Semiclassical Eigenvalue Estimates for the Pauli Operator with Strong Non-Homogeneous Magnetic Fields

We give the leading order semiclassical asymptotics for the sum of the negative eigenvalues of the Pauli operator (in dimension two and three) with a strong non-homogeneous magnetic field. As in [LSY-II] for homogeneous field, this result can be used to prove that the magnetic Thomas-Fermi theory gives the leading order ground state energy of large atoms. We develop a new localization scheme well suited to the anisotropic character of the strong magnetic field. We also use the basic Lieb-Thirring estimate obtained in our companion paper [ES-I]. Received: 11 September 1996 / Accepted: 17 February 1997     

Magnetic Lieb—Thirring Inequalities with Optimal Dependence on the Field Strength

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. Bounds on the sum of the negative eigenvalues are called magnetic Lieb–Thirring (MLT) inequalities. The purpose of this paper is twofold. First, we prove a new MLT inequality in a simple way. Second, we give a short summary of our recent proof of a more refined MLT inequality(8) and we explain the differences between the two results and methods. The main feature of both estimates, compared to earlier results, is that in the large field regime they grow with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain optimal upper bounds on the pointwise density of zero energy eigenfunctions of the Dirac operator.     

A possibility of a new type of the Fulde-Ferrell-Larkin-Ovchinnikov state

We propose a new type of the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state with a cylindrical symmetric order-parameter. We study the Ginzburg-Landau (GL) theory of the strongly Pauli limited type II superconductor with a spherical symmetric fermi surface, near the critical magnetic field of the FFLO state in the ground state. We find that the cylindrical state has a lower energy than the stripe state, which has the lowest energy in the states examined so far.     

束缚磁极化子的性质

采用线性组合算符和幺正变换方法分别导出弱、强耦合情形下束缚磁极化子的振动频率和基态能量.结果表明库仑场的存在使得磁极化子的基态能量的绝对值变小.     

Global Stability of the Normal State of Superconductors in the Presence of a Strong Electric Current

We consider the time-dependent Ginzburg–Landau model of superconductivity in the presence of an electric current flowing through a two-dimensional wire. We show that when the current is sufficiently strong the solution converges in the long-time limit to the normal state. We provide two types of upper bounds for the critical current where such global stability is achieved: by using the principal eigenvalue of the magnetic Laplacian associated with the normal magnetic field, and through the norm of the resolvent of the linearized steady-state operator. In the latter case we estimate the resolvent norm in large domains by the norms of approximate operators defined on the plane and the half-plane. We also obtain a lower bound, in large domains, for the above critical current by obtaining the current for which the normal state looses its local stability.     

The Ground State Energy of the Multi-Polaron in the Strong Coupling Limit

We consider the Fröhlich N-polaron Hamiltonian in the strong coupling limit and bound its ground state energy from below. In particular, our lower bound confirms that the ground state energy of the Fröhlich polaron and the ground state energy of the associated Pekar–Tomasevich variational problem are asymptotically equal in the strong coupling limit. We generalize the operator approach that was used to prove a similar result in the N =  1 case in Lieb and Thomas (Commun. Math. Phys. 183:511–519, 1997) and apply a Feynman–Kac formula to obtain the same result for an arbitrary particle number N ≥  1.     

Quantum mechanical Hamiltonians with large ground-state degeneracy

Nonrelativistic Hamiltonians with large, even infinite, ground-state degeneracy are studied by connecting the degeneracy to the property of a Dirac operator. We then identify a special class of Hamiltonians, for which the full space of degenerate ground states in any spatial dimension can be exhibited explicitly. The two-dimensional version of the latter coincides with the Pauli Hamiltonian, and recently-discussed models leading to higher-dimensional Landau levels are obtained as special cases of the higher-dimensional version of this Hamiltonian. But, in our framework, it is only the asymptotic behavior of the background ‘potential’ that matters for the ground-state degeneracy. We work out in detail the ground states of the three-dimensional model in the presence of a uniform magnetic field and such potential. In the latter case one can see degenerate stacking of all 2d Landau levels along the magnetic field axis.     

