Algebraic reduction of the Ising model

We consider the Ising model on a cylindrical lattice of L columns, with fixed-spin boundary conditions on the top and bottom rows. The spontaneous magnetization can be written in terms of partition functions on this lattice. We show how we can use the Clifford algebra of Kaufman to write these partition functions in terms of L by L determinants, and then further reduce them to m by m determinants, where m is approximately L/2. In this form the results can be compared with those of the Ising case of the superintegrable chiral Potts model. They point to a way of calculating the spontaneous magnetization of that more general model algebraically.Comment: 25 pages, one figure, last reference completed. Various typos fixed. Changes on 12 July 2008: Fig 1, 0 to +1; before (2.1), if to is; after (4.6), from to form; before (4.46), first three to middle two; before (4.46), last to others; Conclusions, 2nd para, insert how ; renewcommand \i to be \rm

Spin operator matrix elements in the superintegrable chiral Potts quantum chain

We derive spin operator matrix elements between general eigenstates of the superintegrable Z_N-symmetric chiral Potts quantum chain of finite length. Our starting point is the extended Onsager algebra recently proposed by R.Baxter. For each pair of spaces (Onsager sectors) of the irreducible representations of the Onsager algebra, we calculate the spin matrix elements between the eigenstates of the Hamiltonian of the quantum chain in factorized form, up to an overall scalar factor. This factor is known for the ground state Onsager sectors. For the matrix elements between the ground states of these sectors we perform the thermodynamic limit and obtain the formula for the order parameters. For the Ising quantum chain in a transverse field (N=2 case) the factorized form for the matrix elements coincides with the corresponding expressions obtained recently by the Separation of Variables Method.Comment: 24 pages, 1 figur

Three-Body Forces Produced by a Similarity Renormalization Group Transformation in a Simple Model

A simple class of unitary renormalization group transformations that force hamiltonians towards a band-diagonal form produce few-body interactions in which low- and high-energy states are decoupled, which can greatly simplify many-body calculations. One such transformation has been applied to phenomenological and effective field theory nucleon-nucleon interactions with success, but further progress requires consistent treatment of at least the three-nucleon interaction. In this paper we demonstrate in an extremely simple model how these renormalization group transformations consistently evolve two- and three-body interactions towards band-diagonal form, and introduce a diagrammatic approach that generalizes to the realistic nuclear problem.Comment: 25 pages, 18 figures, minor typos corrected and references update

A conjecture for the superintegrable chiral Potts model

We adapt our previous results for the ``partition function'' of the superintegrable chiral Potts model with open boundaries to obtain the corresponding matrix elements of e^{-\alpha H}, where H is the associated hamiltonian. The spontaneous magnetization M_r can be expressed in terms of particular matrix elements of e^{-\alpha H} S^r_1 \e^{-\beta H}, where S_1 is a diagonal matrix.We present a conjecture for these matrix elements as an m by m determinant, where m is proportional to the width of the lattice. The author has previously derived the spontaneous magnetization of the chiral Potts model by analytic means, but hopes that this work will facilitate a more algebraic derivation, similar to that of Yang for the Ising model.Comment: 19 pages, one figure; Corrections made between 28 March 2008 and 28 April 2008: (1) 2.10: q to p; (2) 3.1: epsilon to 0 (not infinity); (3) 5.29: p to q; (4) p14: sub-head: p, q to q,p; (5) p15: sub-head: p, q to q,p; (6) 7.5 second theta to -theta ; (7) before 7.6: make more explicit definition of lambda_j. Several other typos fixed late

Molybdenum complexes derived from the oxydianiline [(2-NH₂C₆H₄)₂O] : synthesis, characterization and ε-caprolactone ROP capability

The reaction of Na₂MoO₄ with 2,2′-oxydianiline (2-aminophenylether), (2-NH₂C₆H₄)₂O, LH₄, in DME (DME = 1,2-dimethoxyethane) in the presence of Et₃N and Me₃SiCl afforded either the bis(imido) molybdenum(VI) complex {Mo(L)Cl₂(DME)} (1), where L = (2-NC₆H₄)₂O, or the molybdenum(V) salt [Mo(L′)Cl₄][Et₃NH] (2), where L′ = [(2-NH₂C₆H₄)(2-NC₆H₄)O], depending on the work-up method employed. The same diamine reacted with in situ [Mo(NtBu)₂Cl₂(DME)] afforded a tetra-nuclear complex [Mo₄Cl₃(NtBu)₃(OSiMe₃)(μ₄-O)(L)₂(L′)₂]·2MeCN (3·2MeCN). The crystal structures of 1, 2 and 3·2MeCN have been determined. The structure of the bis(imido) complex 1 contains two unique molecules paired up via weak π-stacking, whereas the structure of 2 contains a chelating amine/imido ligand, and is made up of discrete units of two cations and two anions which are interacting via H-bonding. The tetra-nuclear structure 3 contains four different types of distorted octahedral molybdenum centre, and a bent Me₃SiO group thought to originate from the precursor synthesis. Complexes 1–3 have been screened for their ability to ring open polymerize (ROP) ε-caprolactone. For 1 and 3 (not 2), conversion rates were good (>90%) at high temperatures (100 °C) over 6–24 h, and the polymerization proceeded in a living manner

