49 research outputs found
A homomorphism between link and XXZ modules over the periodic Temperley-Lieb algebra
We study finite loop models on a lattice wrapped around a cylinder. A section
of the cylinder has N sites. We use a family of link modules over the periodic
Temperley-Lieb algebra EPTL_N(\beta, \alpha) introduced by Martin and Saleur,
and Graham and Lehrer. These are labeled by the numbers of sites N and of
defects d, and extend the standard modules of the original Temperley-Lieb
algebra. Beside the defining parameters \beta=u^2+u^{-2} with u=e^{i\lambda/2}
(weight of contractible loops) and \alpha (weight of non-contractible loops),
this family also depends on a twist parameter v that keeps track of how the
defects wind around the cylinder. The transfer matrix T_N(\lambda, \nu) depends
on the anisotropy \nu and the spectral parameter \lambda that fixes the model.
(The thermodynamic limit of T_N is believed to describe a conformal field
theory of central charge c=1-6\lambda^2/(\pi(\lambda-\pi)).)
The family of periodic XXZ Hamiltonians is extended to depend on this new
parameter v and the relationship between this family and the loop models is
established. The Gram determinant for the natural bilinear form on these link
modules is shown to factorize in terms of an intertwiner i_N^d between these
link representations and the eigenspaces of S^z of the XXZ models. This map is
shown to be an isomorphism for generic values of u and v and the critical
curves in the plane of these parameters for which i_N^d fails to be an
isomorphism are given.Comment: Replacement of "The Gram matrix as a connection between periodic loop
models and XXZ Hamiltonians", 31 page
Conformal Curves in Potts Model: Numerical Calculation
We calculated numerically the fractal dimension of the boundaries of the
Fortuin-Kasteleyn clusters of the -state Potts model for integer and
non-integer values of on the square lattice.
In addition we calculated with high accuracy the fractal dimension of the
boundary points of the same clusters on the square domain. Our calculation
confirms that this curves can be described by SLE.Comment: 11 Pages, 4 figure
Deformed strings in the Heisenberg model
We investigate solutions to the Bethe equations for the isotropic S = 1/2
Heisenberg chain involving complex, string-like rapidity configurations of
arbitrary length. Going beyond the traditional string hypothesis of undeformed
strings, we describe a general procedure to construct eigenstates including
strings with generic deformations, discuss general features of these solutions,
and provide a number of explicit examples including complete solutions for all
wavefunctions of short chains. We finally investigate some singular cases and
show from simple symmetry arguments that their contribution to zero-temperature
correlation functions vanishes.Comment: 34 pages, 13 figure
SLE local martingales in logarithmic representations
A space of local martingales of SLE type growth processes forms a
representation of Virasoro algebra, but apart from a few simplest cases not
much is known about this representation. The purpose of this article is to
exhibit examples of representations where L_0 is not diagonalizable - a
phenomenon characteristic of logarithmic conformal field theory. Furthermore,
we observe that the local martingales bear a close relation with the fusion
product of the boundary changing fields.
Our examples reproduce first of all many familiar logarithmic representations
at certain rational values of the central charge. In particular we discuss the
case of SLE(kappa=6) describing the exploration path in critical percolation,
and its relation with the question of operator content of the appropriate
conformal field theory of zero central charge. In this case one encounters
logarithms in a probabilistically transparent way, through conditioning on a
crossing event. But we also observe that some quite natural SLE variants
exhibit logarithmic behavior at all values of kappa, thus at all central
charges and not only at specific rational values.Comment: 40 pages, 7 figures. v3: completely rewritten, new title, new result
Linking Backlund and Monodromy Charges for Strings on AdS_5 x S^5
We find an explicit relation between the two known ways of generating an
infinite set of local conserved charges for the string sigma model on AdS_5 x
S^5: the Backlund and monodromy approaches. We start by constructing the
two-parameter family of Backlund transformations for the string with an
arbitrary world-sheet metric. We then show that only for a special value of one
of the parameters the solutions generated by this transformation are compatible
with the Virasoro constraints. By solving the Backlund equations in a
non-perturbative fashion, we finally show that the generating functional of the
Backlund conservation laws is equal to a certain sum of the quasi-momenta. The
positions of the quasi-momenta in the complex spectral plane are uniquely
determined by the real parameter of the Backlund transform.Comment: 25 pages, 1 figur
Cluster pinch-point densities in polygons
In a statistical cluster or loop model such as percolation, or more generally
the Potts models or O(n) models, a pinch point is a single bulk point where
several distinct clusters or loops touch. In a polygon P harboring such a model
in its interior and with 2N sides exhibiting free/fixed side-alternating
boundary conditions, "boundary" clusters anchor to the fixed sides of P. At the
critical point and in the continuum limit, the density (i.e., frequency of
occurrence) of pinch-point events between s distinct boundary clusters at a
bulk point w in P is proportional to
_P. The
w_i are the vertices of P, psi_1^c is a conformal field theory (CFT) corner
one-leg operator, and Psi_s is a CFT bulk 2s-leg operator. In this article, we
use the Coulomb gas formalism to construct explicit contour integral formulas
for these correlation functions and thereby calculate the density of various
pinch-point configurations at arbitrary points in the rectangle, in the
hexagon, and for the case s=N, in the 2N-sided polygon at the system's critical
point. Explicit formulas for these results are given in terms of algebraic
functions or integrals of algebraic functions, particularly Lauricella
functions. In critical percolation, the result for s=N=2 gives the density of
red bonds between boundary clusters (in the continuum limit) inside a
rectangle. We compare our results with high-precision simulations of critical
percolation and Ising FK clusters in a rectangle of aspect ratio two and in a
regular hexagon and find very good agreement.Comment: 31 pages, 1 appendix, 21 figures. In the second version of this
article, we have improved the organization, figures, and references that
appeared in the first versio
Fusion rules and boundary conditions in the c=0 triplet model
The logarithmic triplet model W_2,3 at c=0 is studied. In particular, we
determine the fusion rules of the irreducible representations from first
principles, and show that there exists a finite set of representations,
including all irreducible representations, that closes under fusion. With the
help of these results we then investigate the possible boundary conditions of
the W_2,3 theory. Unlike the familiar Cardy case where there is a consistent
boundary condition for every representation of the chiral algebra, we find that
for W_2,3 only a subset of representations gives rise to consistent boundary
conditions. These then have boundary spectra with non-degenerate two-point
correlators.Comment: 50 pages; v2: changed formulation in section 1.2.1 and corrected
typos, version to appear in J. Phys.