394 research outputs found
Bond-Propagation Algorithm for Thermodynamic Functions in General 2D Ising Models
Recently, we developed and implemented the bond propagation algorithm for
calculating the partition function and correlation functions of random bond
Ising models in two dimensions. The algorithm is the fastest available for
calculating these quantities near the percolation threshold. In this paper, we
show how to extend the bond propagation algorithm to directly calculate
thermodynamic functions by applying the algorithm to derivatives of the
partition function, and we derive explicit expressions for this transformation.
We also discuss variations of the original bond propagation procedure within
the larger context of Y-Delta-Y-reducibility and discuss the relation of this
class of algorithm to other algorithms developed for Ising systems. We conclude
with a discussion on the outlook for applying similar algorithms to other
models.Comment: 12 pages, 10 figures; submitte
New Results for the Correlation Functions of the Ising Model and the Transverse Ising Chain
In this paper we show how an infinite system of coupled Toda-type nonlinear
differential equations derived by one of us can be used efficiently to
calculate the time-dependent pair-correlations in the Ising chain in a
transverse field. The results are seen to match extremely well long large-time
asymptotic expansions newly derived here. For our initial conditions we use new
long asymptotic expansions for the equal-time pair correlation functions of the
transverse Ising chain, extending an old result of T.T. Wu for the 2d Ising
model. Using this one can also study the equal-time wavevector-dependent
correlation function of the quantum chain, a.k.a. the q-dependent diagonal
susceptibility in the 2d Ising model, in great detail with very little
computational effort.Comment: LaTeX 2e, 31 pages, 8 figures (16 eps files). vs2: Two references
added and minor changes of style. vs3: Corrections made and reference adde
Exact Solution of a Three-Dimensional Dimer System
We consider a three-dimensional lattice model consisting of layers of vertex
models coupled with interlayer interactions. For a particular non-trivial
interlayer interaction between charge-conserving vertex models and using a
transfer matrix approach, we show that the eigenvalues and eigenvectors of the
transfer matrix are related to those of the two-dimensional vertex model. The
result is applied to analyze the phase transitions in a realistic
three-dimensional dimer system.Comment: 11 pages in REVTex with 2 PS figure
Non-intersecting string model and graphical approach: Equivalence with a Potts model
Using a graphical method we establish the exact equivalence of the partition function of a q-state nonintersecting string (NIS) model on an arbitrary planar, even-valenced, lattice with that of a q2-state Potts model on a related lattice. The NIS model considered in this paper is one in which the vertex weights are expressible as sums of those of basic vertex types, and the resulting Ports model generally has multispin interactions. For the square and Kagom6 lattices this leads to the equivalence of a staggered NIS model with Potts models with anisotropic pair interactions, indicating that these NIS models have a first-order transition for q > 2. For the triangular lattice the NIS model turns out to be the five-vertex model of Wu and Lin and it relates to a Potts model with two-and three-site interactions. The most general model we discuss is an oriented NIS model which contains the six-vertex model and the NIS models of Stroganov and Schultz as special cases
Integrability as a consequence of discrete holomorphicity: the Z_N model
It has recently been established that imposing the condition of discrete
holomorphicity on a lattice parafermionic observable leads to the critical
Boltzmann weights in a number of lattice models. Remarkably, the solutions of
these linear equations also solve the Yang-Baxter equations. We extend this
analysis for the Z_N model by explicitly considering the condition of discrete
holomorphicity on two and three adjacent rhombi. For two rhombi this leads to a
quadratic equation in the Boltzmann weights and for three rhombi a cubic
equation. The two-rhombus equation implies the inversion relations. The
star-triangle relation follows from the three-rhombus equation. We also show
that these weights are self-dual as a consequence of discrete holomorphicity.Comment: 11 pages, 7 figures, some clarifications and a reference adde
Long-Time Tails and Anomalous Slowing Down in the Relaxation of Spatially Inhomogeneous Excitations in Quantum Spin Chains
