11,132 research outputs found
Bethe Ansatz Equations for the Broken -Symmetric Model
We obtain the Bethe Ansatz equations for the broken -symmetric
model by constructing a functional relation of the transfer matrix of
-operators. This model is an elliptic off-critical extension of the
Fateev-Zamolodchikov model. We calculate the free energy of this model on the
basis of the string hypothesis.Comment: 43 pages, latex, 11 figure
The quantum bialgebra associated with the eight-vertex R-matrix
The quantum bialgebra related to the Baxter's eight-vertex R-matrix is found
as a quantum deformation of the Lie algebra of sl(2)-valued automorphic
functions on a complex torus.Comment: 4 page
A Generalized Q-operator for U_q(\hat(sl_2)) Vertex Models
In this paper, we construct a Q-operator as a trace of a representation of
the universal R-matrix of over an infinite-dimensional
auxiliary space. This auxiliary space is a four-parameter generalization of the
q-oscillator representations used previously. We derive generalized T-Q
relations in which 3 of these parameters shift. After a suitable restriction of
parameters, we give an explicit expression for the Q-operator of the 6-vertex
model and show the connection with Baxter's expression for the central block of
his corresponding operator.Comment: 22 pages, Latex2e. This replacement is a revised version that
includes a simple explicit expression for the Q matrix for the 6-vertex mode
Corner Transfer Matrix Renormalization Group Method Applied to the Ising Model on the Hyperbolic Plane
Critical behavior of the Ising model is investigated at the center of large
scale finite size systems, where the lattice is represented as the tiling of
pentagons. The system is on the hyperbolic plane, and the recursive structure
of the lattice makes it possible to apply the corner transfer matrix
renormalization group method. From the calculated nearest neighbor spin
correlation function and the spontaneous magnetization, it is concluded that
the phase transition of this model is mean-field like. One parameter
deformation of the corner Hamiltonian on the hyperbolic plane is discussed.Comment: 4 pages, 5 figure
Equations of motion approach to the spin-1/2 Ising model on the Bethe lattice
We exactly solve the ferromagnetic spin-1/2 Ising model on the Bethe lattice
in the presence of an external magnetic field by means of the equations of
motion method within the Green's function formalism. In particular, such an
approach is applied to an isomorphic model of localized Fermi particles
interacting via an intersite Coulomb interaction. A complete set of
eigenoperators is found together with the corresponding eigenvalues. The
Green's functions and the correlation functions are written in terms of a
finite set of parameters to be self-consistently determined. A procedure is
developed, that allows us to exactly fix the unknown parameters in the case of
a Bethe lattice with any coordination number z. Non-local correlation functions
up to four points are also provided together with a study of the relevant
thermodynamic quantities.Comment: RevTex, 29 pages, 13 figure
Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix
We discuss an algebraic method for constructing eigenvectors of the transfer
matrix of the eight vertex model at the discrete coupling parameters. We
consider the algebraic Bethe ansatz of the elliptic quantum group for the case where the parameter satisfies for arbitrary integers , and . When or
is odd, the eigenvectors thus obtained have not been discussed previously.
Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin
chain, some of which are shown to be related to the loop algebra
symmetry of the XXZ spin chain. We show that the dimension of some degenerate
eigenspace of the XYZ spin chain on sites is given by , if
is an even integer. The construction of eigenvectors of the transfer matrices
of some related IRF models is also discussed.Comment: 19 pages, no figure (revisd version with three appendices
Auxiliary matrices for the six-vertex model and the algebraic Bethe ansatz
We connect two alternative concepts of solving integrable models, Baxter's
method of auxiliary matrices (or Q-operators) and the algebraic Bethe ansatz.
The main steps of the calculation are performed in a general setting and a
formula for the Bethe eigenvalues of the Q-operator is derived. A proof is
given for states which contain up to three Bethe roots. Further evidence is
provided by relating the findings to the six-vertex fusion hierarchy. For the
XXZ spin-chain we analyze the cases when the deformation parameter of the
underlying quantum group is evaluated both at and away from a root of unity.Comment: 32 page
Auxiliary matrices on both sides of the equator
The spectra of previously constructed auxiliary matrices for the six-vertex
model at roots of unity are investigated for spin-chains of even and odd
length. The two cases show remarkable differences. In particular, it is shown
that for even roots of unity and an odd number of sites the eigenvalues contain
two linear independent solutions to Baxter's TQ-equation corresponding to the
Bethe ansatz equations above and below the equator. In contrast, one finds for
even spin-chains only one linear independent solution and complete strings. The
other main result is the proof of a previous conjecture on the degeneracies of
the six-vertex model at roots of unity. The proof rests on the derivation of a
functional equation for the auxiliary matrices which is closely related to a
functional equation for the eight-vertex model conjectured by Fabricius and
McCoy.Comment: 22 pages; 2nd version: one paragraph added in the conclusion and some
typos correcte
Variational method and duality in the 2D square Potts model
The ferromagnetic q-state Potts model on a square lattice is analyzed, for
q>4, through an elaborate version of the operatorial variational method. In the
variational approach proposed in the paper, the duality relations are exactly
satisfied, involving at a more fundamental level, a duality relationship
between variational parameters. Besides some exact predictions, the approach is
very effective in the numerical estimates over the whole range of temperature
and can be systematically improved.Comment: 20 pages, 5 EPS figure
Critical dynamics of the k-core pruning process
We present the theory of the k-core pruning process (progressive removal of
nodes with degree less than k) in uncorrelated random networks. We derive exact
equations describing this process and the evolution of the network structure,
and solve them numerically and, in the critical regime of the process,
analytically. We show that the pruning process exhibits three different
behaviors depending on whether the mean degree of the initial network is
above, equal to, or below the threshold _c corresponding to the emergence of
the giant k-core. We find that above the threshold the network relaxes
exponentially to the k-core. The system manifests the phenomenon known as
"critical slowing down", as the relaxation time diverges when tends to
_c. At the threshold, the dynamics become critical characterized by a
power-law relaxation (1/t^2). Below the threshold, a long-lasting transient
process (a "plateau" stage) occurs. This transient process ends with a collapse
in which the entire network disappears completely. The duration of the process
diverges when tends to _c. We show that the critical dynamics of the
pruning are determined by branching processes of spreading damage. Clusters of
nodes of degree exactly k are the evolving substrate for these branching
processes. Our theory completely describes this branching cascade of damage in
uncorrelated networks by providing the time dependent distribution function of
branching. These theoretical results are supported by our simulations of the
-core pruning in Erdos-Renyi graphs.Comment: 12 pages, 10 figure
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