On the Dirac and Pauli Operators with Several Aharonov–Bohm Solenoids

We study the self-adjoint Pauli operators that can be realized as the square of a self-adjoint Dirac operator and correspond to a magnetic field consisting of a finite number of Aharonov–Bohm solenoids and a regular part, and prove an Aharonov–Casher type formula for the number of zero-modes for these operators. We also see that essentially only one of the Pauli operators are spin-flip invariant, and this operator does not have any zero-modes.     

Inequalities for the lowest magnetic Neumann eigenvalue

We study the ground-state energy of the Neumann magnetic Laplacian on planar domains. For a constant magnetic field, we consider the question whether the disc maximizes this eigenvalue for fixed area. More generally, we discuss old and new bounds obtained on this problem.

     

Abrikosov-to-Josephson vortex lattice crossover in heavy fermion CeCoIn5

We present torque magnetization measurements on the quasi-2D heavy fermion superconductor CeCoIn₅ at temperatures down to 20 mK and magnetic fields up to 18?T. At orientations with the magnetic field perpendicular to the conducting planes, a prominent vortex lattice peak effect is present at around 0.5H _c2. The peak effect gradually disappears upon rotating the field into the plane parallel orientation. We interpret the absence of the peak effect for the plane parallel case as a transformation of the Abrikosov lattice into a Josephson vortex state, favored by the Pauli paramagnetic limit in CeCoIn₅ together with the unusually large condensation energy. Additionally, we do not observe flux avalanches as found in organic superconductors and suggest that the complete absence of vortex activity in the plane parallel field orientation is crucial for the formation of Fulde–Ferrell–Larkin–Ovchinnikov superconductivity in CeCoIn₅.     

光学极化子性质对耦合强度和磁场的依赖性

应用线性组合算符和幺正变换方法,研究磁场和耦合强度对光学极化子性质的影响。数值计算表明:当电子接近晶体表面时,光学极化子的振动频率、基态能量和第一激发能仅与磁场有关,且随磁场强度的增强而增大;当电子远离晶体表面时,基态能量和第一激发能与磁场强度和耦合参数均有关,且随磁场强度和耦合参数的增加而增加。     

The pauli principle for a quantum theory on T with magnetic field

We discuss the reducible representation of n particles on a torus with magnetic field, where the gauge automorphisms are unitarily implementable. This representation is of type II_∞, if B is not integer. We show that there is a natural splitting into I_∞ ⊗ II₁ such that the Pauli exclusion principle acts only in the first factor. Under this condition we estimate the ground state energy for several particles and show that in the thermodynamic limit we obtain the correct result.     

Zero Modes for the Magnetic Pauli Operator in Even-Dimensional Euclidean Space

We study the ground state of the Pauli Hamiltonian with a magnetic field in ${\mathbb{R}^{2d}}$ , d > 1. We consider the case where a scalar potential W is present and the magnetic field B is given by ${B=2i\partial\bar{\partial} W}$ . The main result is that there are no zero modes if the magnetic field decays faster than quadratically at infinity. If the magnetic field decays quadratically then zero modes may appear, and we give a lower bound for the number of them. The results in this paper partly correct a mistake in a paper from 1993.     

强耦合半无限极性晶体中磁极化子的激发能量

采用线性组合算符和幺正变换,利用变分法计算了多原子半无限极性晶体中由电子和光学声子强耦合相互作用所产生的磁极化子的第一激发能量及平均声子数,并通过适当的数值计算图示了它们与磁场的关系。结果表明:在不同的磁场条件下,电子无限接近晶体表面和电子处于晶体深处时,磁极化子的第一激发能量和平均声子数都有所不同。     

Gluon condensation and the bag model of hadrons

Migdal's scheme for gluon condensation is applied to a model where gluons are dynamically confined by coloured Higgs fields and are also subjected to ac-number colour field due to external “semiclassical” quarks. We show that, no matter how strong the external field becomes, stabilization of the gluonic ground state is guaranteed by the formation of an inhomogeneous gluon condensate. The contribution to the total energy from the condensate is computed semiclassically using a variational approximation. A physical interpretation of the results obtained is given in terms of a renormalization of the MIT bag model input parameters in particular the zero point energy coefficientZ and the bag constantB.