Exact analytic calculations in spin-1/2 XY chains, show the presence of
long-time tails in the asymptotic dynamics of spatially inhomogeneous
excitations. The decay of inhomogeneities, for , is given in the
form of a power law where the relaxation time
and the exponent depend on the wave vector ,
characterizing the spatial modulation of the initial excitation. We consider
several variants of the XY model (dimerized, with staggered magnetic field,
with bond alternation, and with isotropic and uniform interactions), that are
grouped into two families, whether the energy spectrum has a gap or not. Once
the initial condition is given, the non-equilibrium problem for the
magnetization is solved in closed form, without any other assumption. The
long-time behavior for can be obtained systematically in a form
of an asymptotic series through the stationary phase method. We found that
gapped models show critical behavior with respect to , in the sense that
there exist critical values , where the relaxation time
diverges and the exponent changes discontinuously. At those points, a
slowing down of the relaxation process is induced, similarly to phenomena
occurring near phase transitions. Long-lived excitations are identified as
incommensurate spin density waves that emerge in systems undergoing the Peierls
transition. In contrast, gapless models do not present the above anomalies as a
function of the wave vector .Comment: 25 pages, 2 postscript figures. Manuscript submitted to Physical
Review
Determinant Representations of Correlation Functions for the Supersymmetric t-J Model
Working in the -basis provided by the factorizing -matrix, the scalar
products of Bethe states for the supersymmetric t-J model are represented by
determinants. By means of these results, we obtain determinant representations
of correlation functions for the model.Comment: Latex File, 41 pages, no figure; V2: minor typos corrected, V3: This
version will appear in Commun. Math. Phy
Drinfel'd Realization of Quantum Affine Superalgebra
We obtain Drinfel'd's realization of quantum affine superalgebra
based on the super version of RS construction method and
Gauss decomposition
The Chiral Potts Models Revisited
In honor of Onsager's ninetieth birthday, we like to review some exact
results obtained so far in the chiral Potts models and to translate these
results into language more transparent to physicists, so that experts in Monte
Carlo calculations, high and low temperature expansions, and various other
methods, can use them. We shall pay special attention to the interfacial
tension between the state and the state. By examining
the ground states, it is seen that the integrable line ends at a superwetting
point, on which the relation is satisfied, so that it
is energetically neutral to have one interface or more. We present also some
partial results on the meaning of the integrable line for low temperatures
where it lives in the non-wet regime. We make Baxter's exact results more
explicit for the symmetric case. By performing a Bethe Ansatz calculation with
open boundary conditions we confirm a dilogarithm identity for the
low-temperature expansion which may be new. We propose a new model for
numerical studies. This model has only two variables and exhibits commensurate
and incommensurate phase transitions and wetting transitions near zero
temperature. It appears to be not integrable, except at one point, and at each
temperature there is a point, where it is almost identical with the integrable
chiral Potts model.Comment: J. Stat. Phys., LaTeX using psbox.tex and AMS fonts, 69 pages, 30
figure
Dolan-Grady Relations and Noncommutative Quasi-Exactly Solvable Systems
We investigate a U(1) gauge invariant quantum mechanical system on a 2D
noncommutative space with coordinates generating a generalized deformed
oscillator algebra. The Hamiltonian is taken as a quadratic form in gauge
covariant derivatives obeying the nonlinear Dolan-Grady relations. This
restricts the structure function of the deformed oscillator algebra to a
quadratic polynomial. The cases when the coordinates form the su(2) and sl(2,R)
algebras are investigated in detail. Reducing the Hamiltonian to 1D
finite-difference quasi-exactly solvable operators, we demonstrate partial
algebraization of the spectrum of the corresponding systems on the fuzzy sphere
and noncommutative hyperbolic plane. A completely covariant method based on the
notion of intrinsic algebra is proposed to deal with the spectral problem of
such systems.Comment: 25 pages; ref added; to appear in J. Phys.
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