3,981 research outputs found
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 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
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