Matrix Elements Of Generators Research Articles

A new formulation to calculate general HFB matrix elements through the Pfaffian

A new formula is presented for the calculation of matrix elements between multi-quasiparticle Hartree–Fock–Bogoliubov (HFB) states. The formula is expressed in terms of the Pfaffian, and is derived by using Fermion coherent states with Grassmann numbers. It turns out that the formula corresponds to an extension of the generalized Wickʼs theorem and simplifies the combinatorial complexity resulting from practical applications of the generalized Wickʼs theorem by unifying the transition density and the transition pairing tensor in HFB theory. The resultant formula is simpler and more compact than the traditional description of matrix elements of general many-body operators. In addition, through the derivation of our new formula, we found that the Pfaffian version of the Lewis Carroll formula corresponds to a relation suggested by Balian and Brezin for HFB theory in 1969.

Indecomposability parameters in chiral logarithmic conformal field theory

Work of the last few years has shown that the key algebraic features of Logarithmic Conformal Field Theories (LCFTs) are already present in some finite lattice systems (such as the XXZ spin-1/2 chain) before the continuum limit is taken. This has provided a very convenient way to analyze the structure of indecomposable Virasoro modules and to obtain fusion rules for a variety of models such as (boundary) percolation etc. LCFTs allow for additional quantum numbers describing the fine structure of the indecomposable modules, and generalizing the ‘ b-number’ introduced initially by Gurarie for the c = 0 case. The determination of these indecomposability parameters (or logarithmic couplings) has given rise to a lot of algebraic work, but their physical meaning has remained somewhat elusive. In a recent paper, a way to measure b for boundary percolation and polymers was proposed. We generalize this work here by devising a general strategy to compute matrix elements of Virasoro generators from the numerical analysis of lattice models and their continuum limit. The method is applied to XXZ spin-1/2 and spin-1 chains with open (free) boundary conditions. They are related to gl ( n + m | m ) and osp ( n + 2 m | 2 m ) -invariant superspin chains and to non-linear sigma models with supercoset target spaces. These models can also be formulated in terms of dense and dilute loop gas. We check the method in many cases where the results were already known analytically. Furthermore, we also confront our findings with a construction generalizing Gurarieʼs, where logarithms emerge naturally in operator product expansions to compensate for apparently divergent terms. This argument actually allows us to compute indecomposability parameters in any logarithmic theory. A central result of our study is the construction of a Kac table for the indecomposability parameters of the logarithmic minimal models LM ( 1 , p ) and LM ( p , p + 1 ) .

Unitary irreducible representations of \documentclass[12pt]{minimal}$\mathrm{SL(2,\mathbb {C})}$ SL (2,C) in discrete and continuous \documentclass[12pt]{minimal}$\mathrm{SU(1,1)}$ SU (1,1) bases

We derive the matrix elements of generators of unitary irreducible representations of \documentclass[12pt]{minimal}\begin{document}$\mathrm{SL(2,\mathbb {C})}$\end{document} SL (2,C) with respect to basis states arising from a decomposition into irreducible representations of SU(1,1). This is done with regard to a discrete basis diagonalized by \documentclass[12pt]{minimal}\begin{document}$J^3$\end{document}J3 and a continuous basis diagonalized by \documentclass[12pt]{minimal}\begin{document}$K^1$\end{document}K1, and for both the discrete and continuous series of SU(1,1). For completeness, we also treat the more conventional SU(2) decomposition as a fifth case. The derivation proceeds in a functional/differential framework and exploits the fact that state functions and differential operators have a similar structure in all five cases. The states are defined explicitly and related to SU(1,1) and SU(2) matrix elements.

Factorized finite-size Ising model spin matrix elements from separation of variables

Using the Sklyanin–Kharchev–Lebedev method of separation of variables adapted to the cyclic Baxter–Bazhanov–Stroganov or the τ(2)-model, we derive factorized formulae for general finite-size Ising model spin matrix elements, proving a recent conjecture by Bugrij and Lisovyy.

