1,594 research outputs found

### A Q-operator for the quantum transfer matrix

Baxter's Q-operator for the quantum transfer matrix of the XXZ spin-chain is
constructed employing the representation theory of quantum groups. The spectrum
of this Q-operator is discussed and novel functional relations which describe
the finite temperature regime of the XXZ spin-chain are derived. For
non-vanishing magnetic field the previously known Bethe ansatz equations can be
replaced by a system of quadratic equations which is an important advantage for
numerical studies. For vanishing magnetic field and rational coupling values it
is argued that the quantum transfer matrix exhibits a loop algebra symmetry
closely related to the one of the classical six-vertex transfer matrix at roots
of unity.Comment: 20 pages, v2: some minor style improvement

### Turning the Quantum Group Invariant XXZ Spin-Chain Hermitian: A Conjecture on the Invariant Product

This is a continuation of a previous joint work with Robert Weston on the
quantum group invariant XXZ spin-chain (math-ph/0703085). The previous results
on quasi-Hermiticity of this integrable model are briefly reviewed and then
connected with a new construction of an inner product with respect to which the
Hamiltonian and the representation of the Temperley-Lieb algebra become
Hermitian. The approach is purely algebraic, one starts with the definition of
a positive functional over the Temperley-Lieb algebra whose values can be
computed graphically. Employing the Gel'fand-Naimark-Segal (GNS) construction
for C*-algebras a self-adjoint representation of the Temperley-Lieb algebra is
constructed when the deformation parameter q lies in a special section of the
unit circle. The main conjecture of the paper is the unitary equivalence of
this GNS representation with the representation obtained in the previous paper
employing the ideas of PT-symmetry and quasi-Hermiticity. An explicit example
is presented.Comment: 12 page

### The twisted XXZ chain at roots of unity revisited

The symmetries of the twisted XXZ spin-chain (alias the twisted six-vertex
model) at roots of unity are investigated. It is shown that when the twist
parameter is chosen to depend on the total spin an infinite-dimensional
non-abelian symmetry algebra can be explicitly constructed for all spin
sectors. This symmetry algebra is identified to be the upper or lower Borel
subalgebra of the sl_2 loop algebra. The proof uses only the intertwining
property of the six-vertex monodromy matrix and the familiar relations of the
six-vertex Yang-Baxter algebra.Comment: 10 pages, 2 figures. One footnote and some comments in the
conclusions adde

### Workshop island 3: algebraic aspects of integrability. Introduction to an additional volume of selected papers arising from the conference on algebraic aspects of integrable systems, Island 3, Islay 2007

As did the very first ISLAND workshop, ISLAND 3 took place on the Hebridean island of Islay, providing a beautiful and serene surrounding for the meeting which ran for over four days. Building on the success of the previous meetings, ISLAND 3 saw the largest number (so far) of participants coming from countries all over the world. A complete list can be found below

### Two-particle scattering theory for anyons

We consider potential scattering theory of a nonrelativistic quantum
mechanical 2-particle system in R^2 with anyon statistics. Sufficient
conditions are given which guarantee the existence of wave operators and the
unitarity of the S-matrix. As examples the rotationally invariant potential
well and the delta-function potential are discussed in detail. In case of a
general rotationally invariant potential the angular momentum decomposition
leads to a theory of Jost functions. The anyon statistics parameter gives rise
to an interpolation for angular momenta analogous to the Regge trajectories for
complex angular momenta. Levinson's theorem is adapted to the present context.
In particular we find that in case of a zero energy resonance the statistics
parameter can be determined from the scattering phase.Comment: 42 pages of RevTex and 5 figures (included

### 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

### Auxiliary matrices on both sides of the equator

The spectra of previously constructed auxiliary matrices for the six-vertex
model at roots of unity are investigated for spin-chains of even and odd
length. The two cases show remarkable differences. In particular, it is shown
that for even roots of unity and an odd number of sites the eigenvalues contain
two linear independent solutions to Baxter's TQ-equation corresponding to the
Bethe ansatz equations above and below the equator. In contrast, one finds for
even spin-chains only one linear independent solution and complete strings. The
other main result is the proof of a previous conjecture on the degeneracies of
the six-vertex model at roots of unity. The proof rests on the derivation of a
functional equation for the auxiliary matrices which is closely related to a
functional equation for the eight-vertex model conjectured by Fabricius and
McCoy.Comment: 22 pages; 2nd version: one paragraph added in the conclusion and some
typos correcte

### PT Symmetry of the non-Hermitian XX Spin-Chain: Non-local Bulk Interaction from Complex Boundary Fields

The XX spin-chain with non-Hermitian diagonal boundary conditions is shown to
be quasi-Hermitian for special values of the boundary parameters. This is
proved by explicit construction of a new inner product employing a
"quasi-fermion" algebra in momentum space where creation and annihilation
operators are not related via Hermitian conjugation. For a special example,
when the boundary fields lie on the imaginary axis, we show the spectral
equivalence of the quasi-Hermitian XX spin-chain with a non-local fermion
model, where long range hopping of the particles occurs as the non-Hermitian
boundary fields increase in strength. The corresponding Hamiltonian
interpolates between the open XX and the quantum group invariant XXZ model at
the free fermion point. For an even number of sites the former is known to be
related to a CFT with central charge c=1, while the latter has been connected
to a logarithmic CFT with central charge c=-2. We discuss the underlying
algebraic structures and show that for an odd number of sites the superalgebra
symmetry U(gl(1|1)) can be extended from the unit circle along the imaginary
axis. We relate the vanishing of one of its central elements to the appearance
of Jordan blocks in the Hamiltonian.Comment: 37 pages, 5 figure

### A Q-operator for the twisted XXX model

Taking the isotropic limit in a recent representation theoretic construction
of Baxter's Q-operators for the XXZ model with quasi-periodic boundary
conditions we obtain new results for the XXX model. We show that quasi-periodic
boundary conditions are needed to ensure convergence of the Q-operator
construction and derive a quantum Wronskian relation which implies two
different sets of Bethe ansatz equations, one above the other below the
"equator" of total spin zero. We discuss the limit to periodic boundary
conditions at the end and explain how this construction might be useful in the
context of correlation functions on the infinite lattice. We also identify a
special subclass of solutions to the quantum Wronskian for chains up to a
length of 10 sites and possibly higher.Comment: 19 page

- …