Theory of invariants‐based formulation of Hamiltonians with application to strained zinc‐blende crystals

Group theoretical methods and theory are combined to determine spin‐dependent contributions to the effective conduction band Hamiltonian. To obtain the constants in the effective Hamiltonian, in general all invariants of the Hamiltonian have to be determined. Hence, we present a systematic approach to keep track of all possible invariants and apply it to the Hamiltonian of crystals with zinc‐blende symmetry, in order to find all possible contributions to effective quantities such as effective mass, g‐factor and Dresselhaus constant. Additional spin‐dependent contributions to the effective Hamiltonian arise in the presence of strain. In particular, with regard to the constants C₃ and D which describe spin‐splitting linear in the components of k and ε , considering all possible terms allowed by symmetry is crucial.     

Quantum holonomy theory

We present quantum holonomy theory, which is a non‐perturbative theory of quantum gravity coupled to fermionic degrees of freedom. The theory is based on a ‐algebra that involves holonomy‐diffeo‐morphisms on a 3‐dimensional manifold and which encodes the canonical commutation relations of canonical quantum gravity formulated in terms of Ashtekar variables. Employing a Dirac type operator on the configuration space of Ashtekar connections we obtain a semi‐classical state and a kinematical Hilbert space via its GNS construction. We use the Dirac type operator, which provides a metric structure over the space of Ashtekar connections, to define a scalar curvature operator, from which we obtain a candidate for a Hamilton operator. We show that the classical Hamilton constraint of general relativity emerges from this in a semi‐classical limit and we then compute the operator constraint algebra. Also, we find states in the kinematical Hilbert space on which the expectation value of the Dirac type operator gives the Dirac Hamiltonian in a semi‐classical limit and thus provides a connection to fermionic quantum field theory. Finally, an almost‐commutative algebra emerges from the holonomy‐diffeomorphism algebra in the same limit.     

Symmetry-Induced Emergence of a Pseudo-Qutrit in the Dipolar Coupling of Two Qubits

We investigate a system of two identical and distinguishable spins 1/2, with a direct magnetic dipole–dipole interaction, in an external magnetic field. Constraining the hyperfine tensor to exhibit axial symmetry generates the notable symmetry properties of the corresponding Hamiltonian model. In fact, we show that the reduction of the anisotropy induces the invariance of the Hamiltonian in the 3×3 subspace of the Hilbert space of the two spins in which S^2 invariably assumes its highest eigenvalue of 2. By means of appropriate mapping, it is then possible to choose initial density matrices of the two-spin system that evolve in such a way as to exactly simulate the time evolution of a pseudo-qutrit, in the sense that the the actual two-spin system nests the subdynamics of a qutrit regardless of the strength of the magnetic field. The occurrence of this dynamic similitude is investigated using two types of representation for the initial density matrix of the two spins. We show that the qutrit state emerges when the initial polarizations and probability vectors of the two spins are equal to each other. Further restrictions on the components of the probability vectors are reported and discussed.     

Orbital Ordering in Transition-Metal Compounds: I. The 120-Degree Model

We study the classical version of the 120-model. This is an attractive nearest-neighbor system in three dimensions with XY (rotor) spins and interaction such that only a particular projection of the spins gets coupled in each coordinate direction. Although the Hamiltonian has only discrete symmetries, it turns out that every constant field is a ground state. Employing a combination of spin-wave and contour arguments we establish the existence of long-range order at low temperatures. This suggests a mechanism for a type of ordering in certain models of transition-metal compounds where the very existence of long-range order has heretofore been a matter of some controversy.© 2005 M. Biskup, L. Chayes and Z. Nussinov. Reproduction, by any means, of the entire article for non-commercial purposes is permitted without charge.Acknowledgement The research of M.B. and L.C. was supported by the NSF under the grant NSF DMS-0306167. Parts of this paper were written when M.B. was visiting Microsoft Research in Redmond whose hospitality is gratefully acknowledged. The authors wish to thank Jeroen van den Brink for discussions and clarifications and two anonymous referees for suggestions that led to improvements in the presentation.     

Time evolution automorphisms in generalized mean-field theories

A general class of time evolutions ^Q of infinite quantum systems is rigorously defined. It generalizes thermodynamic limits of polynomial mean-field evolution of quantum spin lattices, the simplest case of which is the strong coupling version of the quasi spin B.C.S.-model of superconductivity. A distinguished feature of the considered type of time evolution is the ^Q -non-invariance of the usually consideredC ^*-algebraA of quasilocal observables of the infinite system. A largerC ^*-algebraC containingA as a subalgebra is introduced in such a way that ^Q has a natural extension to a one parameter group^*-automorphisms ofC. The algebraC contains a commutative subalgebra of classical observables (consisting of the intensive observables of the large quantal system determined by a Lie groupG action(G) ^*-autA) denoted byN which is ^Q invariant and the restriction of ^Q toN reproduces the classical Hamiltonian flow ^Q corresponding to the chosen classical Hamiltonian functionQ on the classical phase space of the intensive observables. The evolution ^Q is determined uniquely by the classical Hamiltonian functionQ as well as by the action(G). Continuity properties of ^Q are considered and reviewed.Presented at the International Conference Selected Topics in Quantum Field Theory and Mathematical Physics, Bechyn, Czechoslovakia, June 23–27, 1986.     