     

Coulomb drag in the quantum Hall state: Role of disorder

We consider Coulomb drag between two layers of two-dimensional electron gases subject to a strong magnetic field, with the Landau level filling factor in each layer being . We find to be very large, as compared to the zero magnetic field case. We attribute this enhancement to the slow decay of density fluctuations in a strong magnetic field. For a clean system, the linear -dependence of the longitudinal conductivity, characteristic of the state, leads a unique temperature dependence – . Within a semiclassical approximation, disorder leads to a decrease of the transresistivity as compared to the clean case, and a temperature dependence of at low temperatures.     

Nuclear ground states dressed with rf photons in Mössbauer–Zeeman 57Fe spectroscopy

We have applied the “dressed” state concept, developed in quantum electronics, to the situation in which spin 1/2 ground-state nuclear levels are coupled by rf photons. In particular, we have studied Mössbauer spectroscopy when there is Zeeman splitting of the nuclear levels and a further interaction due to an applied rf-radiation field when the rf frequency is in the neighborhood of the ground-state splitting. The dressed-state approach treats the coupling of the ground nuclear Zeeman levels, due to a radio frequency field, by considering the total system made up of: nucleus, static magnetic field, and rf field as one global quantum system. The energy levels and corresponding eigenstates of the system are calculated as a function of the rf frequency and the magnitude of the rf magnetic flux density. Mössbauer spectra are calculated for the ⁵⁷Fe case in which the source is subjected to both the static and radiation fields while the absorber nuclear levels are unsplit.

     

On the Born-Oppenheimer expansion for polyatomic molecules

We consider the Schrödinger operatorP(h) for a polyatomic molecule in the semiclassical limit where the mass ratioh ² of electronic to nuclear mass tends to zero. We obtain WKB-type expansions of eigenvalues and eigenfunctions ofP(h) to all orders inh. This allows to treat the splitting of the ground state energy of a non-planar molecule. Our class of potentials covers the physical case of the Coulomb interaction. We use methods ofh-pseudodifferential operators with operator valued symbols, which by use of appropriate coordinate changes in local coordinate patches covering the classically accessible region become applicable even to our class of singular potentials.     

Effects of Pauli paramagnetism on superconducting fluctuations

We discuss the effects of the Pauli paramagnetism on the excess conductivity σ^fl due to fluctuations of the superconducting order parameter. We derived a formula for σ^fl for a thin film placed in a magnetic field of an arbitrary orientation α. It was found that σ^fl has a universal behavior as a function of some parameterp which depends on α and ΔH=H-H _c. If ΔH is kept constant and σ^fl is measured as a function of α, in the absence of the Pauli paramagnetism σ^fl is maximum when the field is parallel to the film and is minimum when the field is perpendicular. But in high field superconductors due to the effect of the Pauli paramagnetism σ^fl becomes maximum at some intermediate field orientation. We also discussed the excess conductivity in magnetic alloys in which impurity spins are aligned by an external magnetic field. It was shown that in this case one should expect, with certain strengths of the external field, the excess conductivity which is non-monotonic in temperature.     

Investigating the hydrogen atom in crossed external fields by the method of moments

Recurrence relations of perturbation theory for the hydrogen ground state are obtained. With the aid of these relations, polarizabilities in constant mutually perpendicular electric and magnetic fields are computed up to the 80th order of perturbation theory. The high-order asymptotic expression is compared with its semiclassical estimate. For the case of an arbitrary relative orientation of external fields, a general sixth-order formula is given. The energy and the width of the ground state are obtained by means of a perturbation-series summation.