2,003 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
PT Symmetry on the Lattice: The Quantum Group Invariant XXZ Spin-Chain
We investigate the PT-symmetry of the quantum group invariant XXZ chain. We
show that the PT-operator commutes with the quantum group action and also
discuss the transformation properties of the Bethe wavefunction. We exploit the
fact that the Hamiltonian is an element of the Temperley-Lieb algebra in order
to give an explicit and exact construction of an operator that ensures
quasi-Hermiticity of the model. This construction relys on earlier ideas
related to quantum group reduction. We then employ this result in connection
with the quantum analogue of Schur-Weyl duality to introduce a dual pair of
C-operators, both of which have closed algebraic expressions. These are novel,
exact results connecting the research areas of integrable lattice systems and
non-Hermitian Hamiltonians.Comment: 32 pages with figures, v2: some minor changes and added references,
version published in JP
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
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
Higher harmonics and ac transport from time dependent density functional theory
We report on dynamical quantum transport simulations for realistic molecular
devices based on an approximate formulation of time-dependent Density
Functional Theory with open boundary conditions. The method allows for the
computation of various properties of junctions that are driven by alternating
bias voltages. Besides the ac conductance for hexene connected to gold leads
via thiol anchoring groups, we also investigate higher harmonics in the current
for a benzenedithiol device. Comparison to a classical quasi-static model
reveals that quantum effects may become important already for small ac bias and
that the full dynamical simulations exhibit a much lower number of higher
harmonics. Current rectification is also briefly discussed.Comment: submitted to J. Comp. Elec. (special issue
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
Quantum cohomology via vicious and osculating walkers
We relate the counting of rational curves intersecting Schubert varieties of the Grassmannian to the counting of certain non-intersecting lattice paths on the cylinder, so-called vicious and osculating walkers. These lattice paths form exactly solvable statistical mechanics models and are obtained from solutions to the Yang–Baxter equation. The eigenvectors of the transfer matrices of these models yield the idempotents of the Verlinde algebra of the gauged u^(n)k -WZNW model. The latter is known to be closely related to the small quantum cohomology ring of the Grassmannian. We establish further that the partition functions of the vicious and osculating walker model are given in terms of Postnikov’s toric Schur functions and can be interpreted as generating functions for Gromov–Witten invariants. We reveal an underlying quantum group structure in terms of Yang–Baxter algebras and use it to give a generating formula for toric Schur functions in terms of divided difference operators which appear in known representations of the nil-Hecke algebra
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
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
- …