5 research outputs found
Complex WKB Analysis of a PT Symmetric Eigenvalue Problem
The spectra of a particular class of PT symmetric eigenvalue problems has
previously been studied, and found to have an extremely rich structure. In this
paper we present an explanation for these spectral properties in terms of
quantisation conditions obtained from the complex WKB method. In particular, we
consider the relation of the quantisation conditions to the reality and
positivity properties of the eigenvalues. The methods are also used to examine
further the pattern of eigenvalue degeneracies observed by Dorey et al. in
[1,2].Comment: 22 pages, 13 figures. Added references, minor revision
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
Lattice fermion models with supersymmetry
We investigate a family of lattice models with manifest N=2 supersymmetry.
The models describe fermions on a 1D lattice, subject to the constraint that no
more than k consecutive lattice sites may be occupied. We discuss the special
properties arising from the supersymmetry, and present Bethe ansatz solutions
of the simplest models. We display the connections of the k=1 model with the
spin-1/2 antiferromagnetic XXZ chain at \Delta=-1/2, and the k=2 model with
both the su(2|1)-symmetric tJ model in the ferromagnetic regime and the
integrable spin-1 XXZ chain at \Delta=-1/\sqrt{2}. We argue that these models
include critical points described by the superconformal minimal models.Comment: 28 pages. v2: added new result on mapping to XXZ chai
T-systems and Y-systems in integrable systems
The T and Y-systems are ubiquitous structures in classical and quantum
integrable systems. They are difference equations having a variety of aspects
related to commuting transfer matrices in solvable lattice models, q-characters
of Kirillov-Reshetikhin modules of quantum affine algebras, cluster algebras
with coefficients, periodicity conjectures of Zamolodchikov and others,
dilogarithm identities in conformal field theory, difference analogue of
L-operators in KP hierarchy, Stokes phenomena in 1d Schr\"odinger problem,
AdS/CFT correspondence, Toda field equations on discrete space-time, Laplace
sequence in discrete geometry, Fermionic character formulas and combinatorial
completeness of Bethe ansatz, Q-system and ideal gas with exclusion statistics,
analytic and thermodynamic Bethe ans\"atze, quantum transfer matrix method and
so forth. This review article is a collection of short reviews on these topics
which can be read more or less independently.Comment: 156 pages. Minor corrections including the last paragraph of sec.3.5,
eqs.(4.1), (5.28), (9.37) and (13.54). The published version (JPA topical
review) also needs these correction
Finite size effects and the supersymmetric sine-Gordon models
We propose nonlinear integral equations to describe the groundstate energy of the fractional supersymmetric sine-Gordon models. The equations encompass the N = 1 supersymmetric sine-Gordon model as well as the phi(id,id,adj) perturbation of the SU(2)(L) x SU(2)(K)/SU(2)(L+K) models at rational level K. A second set of equations is proposed for the groundstate energy of the N = 2 supersymmetric sine-Gordon model