Matrix element evaluation in the unitary group approach to the electron correlation problem

Computationally effective formulations are presented for the evaluation of matrix elements of unitary group generators and products of generators between Gelfand states. These matrix elements are the coefficients of the orbital integrals in the expressions for the Hamiltonian matrix elements in the Gelfand basis, and as such are the key elements in any application of the unitary group approach to wave-function calculations. The present formulations, which, like previous analyses, result in a simple factorization of the generator matrix elements, are based on a graphical representation of the Gelfand basis, and do not require orbital permutations or an interpretation of the Gelfand states in terms of Young tableaus. It is shown that the resulting formalism can lead to very efficient procedures for “direct” configuration-interaction calculations, and probably also for perturbation-theory treatments.

Equations of motion for a spectrum-generating algebra: Lipkin–Meshkov–Glick model

For a spectrum-generating Lie algebra, a generalized equations-of-motion scheme determines numerical values of excitation energies and algebra matrix elements. In the approach to the infinite particle number limit or, more generally, whenever the dimension of the quantum state space is very large, the equations-of-motion method may achieve results that are impractical to obtain by diagonalization of the Hamiltonian matrix. To test the method's effectiveness, we apply it to the well-known Lipkin–Meshkov–Glick (LMG) model to find its low-energy spectrum and associated generator matrix elements in the eigenenergy basis. When the dimension of the LMG representation space is 106, computation time on a notebook computer is a few minutes. For a large particle number in the LMG model, the low-energy spectrum makes a quantum phase transition from a nondegenerate harmonic vibrator to a twofold degenerate harmonic oscillator. The equations-of-motion method computes critical exponents at the transition point.

Correlation-Induced Resonances and Population Switching in a Quantum-Dot Coulomb Valley

Strong correlation effects on electron transport through a spinless quantum dot are considered. When two single-particle levels in the quantum dot are degenerate, a conserved pseudospin degree of freedom appears for generic tunneling matrix elements between the quantum dot and leads. Local fluctuations of the pseudospin in the quantum dot give rise to a pair of asymmetric conductance peaks near the center of a Coulomb valley. An exact relation to the population switching is provided.

ENTROPY AND ENTANGLEMENT OF THE SUPERPOSITIONS OF DISPLACED FOCK STATES WITH A TWO-LEVEL ATOM

The properties of the displaced Fock states (DFS's) superpositions are reviewed. The interaction of these states with a two-level atom in cavity with the presence of additional Kerr medium is studied. Exact general matrix elements of the time-dependent operators of a Jaynes-Cummings model (JCM), in the presence of a Kerr medium, with these states are derived. The atomic inversion and photon number distribution are discussed. The quantum entropy and the entanglement of the atom-field are investigated. The exact results are employed to perform a careful investigation of the temporal evolution of the entropy. The connection between the field entropy and the collapses and revivals of the atomic inversion has been established. The general conclusions reached are illustrated by numerical results.

INTERACTION OF THE SUPERPOSITIONS OF SQUEEZED DISPLACED FOCK STATES WITH ATOMIC SYSTEMS

The properties of the squeezed displaced Fock states (SDFS) superpositions are reviewed. The interaction of these states with a two-level atom in cavity with the presence of additional Kerr medium is studied. Exact general matrix elements of the time-dependent operators of a Jaynes–Cummings model (JCM), in the presence of a Kerr medium, with these states are derived. The atomic inversion and photon number distribution are discussed.

THEORY OF BICRYSTAL c-AXIS TWIST JOSEPHSON JUNCTIONS

We calculate the critical current density [Formula: see text] for Josephson tunneling between identical high temperature superconductors twisted an angle (ϕ0) about the c-axis. Regardless of the shape of the two-dimensional Fermi surface and for very general tunneling matrix elements, an order parameter (OP) with general d-wave symmetry leads to [Formula: see text]. This general result is inconsistent with the data of Li et al. on Bi2Sr2CaCu2O8+δ (Bi2212), which showed [Formula: see text] to be independent of ϕ0. If the momentum parallel to the barrier is conserved in the tunneling process, [Formula: see text] should vary substantially with the twist angle (ϕ0) when the tight-binding Fermi surface appropriate for Bi2212 is taken into account, even if the OP is completely isotropic. We quantify the degree of momentum non-conservation necessary to render [Formula: see text] constant within experimental error for a variety of pair states by interpolating between the coherent and incoherent limits using five specific models to describe the momentum dependence of the tunneling matrix element squared. From the data of Li et al., we conclude that the c-axis tunneling in Bi2212 must be very nearly incoherent, and that the OP must have a non-vanishing Fermi surface average for T≤Tc.

