10 research outputs found
Exactly Solvable Model of Monomer-Monomer Reactions on a Two-Dimensional Random Catalytic Substrate
We present an \textit{exactly solvable} model of a monomer-monomer reaction on a 2D inhomogeneous, catalytic substrate and study the
equilibrium properties of the two-species adsorbate. The substrate contains
randomly placed catalytic bonds of mean density which connect neighboring
adsorption sites. The interacting and (monomer) species undergo
continuous exchanges with corresponding adjacent gaseous reservoirs. A reaction
takes place instantaneously if and particles
occupy adsorption sites connected by a catalytic bond. We find that for the
case of \textit{annealed} disorder in the placement of the catalytic bonds the
reaction model under study can be mapped onto the general spin (GS1)
model. Here we concentrate on a particular case in which the model reduces to
an exactly solvable Blume-Emery-Griffiths (BEG) model (T. Horiguchi, Phys.
Lett. A {\bf 113}, 425 (1986); F.Y. Wu, Phys. Lett. A, {\bf 116}, 245 (1986))
and derive an exact expression for the disorder-averaged equilibrium pressure
of the two-species adsorbate. We show that at equal partial vapor pressures of
the and species this system exhibits a second-order phase transition
which reflects a spontaneous symmetry breaking with large fluctuations and
progressive coverage of the entire substrate by either one of the species.Comment: 4 pages, 2 figures, submitted to Phys. Rev. Let
A set of exactly solvable Ising models with half-odd-integer spin
We present a set of exactly solvable Ising models, with half-odd-integer
spin-S on a square-type lattice including a quartic interaction term in the
Hamiltonian. The particular properties of the mixed lattice, associated with
mixed half-odd-integer spin-(S,1/2) and only nearest-neighbour interaction,
allow us to map this system either onto a purely spin-1/2 lattice or onto a
purely spin-S lattice. By imposing the condition that the mixed
half-odd-integer spin-(S,1/2) lattice must have an exact solution, we found a
set of exact solutions that satisfy the {\it free fermion} condition of the
eight vertex model. The number of solutions for a general half-odd-integer
spin-S is given by . Therefore we conclude that this transformation is
equivalent to a simple spin transformation which is independent of the
coordination number
Optimized t-expansion method for the Rabi Hamiltonian
A polemic arose recently about the applicability of the -expansion method
to the calculation of the ground state energy of the Rabi model. For
specific choices of the trial function and very large number of involved
connected moments, the -expansion results are rather poor and exhibit
considerable oscillations. In this letter, we formulate the -expansion
method for trial functions containing two free parameters which capture two
exactly solvable limits of the Rabi Hamiltonian. At each order of the
-series, is assumed to be stationary with respect to the free
parameters. A high accuracy of estimates is achieved for small numbers (5
or 6) of involved connected moments, the relative error being smaller than
(0.01%) within the whole parameter space of the Rabi Hamiltonian. A
special symmetrization of the trial function enables us to calculate also the
first excited energy , with the relative error smaller than
(1%)
Generalized Transformation for Decorated Spin Models
The paper discusses the transformation of decorated Ising models into an
effective \textit{undecorated} spin models, using the most general Hamiltonian
for interacting Ising models including a long range and high order
interactions. The inverse of a Vandermonde matrix with equidistant nodes
is used to obtain an analytical expression of the transformation. This
kind of transformation is very useful to obtain the partition function of
decorated systems. The method presented by Fisher is also extended, in order to
obtain the correlation functions of the decorated Ising models transforming
into an effective undecorated Ising models. We apply this transformation to a
particular mixed spin-(1/2,1) and (1/2,2) square lattice with only nearest site
interaction. This model could be transformed into an effective uniform spin-
square lattice with nearest and next-nearest interaction, furthermore the
effective Hamiltonian also include combinations of three-body and four-body
interactions, particularly we considered spin 1 and 2.Comment: 16 pages, 4 figure
Exact results of the mixed-spin Ising model on a decorated square lattice with two different decorating spins of integer magnitudes
The mixed-spin Ising model on a decorated square lattice with two different
decorating spins of the integer magnitudes S_B = 1 and S_C = 2 placed on
horizontal and vertical bonds of the lattice, respectively, is examined within
an exact analytical approach based on the generalized decoration-iteration
mapping transformation. Besides the ground-state analysis, finite-temperature
properties of the system are also investigated in detail. The most interesting
numerical result to emerge from our study relates to a striking critical
behaviour of the spontaneously ordered 'quasi-1D' spin system. It was found
that this quite remarkable spontaneous order arises when one sub-lattice of the
decorating spins (either S_B or S_C) tends towards their 'non-magnetic' spin
state S = 0 and the system becomes disordered only upon further single-ion
anisotropy strengthening. The effect of single-ion anisotropy upon the
temperature dependence of the total and sub-lattice magnetization is also
particularly investigated.Comment: 17 pages, 6 figure
Equivalence between non-bilinear spin- Ising model and Wajnflasz model
We propose the mapping of polynomial of degree 2S constructed as a linear
combination of powers of spin- (for simplicity, we called as spin-
polynomial) onto spin-crossover state. The spin- polynomial in general can
be projected onto non-symmetric degenerated spin up (high-spin) and spin down
(low-spin) momenta. The total number of mapping for each general spin- is
given by . As an application of this mapping, we consider a
general non-bilinear spin- Ising model which can be transformed onto
spin-crossover described by Wajnflasz model. Using a further transformation we
obtain the partition function of the effective spin-1/2 Ising model, making a
suitable mapping the non-symmetric contribution leads us to a spin-1/2 Ising
model with a fixed external magnetic field, which in general cannot be solved
exactly. However, for a particular case of non-bilinear spin- Ising model
could become equivalent to an exactly solvable Ising model. The transformed
Ising model exhibits a residual entropy, then it should be understood also as a
frustrated spin model, due to competing parameters coupling of the non-bilinear
spin- Ising model
New variational series expansions for lattice models
For the symmetric two-state vertex model on the honeycomb lattice we construct a series expansion of the free energy which, at finite order, depends on free gauge parameters. We treat these gauge parameters as variational ones, and derive a canonical series of classical approximations which possesses the property of coherent anomaly. As a test model we use the vertex formulation of the Ising antiferromagnet in a field and, within the coherent-anomaly method, determine with a high accuracy its critical frontier and exponent . Numerical checks on the constancy of critical exponents along the phase boundary are presented, too
Solvable weak-graph duals of partially frozen vertex models
A family of -state vertex models on a square lattice is solved exactly using the generalized weak-graph transformation. Concrete physical symmetries of vertex weights turn out to be closely related to abelian groups. The possibility of occurrence of the first-order phase transition is discussed