Some remarks on a generalization of the superintegrable chiral Potts model

The spontaneous magnetization of a two-dimensional lattice model can be expressed in terms of the partition function $W$ of a system with fixed boundary spins and an extra weight dependent on the value of a particular central spin. For the superintegrable case of the chiral Potts model with cylindrical boundary conditions, W can be expressed in terms of reduced hamiltonians H and a central spin operator S. We conjectured in a previous paper that W can be written as a determinant, similar to that of the Ising model. Here we generalize this conjecture to any Hamiltonians that satisfy a more general Onsager algebra, and give a conjecture for the elements of S.Comment: 18 pages, one figur

Manganese coordination chemistry of bis(imino)phenoxide derived [2 + 2] Schiff-base macrocyclic ligands

The [2 + 2] Schiff base macrocycles [2,2'-(CH₂CH₂)(C₆H₄N)₂-2,6-(4-RC₆H₃OH)]₂ (IʳH₂), upon reaction with MnCl₂ (two equivalents) afforded the bimetallic complex [Cl₃Mn(NCMe)][MnCl(IᵗᵇᵘH₂)] (2). Under similar conditions, use of the related [2 + 2] oxy-bridged macrocycle [2,2'-O(C₆H₄N=CH)₂4-RC₆H₃OH] (IIʳH₂), afforded the bimetallic complexes [(MnCl)₂IIʳ] (R = Me 3, tBu 4), whilst the macrocycle derived from 1,2-diaminobenzene and 5,5'-di-tert-butyl-2,2'-dihydroxy-3,3'-methylenedibenzaldehyde (IIIH₄) afforded the complex [(MnCl)₂(III)]·2MeCN (5·2MeCN). For comparative studies, the salt complexes [2,6-(ArNHCH)₂-4-MeC₆H₂O][MnCl₃(NCMe)] (Ar = 2,4-Me₂C₆H₃, 6) and {[2,6-(ArNHCH)₂-4-MeC₆H₂O][MnCl}₂[MnCl₄]·8CH₂Cl₂ (Ar = 4-MeC₆H₄, 7·8CH₂Cl₂) were prepared. The crystal structures of 1 - 7 are reported (synchrotron radiation was necessary for complexes 1, 3 and 5). Complexes 1 - 7 (not 5) were screened for their potential to act as pre-catalysts for the ring opening polymerization (ROP) of ε-caprolactone; 3, 4 and 6, 7 were inactive, whilst 1 and 2 exhibited only poor activity low conversion (<15 %) at temperatures above 60 °C

Generalized Competing Glauber-type Dynamics and Kawasaki-type Dynamics

In this article, we have given a systematic formulation of the new generalized competing mechanism: the Glauber-type single-spin transition mechanism, with probability p, simulates the contact of the system with the heat bath, and the Kawasaki-type spin-pair redistribution mechanism, with probability 1-p, simulates an external energy flux. These two mechanisms are natural generalizations of Glauber's single-spin flipping mechanism and Kawasaki's spin-pair exchange mechanism respectively. On the one hand, the new mechanism is in principle applicable to arbitrary systems, while on the other hand, our formulation is able to contain a mechanism that just directly combines single-spin flipping and spin-pair exchange in their original form. Compared with the conventional mechanism, the new mechanism does not assume the simplified version and leads to greater influence of temperature. The fact, order for lower temperature and disorder for higher temperature, will be universally true. In order to exemplify this difference, we applied the mechanism to 1D Ising model and obtained analytical results. We also applied this mechanism to kinetic Gaussian model and found that, above the critical point there will be only paramagnetic phase, while below the critical point, the self-organization as a result of the energy flux will lead the system to an interesting heterophase, instead of the initially guessed antiferromagnetic phase. We studied this process in details.Comment: 11 pages,1 figure

Kawasaki-type Dynamics: Diffusion in the kinetic Gaussian model

In this article, we retain the basic idea and at the same time generalize Kawasaki's dynamics, spin-pair exchange mechanism, to spin-pair redistribution mechanism, and present a normalized redistribution probability. This serves to unite various order-parameter-conserved processes in microscopic, place them under the control of a universal mechanism and provide the basis for further treatment. As an example of the applications, we treated the kinetic Gaussian model and obtained exact diffusion equation. We observed critical slowing down near the critical point and found that, the critical dynamic exponent z=1/nu=2 is independent of space dimensionality and the assumed mechanism, whether Glauber-type or Kawasaki-type.Comment: accepted for publication in PR