One dimensional S = 1 XXZ chains with uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. The numerical data are analyzed using conformal field theory, the level spectroscopy, phenomenological renormalization group and finite size scaling method. We thus present the first quantitatively reliable ground state phase diagram of this model. The ground states of this model contain the Haldane phase, large-D phase, Néel phase, two XY phases and the ferromagnetic phase. There are four different types of transitions between these phases: the Brezinskii-Kosterlitz-Thouless type transitions, the Gaussian type transitions, the Ising type transitions and the first order transitions. The location of these critical lines are accurately determined.
Basis states and operators composed of n, in general, different nuclear spins of integer or half-integer values are explicitly constructed. Various coupling schemes are discussed, and transformations between them derived. Following this, a complete set of irreducible tensor operators T(k){V} is constructed which can be used as a basis for expanding operators which depend on n nuclear spins. It is further shown that decomposition of the tensor’s components T(k)q{V} into a sum of products of two irreducible tensor components involves transformation matrices between different coupling schemes. Various properties and commutation relations of the T(k)q{V}’s are given along with a discussion of their reduced matrix elements.
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One dimensional S = 1 antiferromagnetic Heisenberg chains with bond alternation and uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. We calculate the ground state phase diagram using the twisted boundary condition method for D ≥ 0. The ground states of this model contain the Haldane state, large-D state and the dimer state. Using conformal field theory and the level spectroscopy method, we calculate the D and δ-dependence of the Luttinger liquid paraments K and the critical exponent ν of the energy gap.KEYWORDS: Heisenberg chain, exact diagonalization, conformal field theory, dimer phase, Haldane phase, large-D phase §1. IntroductionOne dimensional antiferromagnetic Heisenberg chains have been the subject of recent investigations by numerous groups. The various phenomena of current interest are shown to be due to quantum effects. A uniform antiferromagnetic Heisenberg chain is known to have a gapless ground state for half-integer-spin. Especially, the exact solution is available for S = 1/2 chain.1) In contrast, for the integer-spin,2) there is a gap between the first excited state and ground state.One dimensional S = 1 antiferromagnetic Heisenberg chains with bond alternation undergo a transition from the Haldane state to the dimer state as the strength of bond alternation increases. This transition has been studied by many methods.3-5) On the other hand, the phase boundaries and critical properties of S = 1 Heisenberg chains with uniaxial single-ion anisotropy 6) were determined by Glaus and Schneider using a finite size scaling method. The physical picture of both states was clarified by Nijs and Rommelse in terms of string order 7) and by Tasaki using a stochastic geometric representation.8) Using an exact diagonalization method and a phenomenological renormalization group approach, Tonegawa et al. calculated the phase diagram of the S = 1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy. They found the Haldane phase, Néel phase, dimer phase and large-D phase in the ground state. No phase transition was found, however, between the dimer phase and large-D phase.9) In this paper, we calculate the ground state phase diagram of a one demensional S = 1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy using the twisted boundary condition method.5, 10) This improves the accuracy of the transition points. Using conformal field theory and the level spectroscopy method, we obtain the * E-mail: chenwei@riron.ged.saitama-u.ac.jpLuttinger liquid parameter K and the critical exponent ν of the energy gap.This paper is organized as follows. In the next section, the model Hamiltonian is defined and the numerical results are presented. From the exact diagonalization data with the twisted boundary condition, the ground state phase diagram is obtained. The Luttinger liquid parameter K is calculated by conformal field theory and the level spectroscopy method at the transition p...
The rotation operator approach proposed previously is applied to spin dynamics in a time-varying magnetic field. The evolution of the wave function is described, and that of the density operator is also treated in terms of a spherical tensor operator base. It is shown that this formulation provides a straightforward calculation of accumulated phases and probabilities of spin transitions and coherence evolutions. The technique focuses, not on the rotation matrix, but on the three Euler angles and its characteristic equations are equivalent to the Euler geometric equations long known to describe the motion of a rigid body. The method usually depends on numerical calculations, but analytical solutions exist in some situations. In this paper, as examples, a hyperbolic secant pulse is solved analytically, and a Gaussian-shaped pulse is calculated numerically.
1H NMR was used to analyze human cerebrospinal fluid (CSF) from a group of neurological disease controls and from a vitamin B12 deficient patient. The spectra were acquired at either 7.06 or 9.40 T at ambient temperature with CSF freeze dried and reconstituted in 2H2O. 3-Trimethylsilyl propionate was used as an internal chemical shift and concentration reference. All of the CSF samples showed peaks for lactate, L-alanine, acetate, glutamine, citrate, creatine/creatinine and sugar resonances. There was good agreement between the metabolite concentrations as determined by NMR with those obtained using conventional chemical methods. 1D and 2D 1H NMR techniques along with J-coupling and T1 analysis were used to confirm the peak assignments. Methylmalonic acid could be detected and quantitated (ca 150 microM) in the CSF from the vitamin B12 deficient patient.
The relaxation of an I=3/2 spin system in an anisotropic environment characterized by a finite residual quadrupolar splitting ωq is modeled by analytically solving for the density operator from Redfield’s relaxation theory. The resulting equations are cast into the multipole basis in order to describe the tensorial components of the spin density matrix. Included in the relaxation matrix are off-diagonal elements J1 and J2, which account for anisotropic systems with ωq values less than the width of the resonant line. With the Wigner rotation matrices simulating hard pulses, the response to an arbitrary pulse sequence can be determined. An analytical expression for the response to the double quantum filtered (DQF) pulse sequence (π/2)−(τ/2)−π−(τ/2)−θ−δ−θ−AQ for θ=π/2 is presented, showing explicitly the formation of a second rank tensor owing only to the presence of a finite ωq. This second rank tensor displays asymptotic behavior when the (reduced) quadrupole splitting is equal to either of the off-diagonal spectral densities J2 and J1. Line shape simulations for ωq values of less than a linewidth reproduce the general features of some recently reported 23Na DQF line shapes from biological systems. Distinct relaxation dynamics govern each of the tensorial components of the resonant signal revealing the influence of the experimental variables on the line shape.
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