93 research outputs found
Characters of graded parafermion conformal field theory
The graded parafermion conformal field theory at level k is a close cousin of
the much-studied Z_k parafermion model. Three character formulas for the graded
parafermion theory are presented, one bosonic, one fermionic (both previously
known) and one of spinon type (which is new). The main result of this paper is
a proof of the equivalence of these three forms using q-series methods combined
with the combinatorics of lattice paths. The pivotal step in our approach is
the observation that the graded parafermion theory -- which is equivalent to
the coset osp(1,2)_k/ u(1) -- can be factored as (osp(1,2)_k/ su(2)_k) x
(su(2)_k/ u(1)), with the two cosets on the right equivalent to the minimal
model M(k+2,2k+3) and the Z_k parafermion model, respectively. This
factorisation allows for a new combinatorial description of the graded
parafermion characters in terms of the one-dimensional configuration sums of
the (k+1)-state Andrews--Baxter--Forrester model.Comment: 36 page
Ising tricriticality and the dilute A model
Some universal amplitude ratios appropriate to the peturbation
of the c=7/10 minimal field theory, the subleading magnetic perturbation of the
tricritical Ising model, are explicitly demonstrated in the dilute A model,
in regime 1.Comment: 8 pages, LaTeX using iop macro
Dilute Birman--Wenzl--Murakami Algebra and models
A ``dilute'' generalisation of the Birman--Wenzl--Murakami algebra is
considered. It can be ``Baxterised'' to a solution of the Yang--Baxter algebra.
The vertex models are examples of corresponding solvable
lattice models and can be regarded as the dilute version of the
vertex models.Comment: 11 page
The dilute A_L models and the integrable perturbations of unitary minimal CFTs
Recently, a set of thermodynamic Bethe ansatz equations is proposed by Dorey,
Pocklington and Tateo for unitary minimal models perturbed by \phi_{1,2} or
\phi_{2,1} operator. We examine their results in view of the lattice analogues,
dilute A_L models at regime 1 and 2. Taking M_{5,6}+\phi_{1,2} and
M_{3,4}+\phi_{2,1} as the simplest examples, we will explicitly show that the
conjectured TBA equations can be recovered from the lattice model in a scaling
limit.Comment: 14 pages, 2 figure
Scaling Limit of the Ising Model in a Field
The dilute A_3 model is a solvable IRF (interaction round a face) model with
three local states and adjacency conditions encoded by the Dynkin diagram of
the Lie algebra A_3. It can be regarded as a solvable version of an Ising model
at the critical temperature in a magnetic field. One therefore expects the
scaling limit to be governed by Zamolodchikov's integrable perturbation of the
c=1/2 conformal field theory. Indeed, a recent thermodynamic Bethe Ansatz
approach succeeded to unveil the corresponding E_8 structure under certain
assumptions on the nature of the Bethe Ansatz solutions. In order to check
these conjectures, we perform a detailed numerical investigation of the
solutions of the Bethe Ansatz equations for the critical and off-critical
model. Scaling functions for the ground-state corrections and for the lowest
spectral gaps are obtained, which give very precise numerical results for the
lowest mass ratios in the massive scaling limit. While these agree perfectly
with the E_8 mass ratios, we observe one state which seems to violate the
assumptions underlying the thermodynamic Bethe Ansatz calculation. We also
analyze the critical spectrum of the dilute A_3 model, which exhibits massive
excitations on top of the massless states of the Ising conformal field theory.Comment: 29 pages, RevTeX, 11 PostScript figures included by epsf, using
amssymb.sty (v2.2
The Yang-Baxter equation for PT invariant nineteen vertex models
We study the solutions of the Yang-Baxter equation associated to nineteen
vertex models invariant by the parity-time symmetry from the perspective of
algebraic geometry. We determine the form of the algebraic curves constraining
the respective Boltzmann weights and found that they possess a universal
structure. This allows us to classify the integrable manifolds in four
different families reproducing three known models besides uncovering a novel
nineteen vertex model in a unified way. The introduction of the spectral
