1,348 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
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
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
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
XXZ Bethe states as highest weight vectors of the loop algebra at roots of unity
We show that every regular Bethe ansatz eigenvector of the XXZ spin chain at
roots of unity is a highest weight vector of the loop algebra, for some
restricted sectors with respect to eigenvalues of the total spin operator
, and evaluate explicitly the highest weight in terms of the Bethe roots.
We also discuss whether a given regular Bethe state in the sectors generates an
irreducible representation or not. In fact, we present such a regular Bethe
state in the inhomogeneous case that generates a reducible Weyl module. Here,
we call a solution of the Bethe ansatz equations which is given by a set of
distinct and finite rapidities {\it regular Bethe roots}. We call a nonzero
Bethe ansatz eigenvector with regular Bethe roots a {\it regular Bethe state}.Comment: 40pages; revised versio
Noncommutative Schur polynomials and the crystal limit of the U_q sl(2)-vertex model
Starting from the Verma module of U_q sl(2) we consider the evaluation module
for affine U_q sl(2) and discuss its crystal limit (q=0). There exists an
associated integrable statistical mechanics model on a square lattice defined
in terms of vertex configurations. Its transfer matrix is the generating
function for noncommutative complete symmetric polynomials in the generators of
the affine plactic algebra, an extension of the finite plactic algebra first
discussed by Lascoux and Sch\"{u}tzenberger. The corresponding noncommutative
elementary symmetric polynomials were recently shown to be generated by the
transfer matrix of the so-called phase model discussed by Bogoliubov, Izergin
and Kitanine. Here we establish that both generating functions satisfy Baxter's
TQ-equation in the crystal limit by tying them to special U_q sl(2) solutions
of the Yang-Baxter equation. The TQ-equation amounts to the well-known
Jacobi-Trudy formula leading naturally to the definition of noncommutative
Schur polynomials. The latter can be employed to define a ring which has
applications in conformal field theory and enumerative geometry: it is
isomorphic to the fusion ring of the sl(n)_k -WZNW model whose structure
constants are the dimensions of spaces of generalized theta-functions over the
Riemann sphere with three punctures.Comment: 24 pages, 6 figures; v2: several typos fixe
Gross efficiency and cycling performance: a review.
Efficiency, the ratio of work generated to the total metabolic energy cost, has been suggested to be a key determinant of endurance cycling performance. The purpose of this brief review is to evaluate the influence of gross efficiency on cycling power output and to consider whether or not gross efficiency can be modified. In a re-analysis of data from five separate studies, variation in gross efficiency explained ~30% of the variation in power output during cycling time-trials. Whilst other variables, notably VO2max and lactate threshold, have been shown to explain more of the variance in cycling power output, these results confirm the important influence of gross efficiency. Case study, cross-sectional, longitudinal, and intervention research designs have all been used to demonstrate that exercise training can enhance gross efficiency. Whilst improvements have been seen with a wide range of training types (endurance, strength, altitude), it would appear that high intensity training is the most potent stimulus for changes in gross efficiency. In addition to physiological adaptations, gross efficiency might also be improved through biomechanical adaptations. However, âintuitiveâ technique and equipment adjustments may not always be effective. For example, whilst âpedalling in circlesâ allows pedalling to become mechanically more effective, this technique does not result in short term improvements in gross efficiency
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