26 research outputs found
Colour valued Scattering Matrices
We describe a general construction principle which allows to add colour
values to a coupling constant dependent scattering matrix. As a concrete
realization of this mechanism we provide a new type of S-matrix which
generalizes the one of affine Toda field theory, being related to a pair of Lie
algebras. A characteristic feature of this S-matrix is that in general it
violates parity invariance. For particular choices of the two Lie algebras
involved this scattering matrix coincides with the one related to the scaling
models described by the minimal affine Toda S-matrices and for other choices
with the one of the Homogeneous sine-Gordon models with vanishing resonance
parameters. We carry out the thermodynamic Bethe ansatz and identify the
corresponding ultraviolet effective central charges.Comment: 8 pages Latex, example, comment and reference adde
On the universal Representation of the Scattering Matrix of Affine Toda Field Theory
By exploiting the properties of q-deformed Coxeter elements, the scattering
matrices of affine Toda field theories with real coupling constant related to
any dual pair of simple Lie algebras may be expressed in a completely generic
way. We discuss the governing equations for the existence of bound states, i.e.
the fusing rules, in terms of q-deformed Coxeter elements, twisted q-deformed
Coxeter elements and undeformed Coxeter elements. We establish the precise
relation between these different formulations and study their solutions. The
generalized S-matrix bootstrap equations are shown to be equivalent to the
fusing rules. The relation between different versions of fusing rules and
quantum conserved quantities, which result as nullvectors of a doubly
q-deformed Cartan like matrix, is presented. The properties of this matrix
together with the so-called combined bootstrap equations are utilised in order
to derive generic integral representations for the scattering matrix in terms
of quantities of either of the two dual algebras. We present extensive
case-by-case data, in particular on the orbits generated by the various Coxeter
elements.Comment: 57 page
Form factors of the homogeneous sine-Gordon models
We provide general determinant formulae for all n-particle form factors related to the trace of the energy momentum tensor and the analogue of the order and disorder operator in the -homogeneous Sine-Gordon model. We employ the form factors related to the trace of the energy momentum tensor in the application of the c-theorem and find perfect agreement with the physical picture recently obtained by means of the thermodynamic Bethe ansatz. For finite resonance parameter we recover the expected WZNW-coset central charge and for infinite resonance parameter the theory decouples into two free fermions
Large and small Density Approximations to the thermodynamic Bethe Ansatz
We provide analytical solutions to the thermodynamic Bethe ansatz equations
in the large and small density approximations. We extend results previously
obtained for leading order behaviour of the scaling function of affine Toda
field theories related to simply laced Lie algebras to the non-simply laced
case. The comparison with semi-classical methods shows perfect agreement for
the simply laced case. We derive the Y-systems for affine Toda field theories
with real coupling constant and employ them to improve the large density
approximations. We test the quality of our analysis explicitly for the
Sinh-Gordon model and the -affine Toda field theory.Comment: 19 pages Latex, 2 figure
Exactly solvable potentials of Calogero type for q-deformed Coxeter groups
We establish that by parameterizing the configuration space of a
one-dimensional quantum system by polynomial invariants of q-deformed Coxeter
groups it is possible to construct exactly solvable models of Calogero type. We
adopt the previously introduced notion of solvability which consists of
relating the Hamiltonian to finite dimensional representation spaces of a Lie
algebra. We present explicitly the -case for which we construct the
potentials by means of suitable gauge transformations.Comment: 22 pages Late
A spin chain model with non-Hermitian interaction: the Ising quantum spin chain in an imaginary field
We investigate a lattice version of the Yang-Lee model which is characterized by a non-Hermitian quantum spin chain Hamiltonian. We propose a new way to implement PT-symmetry on the lattice, which serves to guarantee the reality of the spectrum in certain regions of values of the coupling constants. In that region of unbroken PT-symmetry we construct a Dyson map, a metric operator and find the Hermitian counterpart of the Hamiltonian for small values of the number of sites, both exactly and perturbatively. Besides the standard perturbation theory about the Hermitian part of the Hamiltonian, we also carry out an expansion in the second coupling constant of the model. Our constructions turns out to be unique with the sole assumption that the Dyson map is Hermitian. Finally we compute the magnetization of the chain in the z and x direction
Non-Hermitian Hamiltonians of Lie algebraic type
We analyse a class of non-Hermitian Hamiltonians, which can be expressed
bilinearly in terms of generators of a sl(2,R)-Lie algebra or their isomorphic
su(1,1)-counterparts. The Hamlitonians are prototypes for solvable models of
Lie algebraic type. Demanding a real spectrum and the existence of a well
defined metric, we systematically investigate the constraints these
requirements impose on the coupling constants of the model and the parameters
in the metric operator. We compute isospectral Hermitian counterparts for some
of the original non-Hermitian Hamiltonian. Alternatively we employ a
generalized Bogoliubov transformation, which allows to compute explicitly real
energy eigenvalue spectra for these type of Hamiltonians, together with their
eigenstates. We compare the two approaches.Comment: 27 page
Non-crystallographic reduction of generalized Calogero-Moser models
We apply a recently introduced reduction procedure based on the embedding of non-crystallographic Coxeter groups into crystallographic ones to Calogero–Moser systems. For rational potentials the familiar generalized Calogero Hamiltonian is recovered. For the Hamiltonians of trigonometric, hyperbolic and elliptic types, we obtain novel integrable dynamical systems with a second potential term which is rescaled by the golden ratio. We explicitly show for the simplest of these non-crystallographic models, how the corresponding classical equations of motion can be derived from a Lie algebraic Lax pair based on the larger, crystallographic Coxeter group
Thermodynamic Bethe Ansatz of the Homogeneous Sine-Gordon models
We apply the thermodynamic Bethe Ansatz to investigate the high energy
behaviour of a class of scattering matrices which have recently been proposed
to describe the Homogeneous sine-Gordon models related to simply laced Lie
algebras. A characteristic feature is that some elements of the suggested
S-matrices are not parity invariant and contain resonance shifts which allow
for the formation of unstable bound states. From the Lagrangian point of view
these models may be viewed as integrable perturbations of WZNW-coset models and
in our analysis we recover indeed in the deep ultraviolet regime the effective
central charge related to these cosets, supporting therefore the S-matrix
proposal. For the -model we present a detailed numerical analysis of
the scaling function which exhibits the well known staircase pattern for
theories involving resonance parameters, indicating the energy scales of stable
and unstable particles. We demonstrate that, as a consequence of the interplay
between the mass scale and the resonance parameter, the ultraviolet limit of
the HSG-model may be viewed alternatively as a massless
ultraviolet-infrared-flow between different conformal cosets. For we
recover as a subsystem the flow between the tricritical Ising and the Ising
model.Comment: 30 pages Latex, two figure
The ultraviolet Behaviour of Integrable Quantum Field Theories, Affine Toda Field Theory
We investigate the thermodynamic Bethe ansatz (TBA) equations for a system of
particles which dynamically interacts via the scattering matrix of affine Toda
field theory and whose statistical interaction is of a general Haldane type. Up
to the first leading order, we provide general approximated analytical
expressions for the solutions of these equations from which we derive general
formulae for the ultraviolet scaling functions for theories in which the
underlying Lie algebra is simply laced. For several explicit models we compare
the quality of the approximated analytical solutions against the numerical
solutions. We address the question of existence and uniqueness of the solutions
of the TBA-equations, derive precise error estimates and determine the rate of
convergence for the applied numerical procedure. A general expression for the
Fourier transformed kernels of the TBA-equations allows to derive the related
Y-systems and a reformulation of the equations into a universal form.Comment: 37 pp Latex, 5 figure