Weight bases of Gelfand-Tsetlin type for representations of classical Lie algebras

This paper completes a series devoted to explicit constructions of finite-dimensional irreducible representations of the classical Lie algebras. Here the case of odd orthogonal Lie algebras (of type B) is considered (two previous papers dealt with C and D types). A weight basis for each representation of the Lie algebra o(2n+1) is constructed. The basis vectors are parametrized by Gelfand--Tsetlin-type patterns. Explicit formulas for the matrix elements of generators of o(2n+1) in this basis are given. The construction is based on the representation theory of the Yangians. A similar approach is applied to the A type case where the well-known formulas due to Gelfand and Tsetlin are reproduced.

A Basis for Representations of Symplectic Lie Algebras

A basis for each finite-dimensional irreducible representation of the symplectic Lie algebra sp(2n) is constructed. The basis vectors are expressed in terms of the Mickelsson lowering operators. Explicit formulas for the matrix elements of generators of sp(2n) in this basis are given. The basis is natural from the viewpoint of the representation theory of the Yangians. The key role in the construction is played by the fact that the subspace of sp(2n-2)-highest vectors in any finite-dimensional irreducible representation of sp(2n) admits a natural structure of a representation of the Yangian Y(gl(2)).

Franck-Condon factors for multidimensional harmonic oscillators

We present a simple formula for the overlap integrals of two sets of multi-dimensional harmonic oscillators. The oscillators have in general different equilibrium points, force constants, and natural vibration modes. The formula expresses the overlap matrix in the one-dimensional case, 〈 m′| n′′〉, as a so-called LU decomposition, 〈m′|n′′〉=〈0′|0′′〉 ∑ L mtU tn , where the summation index has a range 0≤ t≤min( m, n), i.e., it is the matrix product of a lower-triangular matrix L with an upper-triangular U. These matrices are obtained from simple recursion formulae. This form is essentially retained in the multi-dimensional case. General matrix elements are obtained by exact and finite expressions, relating them to matrix elements over a single set of harmonic oscillator wave functions. We present test calculations with error estimates, also comparing with literature examples.

Matrix elements ofU(2n) generators in a multishell spin-orbit basis. II. The two-shell nonzero shift ACCs andU(2n) generator MEs

This is the second in a series of articles whose ultimate goal is the evaluation of the matrix elements (MEs) of the U(2n) generators in a multishell spin–orbit basis. This extends the existing unitary group approach to spin-dependent configuration interaction (CI) and many-body perturbation theory calculations on molecules to systems where there is a natural partitioning of the electronic orbital space. As a necessary preliminary to obtaining the U(2n) generator MEs in a multishell spin–orbit basis, we must obtain a complete set of adjoint coupling coefficients for the two-shell composite Gelfand–Paldus basis. The zero-shift coefficients were obtained in the first article of the series. In this article, we evaluate the nonzero shift adjoint coupling coefficients for the two-shell composite Gelfand–Paldus basis. We then demonstrate that the one-shell versions of these coefficients may be obtained by taking the Gelfand–Tsetlin limit of the two-shell formulas. These coefficients, together with the zero-shift types, then enable us to write down formulas for the U(2n) generator matrix elements in a two-shell spin–orbit basis. Ultimately, the results of the series may be used to determine the many-electron density matrices for a partitioned system. © 1998 John Wiley & Sons, Inc. Int J Quant Chem 66: 345–363, 1998

Matrix elements ofU(2n) generators in a multishell spin-orbit basis. III. General formulas

This is the third and final article in a series directed toward the evaluation of the U(2n) generator matrix elements (MEs) in a multishell spin/orbit basis. Such a basis is required for many-electron systems possessing a partitioned orbital space and where spin-dependence is important. The approach taken is based on the transformation properties of the U(2n) generators as an adjoint tensor operator of U(n) x U(2) and application of the Wigner-Eckart theorem. A complete set of adjoint coupling coefficients for the two-shell composite Gelfand-Paldus basis (which is appropriate to the many-electron problem) were obtained in the first and second articles of this series. Ln the first article we defined zero-shift coupling coefficients. These are proportional to the corresponding two-shell del-operator matrix elements. See P. J. Burton and and M. D. Gould, J. Chem. Phys., 104, 5112 (1996), for a discussion of the del-operator and its properties. Ln the second article of the series, the nonzero shift coupling coefficients were derived. Having obtained all the necessary coefficients, we now apply the formalism developed above to obtain the U(2n) generator MEs in a multishell spin-orbit basis. The methods used are based on the work of Gould et al. (see the above reference). (C) 1998 John Wiley & Sons, Inc.

