Bethe ansatz and boundary energy of the open spin-1/2 XXZ chain

We review recent results on the Bethe ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters α_?, α₊, β_?, β₊ are nonzero. A generalization of the BaxterT-Q equation that involves more than one independentQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.     

Index of a family of Dirac operators on loop space

We use methods of constructive field theory to generalize index theory to an infinite-dimensional setting. We study a family of Dirac operatorsQ on loop space. These operators arise in the context of supersymmetric nonlinear quantum field models with HamiltoniansH=Q ². In these modelsQ is self-adjoint and Fredholm. A natural grading operator Γ exists such that ΓQ+QΓ=0. We studyQ ₊=P _? QP ₊, whereP _±=1/2 (1±Γ) are the orthogonal projections onto the eigenspaces of Γ. We calculate the indexi(Q ₊) for Wess-Zumino models defined by a superpotentialV(ω). HereV is a polynomial of degreen≧2. We establish thati(Q ₊)=n?1=degδV. In particular, the field theory models have unbroken supersymmetry, and (forn≧3) they have degenerate vacua. We believe that this is the first index theorem for a Dirac operator that couples infinitely many degrees of freedom.     

Spectra of random contractions and scattering theory for discrete-time systems

Random contractions (subunitary random matrices) appear naturally when considering quantized chaotic maps within a general theory of open linear stationary systems with discrete time. We analyze statistical properties of complex eigenvalues of generic N × N random matrices  of such a type, corresponding to systems with broken time reversal invariance. Deviations from unitarity are characterized by rank M≤N and a set of eigenvalues 0<T _i≤1, i=1,..., M of the matrix $\hat T = \hat 1 - \hat A^\dag \hat A$ . We solve the problem completely by deriving the joint probability density of N complex eigenvalues and calculating all n-point correlation functions. In the limit N?M, n, the correlation functions acquire the universal form found earlier for weakly non-Hermitian random matrices.     

Lattices of matrices

We study a new class of matrix models, formulated on a lattice. On each site are N states with random energies governed by a gaussian random matrix hamiltonian. The states on different sites are coupled randomly. We calculate the density of and correlation between the eigenvalues of the total hamiltonian in the large-N limit. We find that this correlation exhibits the same type of universal behavior we discovered recently. Several derivations of this result are given. This class of random matrices allows us to model the transition between the “localized” and “extended” regimes within the limited context of random matrix theory.     

The N-mode squeezed state with enhanced squeezing

It is known that exp [iλ (Q1P1i/2)] is a unitary single-mode squeezing operator,where Q1,P1 are the coordinate and momentum operators,respectively.In this paper we employ Dirac’s coordinate representation to prove that the exponential operator S n ≡ exp [iλ sum((QiPi+1+Qi+1Pi))) from i=1 to n ],(Qn+1=Q1,Pn+1=P1),is an n-mode squeezing operator which enhances the standard squeezing.By virtue of the technique of integration within an ordered product of operators we derive S n ’s normally ordered expansion and obtain new n-mode squeezed vacuum states,its Wigner function is calculated by using the Weyl ordering invariance under similar transformations.     

Inelastic 2-prong events in 147 GeV/cπ−p collisions

An SU(3) gauge-field non-topological soliton in three space-dimensions is discussed. Let N be the largest among the magnitudes of eigenvalues of the matrix Σ₁⁸λ_aQ_a, where the Q_a's are the eight conserved charges in the classical SU(3) gauge theory, and the λ_a's the usual 3 × 3 Gell-Mann matrices. We show that stable solitons exist if N > a critical value N_S, and if the ratio between the mass of the SU(3) gauge field and that of a relevant spin-0 field is >√6. Our trial function analysis suggests that, for large N, states with dets (Σ₁⁸λ_aQ_a) ≠ 0 are unstable against fission.     

Non-diffractive production of meson resonances

We study the peripheral cross sections of resonances that cannot be produced by π-exchange. In particular, we concentrate on the four meson nonets expected as L = 1 quark states (i.e., the J^P = 0⁺π_N(980); J^P = 1⁺ A₁, B; J^P = 2⁺ A₂). We use SU(3), Regge poles, factorization, exchange degeneracy, pole extrapolation, and the vector-meson-photon analogy. We predict the cross sections in both photoproduction and non-diffractive hadronic reactions. In passing, we discuss the large unnatural-parity (B, K?Q_B) exchange contributions and even the possibility of studying ππ → πω while avoiding the B production background.     

