611 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
A possible combinatorial point for XYZ-spin chain
We formulate and discuss a number of conjectures on the ground state vectors
of the XYZ-spin chains of odd length with periodic boundary conditions and a
special choice of the Hamiltonian parameters. In particular, arguments for the
validity of a sum rule for the components, which describes in a sense the
degree of antiferromagneticity of the chain, are given.Comment: AMSLaTeX, 15 page
On Z-graded loop Lie algebras, loop groups, and Toda equations
Toda equations associated with twisted loop groups are considered. Such
equations are specified by Z-gradations of the corresponding twisted loop Lie
algebras. The classification of Toda equations related to twisted loop Lie
algebras with integrable Z-gradations is discussed.Comment: 24 pages, talk given at the Workshop "Classical and Quantum
Integrable Systems" (Dubna, January, 2007
The Wave Functions for the Free-Fermion Part of the Spectrum of the Quantum Spin Models
We conjecture that the free-fermion part of the eigenspectrum observed
recently for the Perk-Schultz spin chain Hamiltonian in a finite
lattice with is a consequence of the existence of a
special simple eigenvalue for the transfer matrix of the auxiliary
inhomogeneous vertex model which appears in the nested Bethe ansatz
approach. We prove that this conjecture is valid for the case of the SU(3) spin
chain with periodic boundary condition. In this case we obtain a formula for
the components of the eigenvector of the auxiliary inhomogeneous 6-vertex model
(), which permit us to find one by one all components of
this eigenvector and consequently to find the eigenvectors of the free-fermion
part of the eigenspectrum of the SU(3) spin chain. Similarly as in the known
case of the case at our numerical and analytical
studies induce some conjectures for special rates of correlation functions.Comment: 25 pages and no figure
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
Sum rules for the ground states of the O(1) loop model on a cylinder and the XXZ spin chain
The sums of components of the ground states of the O(1) loop model on a
cylinder or of the XXZ quantum spin chain at Delta=-1/2 (of size L) are
expressed in terms of combinatorial numbers. The methods include the
introduction of spectral parameters and the use of integrability, a mapping
from size L to L+1, and knot-theoretic skein relations.Comment: final version to be publishe
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
Perturbative quantum gauge invariance: Where the ghosts come from
A condensed introduction to quantum gauge theories is given in the
perturbative S-matrix framework; path integral methods are used nowhere. This
approach emphasizes the fact that it is not necessary to start from classical
gauge theories which are then subject to quantization, but it is also possible
to recover the classical group structure and coupling properties from purely
quantum mechanical principles. As a main tool we use a free field version of
the Becchi-Rouet-Stora-Tyutin gauge transformation, which contains no
interaction terms related to a coupling constant. This free gauge
transformation can be formulated in an analogous way for quantum
electrodynamics, Yang-Mills theories with massless or massive gauge bosons and
quantum gravity.Comment: 28 pages, LATEX. Some typos corrected, version to be publishe
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
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