Quantum and semiclassical Green's functions in chemical reaction dynamics

A variety of quantities related to chemical reaction dynamics (state-selected and cumulative probabilities for chemical reactions, photo-dissociation or -detachment cross-sections, and others) can be expressed compactly (and exactly) in terms of the quantum mechanical Green's function (actually an operator) Ĝ(E) ≡ (E + i − Ĥ) −1 , where Ĥ is the Hamiltonian for the molecular system and an absorbing potential. It is emphasized that these ‘formal’ quantum expressions can serve as the basis for practical calculations by utilizing a straightforward L 2 matrix representation of the operator (E + i − Ĥ). It is also shown how the semiclassical initial value representation (IVR) can be used to construct approximations for general matrix elements of the Green's function, so that these same formally exact quantum expressions can also be used to provide semiclassical approximations for all of these dynamical quantities. Recent applications of the quantum and semiclassical methodologies are discussed.

Character states and generator matrix elements for Sp(4)⊃SU(2)×U(1)

A new set of polynomial states (to be called character states) are derived for Sp(4) reduced to its SU(2)×U(1) subgroup, and the relevant generator matrix elements are evaluated for generic representations (a,b) of Sp(4). [The degenerate representations (a,0) and (0,b) were treated in our previous work and are also given in this paper]. The group–subgroup in question is that of the seniority model of nuclear physics.

Nonlocal Coulomb interactions and the magnetic anisotropies in the low-temperature phases of the cuprates.

We have extended and generalized earlier studies of the magnetic anisotropies of the Cu spins in the copper-oxide planes generated by the spin-orbit coupling on the coppers and by the nonlocal Coulomb interactions between neighboring ions. Our derivation of the anisotropic magnetic Hamiltonian includes the contributions from general matrix elements of the Coulomb interaction both between the copper ions of a single bond and between the adjacent copper and oxygen sites. The results apply to a general rotation axis of the oxygen octahedra, with the assumption of tetragonal site symmetry of the copper. This mechanism leads to a form of the anisotropic spin interaction between the Cu ions which is identical to the one generated by kinetic superexchange in the presence of purely local Coulomb interactions. Together with the numerical estimates for the resulting spin couplings, this indicates that the Coulomb direct exchange leads to significant contributions to the isotropic spin interaction and the out-of-plane anisotropy, but does not affect qualitatively the magnetic order in the ground state. \textcopyright{} 1996 The American Physical Society.

Homodyne detection of the density matrix of the radiation field.

We provide a simple analytic relation that connects the density operator \ensuremath{\rho}^ of the electromagnetic field with the tomographic homodyne probabilities for generic quantum efficiency \ensuremath{\eta} of detectors. The problem of experimentally ``sampling'' a general matrix element 〈\ensuremath{\psi}\ensuremath{\Vert}\ensuremath{\rho}^\ensuremath{\Vert}\ensuremath{\Phi}〉 is addressed in the statistically rigorous sense of the central-limit theorem. We show that experimental sampling is possible also for nonunit efficiency, provided that \ensuremath{\eta} satisfies a lower bound related to the ``resolutions'' of vectors \ensuremath{\Vert}\ensuremath{\psi}〉 and \ensuremath{\Vert}\ensuremath{\Phi}〉 in the quadrature representations. For coherent and number states the bound is \ensuremath{\eta}\ensuremath{\gtrsim}1/2. On the basis of computer-simulated experiments we show the feasibility of detecting delicate quantum probability oscillations, which otherwise would be smeared out by inefficient detection.

Tomographic measurement of the density matrix of the radiation field

Recently a novel homodine tomographic technique has been proposed which allows us to detect the density matrix experimentally in terms of averages on data. The method has been further extended to a general matrix element ( psi mod Q mod phi ) of the density operator Q, along with the possibility of using low-efficiency detectors. In this paper this technique is reconsidered as a new genuine quantum measurement. Numerical results for measured probabilities are given, and the mechanism underlying the generation of statistical errors in the measured matrix is illustrated. It is shown that the data processing of the novel technique can also be used as a new imaging algorithm for conventional tomography.