Transfer Matrix Functional Relations for the Generalized τ2 (t q ) Model

The N-state chiral Potts model in lattice statistical mechanics can be obtained as a “descendant” of the six-vertex model, via an intermediate “Q” or “τ₂ (t _q )” model. Here we generalize this to obtain a column-inhomogeneous τ₂ (t _q ) model, and derive the functional relations satisfied by its row-to-row transfer matrix. We do not need the usual chiral Potts relations between the Nth powers of the rapidity parameters a _p , b _p , c _p , d _p of each column. This enables us to readily consider the case of fixed-spin boundary conditions on the left and right-most columns. We thereby re-derive the simple direct product structure of the transfer matrix eigenvalues of this model, which is closely related to the superintegrable chiral Potts model with fixed-spin boundary conditions.     

Heavy neutrino mixing and single production at linear collider

We study the single production of heavy neutrinos via the processes e⁻e⁺ → νN and e⁻γ → W⁻N at future linear colliders. As a base of our considerations we take a wide class of models, both with vanishing and non-vanishing left-handed Majorana neutrino mass matrix m_L. We perform a model independent analyses of the existing experimental data and find connections between the characteristic of heavy neutrinos (masses, mixings, CP eigenvalues) and the m_L parameters. We show that with the present experimental constraints heavy neutrino masses almost up to the collision energy can be tested in the future experiments.     

Multispectrum analysis of the ν9 band of C2H6: Positions, intensities, self- and N2-broadened half-width coefficients

Line positions, intensities, Lorentz self- and N₂-broadened half-width coefficients have been measured for ^PQ₃, ^PQ₂, ^PQ₁, ^RQ₀, ^RQ₁, ^RQ₂, and ^RQ₃ sub-band transitions in the ν₉ fundamental band of ¹²C₂H₆. A multispectrum nonlinear least-squares fitting technique was used to fit up to 17 high-resolution (∼0.00156 cm⁻¹), room temperature absorption spectra of pure (99.99% chemical purity) natural sample of ethane and lean mixtures of the high-purity ethane diluted with N₂. A Bruker IFS 120HR Fourier transform spectrometer located at the Pacific Northwest National Laboratory (PNNL), in Richland, Washington was used to record the data. A standard Voigt line shape was assumed to fit all the data since no line mixing or other non Voigt line shapes were required to fit any of the spectra used in the analysis. Short spectral intervals (∼2-2.5 cm⁻¹) of all 17 spectra covering a specific ^PQ or ^RQ sub-band were fit simultaneously. For the first time in an ethane band, pressure-broadened half-width coefficients were determined for the torsional-split components. However, for better reliability of the retrieved coefficients for the weaker components (transitions with large intensity ratios of 4:1 or 3:1 for most K levels between the strong and weak components), constraints were used such that the half-width coefficients of both torsional-split components for a given J were identical for a specific broadening gas. No pressure-induced shift coefficients were necessary to fit the spectra to their noise level. The present study revealed for the first time the dependence of self- and N₂-broadened half-width coefficients upon the J, K quantum numbers of the transitions in ethane. A number of transitions belonging to the ν₉+ν₄−ν₄ and the ν₉+2ν₄−2ν₄ hot bands were also observed in the fitted regions and measurements were made when possible.     

Line mixing in the ν6Q branches of self- and nitrogen-broadened methyl bromide

Line-mixing effects are studied in the ν₆^RQ_K and ^PQ_K (K=0-6) branches of methyl bromide (CH₃Br) perturbed by nitrogen (N₂). Laboratory Fourier transform spectra have been obtained at room temperature, and for a large range of pressure values of atmospheric interest. In order to accurately model these spectra, a theoretical approach accounting for line-mixing effects is proposed. This model is based on the use of the state-to-state rotational cross-sections calculated by a statistical modified exponential-gap fitting law depending on a few empirical parameters. These parameters are deduced by adjusting the calculated diagonal elements of the relaxation matrix to the N₂-broadening coefficients, known from accurate previous measurements. Comparisons between experimental and calculated profiles for various Q branches and under various pressure conditions (0.2-1 atm) demonstrate the adequacy and consistency of the proposed model. To allow accurate laboratory measurements, line-mixing effects are also modeled in the case of self-perturbed CH₃Br.     

Global and local condensate and superfluid fractions of a few hard core bosons in a combined harmonic optical cubic lattice in continuous space

We explore the global and local condensate and superfluid (SF) fractions in a system of a few hard core (HC) bosons (N = 8 and N = 40) trapped inside a combined harmonic optical cubic lattice (CHOCL) in continuous space at T = 0 K. The local condensate fraction (CF) is computed for individual lattice wells by separating the one-body density matrix (OBDM) of the whole system into components at the various lattice sites. Then each ??lattice-site?? component is diagonalized to find its eigenvalues. The eigenvalues are obtained by a method presented earlier [J.L. DuBois, H.R. Glyde, Phys. Rev. A 63, 023602 (2001)]. The effects of overlap between the condensates in the lattice wells on the CF in one well is also investigated. The SF fraction (SFF) is calculated for N = 40 only by using the diffusion formula of Pollock and Ceperley [Phys. Rev. B 36, 8343 (1987)]. Our chief result is an opposing behavior of the global and local CF and SFF with increasing lattice wave vector k. In addition, the CF in a lattice well is enhanced by the overlap with its neighbor wells beyond the result when the overlap is neglected. The global SF is depleted with a rise of the repulsion between the bosons, yet at very strong interaction superfluidity is still present. The global CF remains almost constant with increasing HC repulsion.     

