23 research outputs found
The Q-operator and Functional Relations of the Eight-vertex Model at Root-of-unity for odd N
Following Baxter's method of producing Q_{72}-operator, we construct the
Q-operator of the root-of-unity eight-vertex model for the crossing parameter
with odd where Q_{72} does not exist. We use this
new Q-operator to study the functional relations in the Fabricius-McCoy
comparison between the root-of-unity eight-vertex model and the superintegrable
N-state chiral Potts model. By the compatibility of the constructed Q-operator
with the structure of Baxter's eight-vertex (solid-on-solid) SOS model, we
verify the set of functional relations of the root-of-unity eight-vertex model
using the explicit form of the Q-operator and fusion weights of SOS model.Comment: Latex 28 page; Typos corrected, minor changes in presentation,
References added and updated-Journal versio
The Q-operator for Root-of-Unity Symmetry in Six Vertex Model
We construct the explicit -operator incorporated with the
-loop-algebra symmetry of the six-vertex model at roots of unity. The
functional relations involving the -operator, the six-vertex transfer matrix
and fusion matrices are derived from the Bethe equation, parallel to the
Onsager-algebra-symmetry discussion in the superintegrable -state chiral
Potts model. We show that the whole set of functional equations is valid for
the -operator. Direct calculations in certain cases are also given here for
clearer illustration about the nature of the -operator in the symmetry study
of root-of-unity six-vertex model from the functional-relation aspect.Comment: Latex 26 Pages; Typos and small errors corrected, Some explanations
added for clearer presentation, References updated-Journal version with
modified labelling of sections and formula
On the occurrence of oscillatory modulations in the power-law behavior of dynamic and kinetic processes in fractals
The dynamic and kinetic behavior of processes occurring in fractals with
spatial discrete scale invariance (DSI) is considered. Spatial DSI implies the
existence of a fundamental scaling ratio (b_1). We address time-dependent
physical processes, which as a consequence of the time evolution develop a
characteristic length of the form , where z is the dynamic
exponent. So, we conjecture that the interplay between the physical process and
the symmetry properties of the fractal leads to the occurrence of time DSI
evidenced by soft log-periodic modulations of physical observables, with a
fundamental time scaling ratio given by . The conjecture is
tested numerically for random walks, and representative systems of broad
universality classes in the fields of irreversible and equilibrium critical
phenomena.Comment: 6 pages, 3 figures. Submitted to EP
Finite temperature Drude weight of the one dimensional spin 1/2 Heisenberg model}
Using the Bethe ansatz method, the zero frequency contribution (Drude weight)
to the spin current correlations is analyzed for the easy plane
antiferromagnetic Heisenberg model. The Drude weight is a monotonically
decreasing function of temperature for all 0<Delta< 1, it approaches the zero
temperature value with a power law and it appears to vanish for all finite
temperatures at the isotropic Delta=1 point.Comment: 5 pages, 2 Postscript figure
The open XXZ-chain: Bosonisation, Bethe ansatz and logarithmic corrections
We calculate the bulk and boundary parts of the free energy for an open
spin-1/2 XXZ-chain in the critical regime by bosonisation. We identify the
cutoff independent contributions and determine their amplitudes by comparing
with Bethe ansatz calculations at zero temperature T. For the bulk part of the
free energy we find agreement with Lukyanov's result [Nucl.Phys.B 522, 533
(1998)]. In the boundary part we obtain a cutoff independent term which is
linear in T and determines the temperature dependence of the boundary
susceptibility in the attractive regime for . We further show that at
particular anisotropies where contributions from irrelevant operators with
different scaling dimensions cross, logarithmic corrections appear. We give
explicit formulas for these terms at those anisotropies where they are most
important. We verify our results by comparing with extensive numerical
calculations based on a numerical solution of the T=0 Bethe ansatz equations,
the finite temperature Bethe ansatz equations in the quantum-transfer matrix
formalism, and the density-matrix renormalisation group applied to transfer
matrices.Comment: 35 pages, 8 figure
Fusion Operators in the Generalized -model and Root-of-unity Symmetry of the XXZ Spin Chain of Higher Spin
We construct the fusion operators in the generalized -model using
the fused -operators, and verify the fusion relations with the truncation
identity. The algebraic Bethe ansatz discussion is conducted on two special
classes of which include the superintegrable chiral Potts model.
We then perform the parallel discussion on the XXZ spin chain at roots of
unity, and demonstrate that the -loop-algebra symmetry exists for the
root-of-unity XXZ spin chain with a higher spin, where the evaluation
parameters for the symmetry algebra are identified by the explicit
Fabricius-McCoy current for the Bethe states. Parallels are also drawn to the
comparison with the superintegrable chiral Potts model.Comment: Latex 33 Pages; Typos and errors corrected, New improved version by
adding explanations for better presentation. Terminology in the content and
the title refined. References added and updated-Journal versio
Irreducibility criterion for a finite-dimensional highest weight representation of the sl(2) loop algebra and the dimensions of reducible representations
We present a necessary and sufficient condition for a finite-dimensional
highest weight representation of the loop algebra to be irreducible. In
particular, for a highest weight representation with degenerate parameters of
the highest weight, we can explicitly determine whether it is irreducible or
not. We also present an algorithm for constructing finite-dimensional highest
weight representations with a given highest weight. We give a conjecture that
all the highest weight representations with the same highest weight can be
constructed by the algorithm. For some examples we show the conjecture
explicitly. The result should be useful in analyzing the spectra of integrable
lattice models related to roots of unity representations of quantum groups, in
particular, the spectral degeneracy of the XXZ spin chain at roots of unity
associated with the loop algebra.Comment: 32 pages with no figure; with corrections on the published versio
Traces on the Sklyanin algebra and correlation functions of the eight-vertex model
We propose a conjectural formula for correlation functions of the Z-invariant
(inhomogeneous) eight-vertex model. We refer to this conjecture as Ansatz. It
states that correlation functions are linear combinations of products of three
transcendental functions, with theta functions and derivatives as coefficients.
The transcendental functions are essentially logarithmic derivatives of the
partition function per site. The coefficients are given in terms of a linear
functional on the Sklyanin algebra, which interpolates the usual trace on
finite dimensional representations. We establish the existence of the
functional and discuss the connection to the geometry of the classical limit.
We also conjecture that the Ansatz satisfies the reduced qKZ equation. As a
non-trivial example of the Ansatz, we present a new formula for the
next-nearest neighbor correlation functions.Comment: 35 pages, 2 figures, final versio
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