84 research outputs found
Eight-vertex model and non-stationary Lame equation
We study the ground state eigenvalues of Baxter's Q-operator for the
eight-vertex model in a special case when it describes the off-critical
deformation of the six-vertex model. We show that these
eigenvalues satisfy a non-stationary Schrodinger equation with the
time-dependent potential given by the Weierstrass elliptic P-function where the
modular parameter plays the role of (imaginary) time. In the scaling
limit the equation transforms into a ``non-stationary Mathieu equation'' for
the vacuum eigenvalues of the Q-operators in the finite-volume massive
sine-Gordon model at the super-symmetric point, which is closely related to the
theory of dilute polymers on a cylinder and the Painleve III equation.Comment: 11 pages, LaTeX, minor misprints corrected, references adde
The eight-vertex model and Painleve VI
In this letter we establish a connection of Picard-type elliptic solutions of
Painleve VI equation with the special solutions of the non-stationary Lame
equation. The latter appeared in the study of the ground state properties of
Baxter's solvable eight-vertex lattice model at a particular point,
, of the disordered regime.Comment: 9 pages, LaTeX, submitted to the special issue on Painleve VI,
Journal of Physics
Eight-vertex model and Painlev\'e VI equation. II. Eigenvector results
We study a special anisotropic XYZ-model on a periodic chain of an odd length
and conjecture exact expressions for certain components of the ground state
eigenvectors. The results are written in terms of tau-functions associated with
Picard's elliptic solutions of the Painlev\'e VI equation. Connections with
other problems related to the eight-vertex model are briefly discussed.Comment: 18 page
Quantum Geometry of 3-Dimensional Lattices and Tetrahedron Equation
We study geometric consistency relations between angles of 3-dimensional (3D)
circular quadrilateral lattices -- lattices whose faces are planar
quadrilaterals inscribable into a circle. We show that these relations generate
canonical transformations of a remarkable "ultra-local" Poisson bracket algebra
defined on discrete 2D surfaces consisting of circular quadrilaterals.
Quantization of this structure allowed us to obtain new solutions of the
tetrahedron equation (the 3D analog of the Yang-Baxter equation) as well as
reproduce all those that were previously known. These solutions generate an
infinite number of non-trivial solutions of the Yang-Baxter equation and also
define integrable 3D models of statistical mechanics and quantum field theory.
The latter can be thought of as describing quantum fluctuations of lattice
geometry.Comment: Plenary talk at the XVI International Congress on Mathematical
Physics, 3-8 August 2009, Prague, Czech Republi
An integrable 3D lattice model with positive Boltzmann weights
In this paper we construct a three-dimensional (3D) solvable lattice model
with non-negative Boltzmann weights. The spin variables in the model are
assigned to edges of the 3D cubic lattice and run over an infinite number of
discrete states. The Boltzmann weights satisfy the tetrahedron equation, which
is a 3D generalisation of the Yang-Baxter equation. The weights depend on a
free parameter 0<q<1 and three continuous field variables. The layer-to-layer
transfer matrices of the model form a two-parameter commutative family. This is
the first example of a solvable 3D lattice model with non-negative Boltzmann
weights.Comment: HyperTex is disabled due to conflicts with some macro
Exact solution of the Faddeev-Volkov model
The Faddeev-Volkov model is an Ising-type lattice model with positive
Boltzmann weights where the spin variables take continuous values on the real
line. It serves as a lattice analog of the sinh-Gordon and Liouville models and
intimately connected with the modular double of the quantum group U_q(sl_2).
The free energy of the model is exactly calculated in the thermodynamic limit.
In the quasi-classical limit c->infinity the model describes quantum
fluctuations of discrete conformal transformations connected with the
Thurston's discrete analogue of the Riemann mappings theorem. In the
strongly-coupled limit c->1 the model turns into a discrete version of the D=2
Zamolodchikov's ``fishing-net'' model.Comment: 4 page
Zamolodchikov's Tetrahedron Equation and Hidden Structure of Quantum Groups
The tetrahedron equation is a three-dimensional generalization of the
Yang-Baxter equation. Its solutions define integrable three-dimensional lattice
models of statistical mechanics and quantum field theory. Their integrability
is not related to the size of the lattice, therefore the same solution of the
tetrahedron equation defines different integrable models for different finite
periodic cubic lattices. Obviously, any such three-dimensional model can be
viewed as a two-dimensional integrable model on a square lattice, where the
additional third dimension is treated as an internal degree of freedom.
Therefore every solution of the tetrahedron equation provides an infinite
sequence of integrable 2d models differing by the size of this "hidden third
dimension". In this paper we construct a new solution of the tetrahedron
equation, which provides in this way the two-dimensional solvable models
related to finite-dimensional highest weight representations for all quantum
affine algebra , where the rank coincides with the size
of the hidden dimension. These models are related with an anisotropic
deformation of the -invariant Heisenberg magnets. They were extensively
studied for a long time, but the hidden 3d structure was hitherto unknown. Our
results lead to a remarkable exact "rank-size" duality relation for the nested
Bethe Ansatz solution for these models. Note also, that the above solution of
the tetrahedron equation arises in the quantization of the "resonant three-wave
scattering" model, which is a well-known integrable classical system in 2+1
dimensions.Comment: v2: references adde
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