452 research outputs found
On the supersymmetric vacua of the Veneziano-Wosiek model
We study the supersymmetric vacua of the Veneziano-Wosiek model in sectors
with fermion number F=2, 4 at finite 't Hooft coupling lambda. We prove that
for F=2 there are two zero energy vacua for lambda > lambda_c = 1 and none
otherwise. We give the analytical expressions of both vacua. One of them was
previously known, the second one is obtained by solving the cohomology of the
supersymmetric charges. At F=4 we compute the would-be supersymmetric vacua at
high order in the the strong coupling expansion and provide strong support to
the conclusion that lambda = 1 is a critical point in this sector too. It
separates a strong coupling phase with two symmetric vacua from a weak coupling
phase with positive spectrum.Comment: 17 pages, 2 eps figure
On the symmetry of the partition function of some square ice models
We consider the partition function Z(N;x_1,...,x_N,y_1,...,y_N) of the square
ice model with domain wall boundary. We give a simple proof of the symmetry of
Z with respect to all its variables when the global parameter a of the model is
set to the special value a=exp(i\pi/3). Our proof does not use any
determinantal interpretation of Z and can be adapted to other situations (for
examples to some symmetric ice models).Comment: 8 page
Spin chains and combinatorics: twisted boundary conditions
The finite XXZ Heisenberg spin chain with twisted boundary conditions was
considered. For the case of even number of sites , anisotropy parameter -1/2
and twisting angle the Hamiltonian of the system possesses an
eigenvalue . The explicit form of the corresponding eigenvector was
found for . Conjecturing that this vector is the ground state of the
system we made and verified several conjectures related to the norm of the
ground state vector, its component with maximal absolute value and some
correlation functions, which have combinatorial nature. In particular, the
squared norm of the ground state vector is probably coincides with the number
of half-turn symmetric alternating sign matrices.Comment: LaTeX file, 7 page
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
Dependent coordinates in path integral measure factorization
The transformation of the path integral measure under the reduction procedure
in the dynamical systems with a symmetry is considered. The investigation is
carried out in the case of the Wiener--type path integrals that are used for
description of the diffusion on a smooth compact Riemannian manifold with the
given free isometric action of the compact semisimple unimodular Lie group. The
transformation of the path integral, which factorizes the path integral
measure, is based on the application of the optimal nonlinear filtering
equation from the stochastic theory. The integral relation between the kernels
of the original and reduced semigroup are obtained.Comment: LaTeX2e, 28 page
Multidimensional Toda type systems
On the base of Lie algebraic and differential geometry methods, a wide class
of multidimensional nonlinear systems is obtained, and the integration scheme
for such equations is proposed.Comment: 29 pages, LaTeX fil
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
The Importance of being Odd
In this letter I consider mainly a finite XXZ spin chain with periodic
boundary conditions and \bf{odd} \rm number of sites. This system is described
by the Hamiltonian . As it turned out, its ground state
energy is exactly proportional to the number of sites for a special
value of the asymmetry parameter . The trigonometric polynomial
, zeroes of which being the parameters of the ground state Bethe
eigenvector is explicitly constructed. This polynomial of degree
satisfy the Baxter T-Q equation. Using the second independent solution of this
equation corresponding to the same eigenvalue of the transfer matrix, it is
possible to find a derivative of the ground state energy w.r.t. the asymmetry
parameter. This derivative is closely connected with the correlation function
. In its turn this correlation
function is related to an average number of spin strings for the ground state
of the system under consideration: . I would like
to stress once more that all these simple formulas are \bf wrong \rm in the
case of even number of sites. Exactly this case is usually considered.Comment: 9 pages, based on the talk given at NATO Advanced Research Workshop
"Dynamical Symmetries in Integrable Two-dimensional Quantum Field Theories
and Lattice Models", 25-30 September 2000, Kyiv, Ukraine. New references are
added plus some minor correction
Free Energy of the Eight Vertex Model with an Odd Number of Lattice Sites
We calculate the bulk contribution for the doubly degenerated largest
eigenvalue of the transfer matrix of the eight vertex model with an odd number
of lattice sites N in the disordered regime using the generic equation for
roots proposed by Fabricius and McCoy. We show as expected that in the
thermodynamic limit the result coincides with the one in the N even case.Comment: 11 pages LaTeX New introduction, Method change
- …