The free energy of quantum spin systems and large deviations

For a quantum system ofn identical spins of magnitudej, we introduce an integrated density of states of definite total spin angular momentum. The underlying sequence of probability measures satisfies Varadhan's large deviation principle, and converges to a degenerate distribution. We use the Berezin-Lieb Inequalities to obtain upper and lower bounds for the limiting specific free-energy of the spins interacting with a second quantum system under specified conditions on the Hamiltonian. The method is illustrated by applications to the BCS model and to the Dicke maser model.     

The classical limit of quantum partition functions

We extend Lieb's limit theorem [which asserts that SO(3) quantum spins approachS ² classical spins asL] to general compact Lie groups. We also discuss the classical limit for various continuum systems. To control the compact group case, we discuss coherent states built up from a maximal weight vector in an irreducible representation and we prove that every bounded operator is an integral of projections onto coherent vectors (i.e. every operator has diagonal form).Supported by USNSF Grant MCS-78-01885     

Target-space duality between simple compact Lie groups and Lie algebras under the Hamiltonian formalism: I. Remnants of duality at the classical level

It has been suggested that a possible classical remnant of the phenomenon of target-space duality (T-duality) would be the equivalence of the classical string Hamiltonian systems. Given a simple compact Lie groupG with a bi-invariant metric and a generating function suggested in the physics literature, we follow the above line of thought and work out the canonical transformation generated by together with an Ad-invariant metric and a B-field on the associated Lie algebra ofG so thatG and form a string target-space dual pair at the classical level under the Hamiltonian formalism. In this article, some general features of this Hamiltonian setting are discussed. We study properties of the canonical transformation including a careful analysis of its domain and image. The geometry of the T-dual structure on is lightly touched. We leave the task of tracing back the Hamiltonian formalism at the quantum level to the sequel of this paper.     

Logicoalgebraic foundations of contact mechanics

The logicoalgebraic foundations of the Lagrangian and Hamiltonian techniques of contact mechanics are exhibited, by starting axiomatically with a classical system whose logic is a Boolean algebra.     

The Schrieffer-Wolff transformation for the underscreened Anderson lattice

We present a derivation of the Schrieffer-Wolff transformation for the Anderson Lattice Hamiltonian with a two-fold degenerate f-level in each site. The degeneracy of the f-electrons has been taken into account in order to describe uranium and other actinide magnetic compounds with a spin larger than , for example a total S=1 spin for the f-electrons. The transformed Hamiltonian has several terms as in the classical case, but we have obtained here both an exchange (Kondo) interaction between the S=1 f-spins and the spins of the conduction electrons, and also an effective f-band term. This f-band term describes better the underscreened Kondo lattice model which has been recently developed to explain the Kondo-ferromagnetism coexistence observed in uranium compounds such as UTe [N.B. Perkins, M.D. Nunez-Regueiro, J. R. Iglesias, B. Coqblin, Phys. Rev. B 76 (2007) 125101].     

Reduction of dipolar interaction by off-resonance phonon avalanche

An off-resonance phonon avalanche is shown to reduce effectively the magnetic dipolar interaction between paramagnetic spins

$\text{1}\text{2}$

in a crystal and to change the Larmor frequency of the spins. Both effects are obtained by a unitary transformation which eliminates approximately the phonon field from the Hamiltonian of the system.     

Equivalence and hermiticity of Dirac Hamiltonians in the Kerr gravitational field

In the paper, for the Kerr field, we prove that Chandrasekhar's Dirac Hamiltonian and the self‐adjoint Hamiltonian with a flat scalar product of the wave functions are physically equivalent. Operators of transformation of Chandrasekhar's Hamiltonian and wave functions to the η representation with a flat scalar product are defined explicitly. If the domain of the wave functions of Dirac's equation in the Kerr field is bounded by two‐dimensional surfaces of revolution around the z axis, Chandrasekhar's Hamiltonian and the self‐adjoint Hamiltonian in the η representation are Hermitian with equality of the scalar products, .