parameter on the weights is made via the parameterization of the fundamental
algebraic curve which is a conic. The diagonalization of the transfer matrix of
the new vertex model and its thermodynamic limit properties are discussed. We
point out a connection between the form of the main curve and the nature of the
excitations of the corresponding spin-1 chains.Comment: 43 pages, 6 figures and 5 table
Correction induced by irrelevant operators in the correlators of the 2d Ising model in a magnetic field
We investigate the presence of irrelevant operators in the 2d Ising model
perturbed by a magnetic field, by studying the corrections induced by these
operators in the spin-spin correlator of the model. To this end we perform a
set of high precision simulations for the correlator both along the axes and
along the diagonal of the lattice. By comparing the numerical results with the
predictions of a perturbative expansion around the critical point we find
unambiguous evidences of the presence of such irrelevant operators. It turns
out that among the irrelevant operators the one which gives the largest
correction is the spin 4 operator T^2 + \bar T^2 which accounts for the
breaking of the rotational invariance due to the lattice. This result agrees
with what was already known for the correlator evaluated exactly at the
critical point and also with recent results obtained in the case of the thermal
perturbation of the model.Comment: 28 pages, no figure
Polynomial Identities, Indices, and Duality for the N=1 Superconformal Model SM(2,4\nu)
We prove polynomial identities for the N=1 superconformal model SM(2,4\nu)
which generalize and extend the known Fermi/Bose character identities. Our
proof uses the q-trinomial coefficients of Andrews and Baxter on the bosonic
side and a recently introduced very general method of producing recursion
relations for q-series on the fermionic side. We use these polynomials to
demonstrate a dual relation under q \rightarrow q^{-1} between SM(2,4\nu) and
M(2\nu-1,4\nu). We also introduce a generalization of the Witten index which is
expressible in terms of the Rogers false theta functions.Comment: 41 pages, harvmac, no figures; new identities, proofs and comments
added; misprints eliminate
Higher string functions, higher-level Appell functions, and the logarithmic ^sl(2)_k/u(1) CFT model
We generalize the string functions C_{n,r}(tau) associated with the coset
^sl(2)_k/u(1) to higher string functions A_{n,r}(tau) and B_{n,r}(tau)
associated with the coset W(k)/u(1) of the W-algebra of the logarithmically
extended ^sl(2)_k conformal field model with positive integer k. The higher
string functions occur in decomposing W(k) characters with respect to level-k
theta and Appell functions and their derivatives (the characters are neither
quasiperiodic nor holomorphic, and therefore cannot decompose with respect to
only theta-functions). The decomposition coefficients, to be considered
``logarithmic parafermionic characters,'' are given by A_{n,r}(tau),
B_{n,r}(tau), C_{n,r}(tau), and by the triplet \mathscr{W}(p)-algebra
characters of the (p=k+2,1) logarithmic model. We study the properties of
A_{n,r} and B_{n,r}, which nontrivially generalize those of the classic string
functions C_{n,r}, and evaluate the modular group representation generated from
A_{n,r}(tau) and B_{n,r}(tau); its structure inherits some features of modular
transformations of the higher-level Appell functions and the associated
transcendental function Phi.Comment: 34 pages, amsart++, times. V2: references added; minor changes; some
nonsense in B.3.3. correcte
Surface Critical Phenomena and Scaling in the Eight-Vertex Model
We give a physical interpretation of the entries of the reflection
-matrices of Baxter's eight-vertex model in terms of an Ising interaction at
an open boundary. Although the model still defies an exact solution we
nevertheless obtain the exact surface free energy from a crossing-unitarity
relation. The singular part of the surface energy is described by the critical
exponents and , where controls the strength of the four-spin
interaction. These values reduce to the known Ising exponents at the decoupling
point and confirm the scaling relations
and .Comment: 12 pages, LaTeX with REVTEX macros needed. To appear in Physical
Review Letter
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