Eigenvalue spectrum with chebyshev polynomial approximation of the transport equation in slab geometry

Using certain well-known properties of chebyshev polynomials, a simple and highly efficient approach to evaluate eigenvalue in radiation transport is presented. The spectrum of eigenvalues has been studied for slabs with isotropic scattering of different magnitudes of the cross section parameter c (i.e., the mean number of neutrons emitted per collision). It is shown that in the presence of the chebyshev polynomial approximation (T_N) there are both discrete and continuum of eigenvalues. It is found that the T_N method gives very good agreement with conventional spherical harmonics approximation (P_N).     

Measurements of inclusive pion electroproduction

Measurements of the inclusive pion electroproduction reaction e^?+p → e^?+π^±+X made at the Wilson Synchrotron Laboratory are reported. Data were taken at points with nominal (Q², W) values of (0.6, 2.67), (1.2, 2.67), (2.0, 2.67) and (1.2, 2.15) in (GeV², GeV). A more complete angular distribution was taken at (1.2, 2.67). The virtual photoproduction cross section for π⁺ + missing mass is studied as a function of missing mass, Feynman scaling variables, Q² and W. The invariant cross section show a simple behavior as a function of Q² and W. Its dependenc on P_⊥², x′ and φ is determined.     

Entanglement dynamics in Coriolis coupled rovibrational states of formaldehyde

The entanglement dynamics of two vibrational modes of a polyatomic molecule coupled by Coriolis interaction to overall molecular rotation is studied in terms of two negativities, N(t) and N_s(t), respectively, defined by the minimum of the eigenvalues and by the sum of the negative eigenvalues of the partial transpose of a density matrix. Various initial states are the products of Dicke states and the products of coherent states of vibrations and rotations. Formaldehyde is taken as an example, and the von Neumann entropy s(t) is simulated for the comparison with both negativities. It is shown that negativity N_s(t) is positively correlated with entropy s(t), and the correlated behavior between negativity N(t) and entropy s(t) strongly depends on initial states. However, these three indicators of entanglement display a dominantly positive correlation for the coherent states with small or large parameters. In addition, for the latter state two quantities N(t) and s(t) are nearly unchanged for a long time. This time can be further increased by the increasing of vibrational quantum number so that molecular information processing and quantum computing is allowed. These results are useful in quantum information theory.     

Excitation transfer and quenching induced in inelastic collisions of Zn 41P1 and 43P1 atoms

Quenching of excited Zn(4¹P₁) and Zn(4³P₁) atoms by collisions with Ar, N₂, H₂, CO and CO₂ has been investigated using methods of fluorescence spectroscopy. Mixtures of Zn vapor and quenching gases topped-up with Ar to maintain constant pressure were irradiated in a quartz cell with 2139 or 3076 Å resonance radiation which excited the 4¹P₁ or the 4³P₁ state, respectively. The resulting resonance fluorescence and sensitized fluorescence were monitored in relation to the gas density. Intensity measurements yielded the following cross sections Q_tot for the overall collisional deexcitation of the ¹P₁ state, Q₃ for quenching of the ³P₁ state, and Q₁₃ for ¹P₁→³P_J excitation transfer.For N₂: Q_tot = 26, Q₃ = 4.4, Q₁₃ = 5.8; for H₂: Q_tot = 12, Q₃ = 21, Q₁₃ < 5 × 10^-3; for CO: Q_tot = 28, Q₃ = 14, Q₁₃ = 17; for CO₂: Q_tot = 76, Q₃ = 23. All values are in Ų.     

A Generalization of Wigner’s Law

We present a generalization of Wigner’s semicircle law: we consider a sequence of probability distributions , with mean value zero and take an N × N real symmetric matrix with entries independently chosen from p _N and analyze the distribution of eigenvalues. If we normalize this distribution by its dispersion we show that as N → ∞ for certain p _N the distribution weakly converges to a universal distribution. The result is a formula for the moments of the universal distribution in terms of the rate of growth of the k ^th moment of p _N (as a function of N), and describe what this means in terms of the support of the distribution. As a corollary, when p _N does not depend on N we obtain Wigner’s law: if all moments of a distribution are finite, the distribution of eigenvalues is a semicircle.