108 research outputs found
Bethe roots and refined enumeration of alternating-sign matrices
The properties of the most probable ground state candidate for the XXZ spin
chain with the anisotropy parameter equal to -1/2 and an odd number of sites is
considered. Some linear combinations of the components of the considered state,
divided by the maximal component, coincide with the elementary symmetric
polynomials in the corresponding Bethe roots. It is proved that those
polynomials are equal to the numbers providing the refined enumeration of the
alternating-sign matrices of order M+1 divided by the total number of the
alternating-sign matrices of order M, for the chain of length 2M+1.Comment: LaTeX 2e, 12 pages, minor corrections, references adde
Three-coloring statistical model with domain wall boundary conditions. I. Functional equations
In 1970 Baxter considered the statistical three-coloring lattice model for
the case of toroidal boundary conditions. He used the Bethe ansatz and found
the partition function of the model in the thermodynamic limit. We consider the
same model but use other boundary conditions for which one can prove that the
partition function satisfies some functional equations similar to the
functional equations satisfied by the partition function of the six-vertex
model for a special value of the crossing parameter.Comment: 16 pages, notations changed for consistency with the next part,
appendix adde
Bethe Equations "on the Wrong Side of Equator"
We analyse the famous Baxter's equations for () spin chain
and show that apart from its usual polynomial (trigonometric) solution, which
provides the solution of Bethe-Ansatz equations, there exists also the second
solution which should corresponds to Bethe-Ansatz beyond . This second
solution of Baxter's equation plays essential role and together with the first
one gives rise to all fusion relations.Comment: 13 pages, original paper was spoiled during transmissio
Polynomial solutions of qKZ equation and ground state of XXZ spin chain at Delta = -1/2
Integral formulae for polynomial solutions of the quantum
Knizhnik-Zamolodchikov equations associated with the R-matrix of the six-vertex
model are considered. It is proved that when the deformation parameter q is
equal to e^{+- 2 pi i/3} and the number of vertical lines of the lattice is
odd, the solution under consideration is an eigenvector of the inhomogeneous
transfer matrix of the six-vertex model. In the homogeneous limit it is a
ground state eigenvector of the antiferromagnetic XXZ spin chain with the
anisotropy parameter Delta equal to -1/2 and odd number of sites. The obtained
integral representations for the components of this eigenvector allow to prove
some conjectures on its properties formulated earlier. A new statement relating
the ground state components of XXZ spin chains and Temperley-Lieb loop models
is formulated and proved.Comment: v2: cosmetic changes, new section on refined TSSCPPs vs refined ASM
Direct SIMS Determination of the InxGa1-xN Mole Fraction
We demonstrate that our secondary mass ion spectroscopy (SIMS) method for the determination of the mole fraction in solid InxGa1-xN solutions is accurate and reproduceable without need of reference samples. The method is based on measuring relative current values of CsM+ (M=Ga, In) secondary ions. The claim of reliable SIMS determination without reference samples was confirmed by four independent analytical methods on the same samples with a relative error in the InN mole fraction determination below 15
The role of orthogonal polynomials in the six-vertex model and its combinatorial applications
The Hankel determinant representations for the partition function and
boundary correlation functions of the six-vertex model with domain wall
boundary conditions are investigated by the methods of orthogonal polynomial
theory. For specific values of the parameters of the model, corresponding to
1-, 2- and 3-enumerations of Alternating Sign Matrices (ASMs), these
polynomials specialize to classical ones (Continuous Hahn, Meixner-Pollaczek,
and Continuous Dual Hahn, respectively). As a consequence, a unified and
simplified treatment of ASMs enumerations turns out to be possible, leading
also to some new results such as the refined 3-enumerations of ASMs.
Furthermore, the use of orthogonal polynomials allows us to express, for
generic values of the parameters of the model, the partition function of the
(partially) inhomogeneous model in terms of the one-point boundary correlation
functions of the homogeneous one.Comment: Talk presented by F.C. at the Short Program of the Centre de
Recherches Mathematiques: Random Matrices, Random Processes and Integrable
Systems, Montreal, June 20 - July 8, 200
Refined Razumov-Stroganov conjectures for open boundaries
Recently it has been conjectured that the ground-state of a Markovian
Hamiltonian, with one boundary operator, acting in a link pattern space is
related to vertically and horizontally symmetric alternating-sign matrices
(equivalently fully-packed loop configurations (FPL) on a grid with special
boundaries).We extend this conjecture by introducing an arbitrary boundary
parameter. We show that the parameter dependent ground state is related to
refined vertically symmetric alternating-sign matrices i.e. with prescribed
configurations (respectively, prescribed FPL configurations) in the next to
central row.
We also conjecture a relation between the ground-state of a Markovian
Hamiltonian with two boundary operators and arbitrary coefficients and some
doubly refined (dependence on two parameters) FPL configurations. Our
conjectures might be useful in the study of ground-states of the O(1) and XXZ
models, as well as the stationary states of Raise and Peel models.Comment: 11 pages LaTeX, 8 postscript figure
Exact expressions for correlations in the ground state of the dense O(1) loop model
Conjectures for analytical expressions for correlations in the dense O
loop model on semi infinite square lattices are given. We have obtained these
results for four types of boundary conditions. Periodic and reflecting boundary
conditions have been considered before. We give many new conjectures for these
two cases and review some of the existing results. We also consider boundaries
on which loops can end. We call such boundaries ''open''. We have obtained
expressions for correlations when both boundaries are open, and one is open and
the other one is reflecting. Also, we formulate a conjecture relating the
ground state of the model with open boundaries to Fully Packed Loop models on a
finite square grid. We also review earlier obtained results about this relation
for the three other types of boundary conditions. Finally, we construct a
mapping between the ground state of the dense O loop model and the XXZ
spin chain for the different types of boundary conditions.Comment: 25 pages, version accepted by JSTA
On two-point boundary correlations in the six-vertex model with DWBC
The six-vertex model with domain wall boundary conditions (DWBC) on an N x N
square lattice is considered. The two-point correlation function describing the
probability of having two vertices in a given state at opposite (top and
bottom) boundaries of the lattice is calculated. It is shown that this
two-point boundary correlator is expressible in a very simple way in terms of
the one-point boundary correlators of the model on N x N and (N-1) x (N-1)
lattices. In alternating sign matrix (ASM) language this result implies that
the doubly refined x-enumerations of ASMs are just appropriate combinations of
the singly refined ones.Comment: v2: a reference added, typos correcte
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