection with the domain of the linear operator. In this note, we provide an affirmative answer to Simons’ question. In fact, we show that the sum theorem is true for a maximal monotone linear relation and a normal cone operator. The proof relies on Rockafellar’s formula for the Fenchel conjugate of the sum as well as some results featuring the Fitzpatrick function.      

Spontaneous fermion creation in the coulomb field and Aharonov-Bohm potential in 2+1 dimensions

We find exact solutions of the Dirac equation and the fermion energy spectrum in the Coulomb (vector and scalar) potential and Aharonov-Bohm potential in 2+1 dimensions taking the particle spin into account. We describe the fermion spin using the two-component Dirac equation with the additional (spin) parameter introduced by Hagen. We consider the effect of creation of fermion pairs from the vacuum by a strong Coulomb field in the Aharonov-Bohm potential in 2+1 dimensions. We obtain transcendental equations implicitly determining the electron energy spectrum near the boundary of the lower energy continuum and the critical charge. We numerically solve the equation for the critical charge. We show that for relatively weak magnetic flows, the critical charge decreases (compared with the case with no magnetic field) if the energy of interaction of the electron spin magnetic moment with the magnetic field is negative and increases if this energy is positive. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 2, pp. 250–262, February, 2009.     

A quantum model with one fermionic degree of freedom

A quantum model with one fermionic degree of freedom is discussed in detail. The operator action of the model has local operator gauge symmetry. A group of constrains on operator gauge potentialB ₀ and gauge transformation operatorU from some physical requirement are obtained. The Euler-Lagrange equation of motion of fermionic operator φ is just the usual equation of motion of fermion type. And the Euler-Lagrange equation of motion of operator gauge potentialB ₀ is just a constraint, which is just. the canonical quantization condition of fermion.     

Minimizers near the first critical field for the nonself-dual Chern–Simons–Higgs energy

The Chern–Simons–Higgs energy serves as a model for high temperature superconductivity. We show the existence of weak solutions to the CSH equations that are minimizers of the CSH energy. The solutions are vortexless for an applied magnetic field h _ex below the critical field strength, whereas vortices appear when h _ex exceeds the critical field strength. D. Spirn was supported in part by NSF grants DMS-0510121 and DMS-0707714. X. Yan was supported in part by NSF grants DMS-0700966 and DMS-0401048.     

Classification of linearly compact simple algebraic N = 6 3-algebras

N ≤ 8 3-algebras have recently appeared in N-supersymmetric 3-dimensional Chern–Simons gauge theories. In our previous paper we classified linearly compact simple N = 8 n-algebras for any n ≥ 3. In the present paper we classify algebraic linearly compact simple N = 6 3-algebras over an algebraically closed field \mathbbF \mathbb{F} of characteristic 0, using their correspondence with simple linearly compact Lie superalgebras with a consistent short \mathbbZ \mathbb{Z} -grading, endowed with a graded conjugation. We also briey discuss N = 5 3-algebras.     

Gibbsianness of fermion random point fields

We consider fermion (or determinantal) random point fields on Euclidean space ℝ^d. Given a bounded, translation invariant, and positive definite integral operator J on L²(ℝ^d), we introduce a determinantal interaction for a system of particles moving on ℝ^d as follows: the n points located at x₁,· · ·,x_n ∈ ℝ^d have the potential energy given by where j(x−y) is the integral kernel function of the operator J. We show that the Gibbsian specification for this interaction is well-defined. When J is of finite range in addition, and for d≥2 if the intensity is small enough, we show that the fermion random point field corresponding to the operator J(I+J)⁻¹ is a Gibbs measure admitted to the specification.     

On verifiable sufficient conditions for sparse signal recovery via <Emphasis Type="Italic">ℓ</Emphasis><Subscript>1</Subscript> minimization

We discuss necessary and sufficient conditions for a sensing matrix to be “s-good”—to allow for exact ℓ ₁-recovery of sparse signals with s nonzero entries when no measurement noise is present. Then we express the error bounds for imperfect ℓ ₁-recovery (nonzero measurement noise, nearly s-sparse signal, near-optimal solution of the optimization problem yielding the ℓ ₁-recovery) in terms of the characteristics underlying these conditions. Further, we demonstrate (and this is the principal result of the paper) that these characteristics, although difficult to evaluate, lead to verifiable sufficient conditions for exact sparse ℓ ₁-recovery and to efficiently computable upper bounds on those s for which a given sensing matrix is s-good. We establish also instructive links between our approach and the basic concepts of the Compressed Sensing theory, like Restricted Isometry or Restricted Eigenvalue properties.     

Positive undecidable numberings in the Ershov hierarchy

A sufficient condition is given under which an infinite computable family of Σ^-1 _a -sets has computable positive but undecidable numberings, where a is a notation for a nonzero computable ordinal. This extends a theorem proved for finite levels of the Ershov hierarchy in [1]. As a consequence, it is stated that the family of all Σ^-1 _a -sets has a computable positive undecidable numbering. In addition, for every ordinal notation a > 1, an infinite family of Σ^-1 _a -sets is constructed which possesses a computable positive numbering but has no computable Friedberg numberings. This answers the question of whether such families exist at any—finite or infinite—level of the Ershov hierarchy, which was originally raised by Badaev and Goncharov only for the finite levels bigger than 1.     

I sottogruppi del gruppo modulare con coefficienti del corpo di Jacobi-Eisenstein e un teorema sui gruppi finiti

Sommario Introduzione — § 1 – 1. L'indice μ(n) dei sottogruppi Г_μ(n) del gruppo Γ di sostituzioni lineari unimodulari con coefficienti del campo diJacobi-Eisenstein — 2. Il poliedro fondamentale del sottogruppo Г_μ(1−ε) — § 2 – 3. I campi fondamentali dei gruppi Г_μ(n) — 4. Impossibilità di limitare con un numero finito di piani e sfere di riflessione i poliedri fondamentali dei gruppi Г_μ(n), conn intero razionale pari, diverso da 2 — § 3 – 5. Relazioni fondamentali fra le sostituzioni generatrici del gruppo di sostituzioni lineari con coefficienti del corpo K_ε con determinante ±1 — § 4 – 6. Sulla indipendenza delle sostituzioniS,T,U, generatrici del gruppo finito G_2μ(n) e sulle loro relazioni caratteristiche nel gruppo G_2μ(n) — § 5 – 7. Dimostrazione del teorema fondamentale sui gruppi G_2μ(n). Lemmi preliminari — 8, Dimostrazione del teorema fondamentale nel caso di moduli primi con 2(1−ε) — § 6 – 9. Il teorema fondamentale per i modulim(1−ε), 3m, 2m, 2m(1−ε), 6m conm primo con 6 – 10. Immagine geometrica dei gruppi G_2μ(1−ε) — § 7 – 11. Il gruppo delle sostituzioni unimodulari , [c/1+4ma]=+1, e il caso eccezionale dei moduli 4m – 12. Il gruppo delle sostituzioni unimodulari [c/1+3m(1−ε)a]₃=+1 e il caso eccezionale dei moduli 3(1−ε)m.     

One Loop Approximation of the Chern–Simons Integral

It is proven that the one loop approximation of the Wilson line integral in a perturbative SU(2) Chern–Simons theory is localized around the critical point in the large level.