     

Information entropic measures of a quantum harmonic oscillator in symmetric and asymmetric confinement within an impenetrable box

Information‐based uncertainty measures like Shannon entropy, Onicescu energy and Fisher information (in position and momentum space) are employed to understand the effect of symmetric and asymmetric confinement in a quantum harmonic oscillator. Also, the transformation of the Hamiltonian into a dimensionless form gives an idea of the composite effect of force constant and confinement length (x_c). In the symmetric case, a wide range of x_c has been taken up, whereas asymmetric confinement is dealt with by shifting the minimum of the potential from the origin keeping box length and boundary fixed. Eigenvalues and eigenvectors for these systems are obtained quite accurately via an imaginary‐time propagation scheme. For asymmetric confinement, a variation‐induced exact diagonalization procedure is also introduced, which produces very high quality results. One finds that, in symmetric confinement, after a certain characteristic x_c, all these properties converge to respective values of a free harmonic oscillator. In the asymmetric situation, excited‐state energies always pass through a maximum. For this potential, the classical turning point decreases, whereas well depth increases with the strength of asymmetry. A study of these uncertainty measures reveals that localization increases with an increase of the asymmetry parameter.

     

Peculiarities of resonance fluorescence statistics for a two‐level atom in frequency selective feedback loop

A theoretical analysis of the resonance fluorescence of a two‐level atom in a classical monochromatic field with feedback phase switching depending on the fluorescence triplet component which the last spontaneously emitted photon belongs to is presented. The considered feedback loop is a hybrid quantum‐classical system. Statistics of photoemissions into the triplet components is investigated as well as correlations between the components. In contrast to the well‐known resonance fluorescence of a two‐level atom without feedback phase switching, a bunching of photocounts is predicted in each side‐band, and successive photoemissions into different side‐bands manifest antibunching. The type of the statistics can efficiently be controlled by the frequency detuning of the external field. In many points the considered feedback scheme provides drastically different statistical features of fluorescence when compared with the scheme of frequency‐unselective feedback phase switching.

     

All 36 exactly solvable solutions of eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor with expanded characteristic equation listing

This paper derives all 36 analytical solutions of the energy eigenvalues for nuclear electric quadrupole interaction Hamiltonian and equivalent rigid asymmetric rotor for polynomial degrees 1 through 4 using classical algebraic theory. By the use of double-parameterization the full general solution sets are illustrated in a compact, symmetric, structural, and usable form that is valid for asymmetry parameter $\eta \in \left({- \infty , + \infty}\right)$ . These results are useful for code developers in the area of Perturbed Angular Correlation (PAC), Nuclear Quadrupole Resonance (NQR) and rotational spectroscopy who want to offer exact solutions whenever possible, rather that resorting to numerical solutions. In addition, by using standard linear algebra methods, the characteristic equations of all integer and half-integer spins I from 0 to 15, inclusive are represented in a compact and naturally parameterized form that illustrates structure and symmetries. This extends Nielson’s?[1] listing of characteristic equations for integer spins out to I?=?15, inclusive.     

Energetic damping in electronic transport simulations on finite systems

In this paper, an implementation of energetic damping for fermionic transport simulations which respects particle conservation is presented. For this, nonhermitian terms in the Hamiltonian of the system are used. After an explanation of the method, it is demonstrated studying the current over time and I/V characteristics in the noninteracting resonant level model for spinless fermions.

     

Free‐space coupling of nanoantennas and whispering‐gallery microcavities with narrowed linewidth and enhanced sensitivity

Due to the broad scattering spectral profiles, localized surface plasmon resonances (LSPRs) of Pd nanoparticles have low resolution and limited sensitivity for hydrogen detection. In this work, we use a simple light‐irradiation method to demonstrate that free‐space light can be efficiently coupled into and from the microfiber whispering‐gallery modes (WGMs) by the Pd nanoantennas. The nanoantenna–microfiber cavity system provides strong intermodal coupling between LSPRs and WGMs, and induces significant modulation of the scattering spectra. A measured full width at half‐maximum of 3.2 nm at 622.7 nm is obtained, which is the narrowest in Pd nanoparticle‐based LSPR structures reported up to now. The ultranarrow resonances offer enhanced sensitivity to hydrogen gas detection with a figure of merit reaching ∼2.22. Other advantages of the Pd nanoantenna–microfiber cavity system including independence of precise alignment of excitation light, large tunability of the resonant wavelengths, easy and low‐cost fabrication of the system, have also been demonstrated.

     

On causality in nonlinear vacuum electrodynamics of the Plebański class

We investigate the Plebański class of electrodynamical theories, i.e., theories of nonlinear vacuum electrodynamics that derive from a Lorentz‐invariant Lagrangian (or Hamiltonian). In any such theory the light rays are the lightlike geodesics of two optical metrics that depend on the electromagnetic background field. A set of necessary and sufficient conditions is found whose fulfillment secures that the optical metrics are causal in the sense that the light rays are lightlike or timelike with respect to the underlying space‐time metric. Thereupon we derive conditions on the Lagrangian, or the Hamiltonian, of the theory such that the causality conditions are satisfied for all allowed background fields. (The allowed values of the field strength tensor are those for which the excitation tensor is finite and real.) The general results are illustrated with several examples.