240 research outputs found
Dilogarithm Identities in Conformal Field Theory and Group Homology
Recently, Rogers' dilogarithm identities have attracted much attention in the
setting of conformal field theory as well as lattice model calculations. One of
the connecting threads is an identity of Richmond-Szekeres that appeared in the
computation of central charges in conformal field theory. We show that the
Richmond-Szekeres identity and its extension by Kirillov-Reshetikhin can be
interpreted as a lift of a generator of the third integral homology of a finite
cyclic subgroup sitting inside the projective special linear group of all real matrices viewed as a {\it discrete} group. This connection
allows us to clarify a few of the assertions and conjectures stated in the work
of Nahm-Recknagel-Terhoven concerning the role of algebraic -theory and
Thurston's program on hyperbolic 3-manifolds. Specifically, it is not related
to hyperbolic 3-manifolds as suggested but is more appropriately related to the
group manifold of the universal covering group of the projective special linear
group of all real matrices viewed as a topological group. This
also resolves the weaker version of the conjecture as formulated by Kirillov.
We end with the summary of a number of open conjectures on the mathematical
side.Comment: 20 pages, 2 figures not include
Vertex--IRF correspondence and factorized L-operators for an elliptic R-operator
As for an elliptic -operator which satisfies the Yang--Baxter equation,
the incoming and outgoing intertwining vectors are constructed, and the
vertex--IRF correspondence for the elliptic -operator is obtained. The
vertex--IRF correspondence implies that the Boltzmann weights of the IRF model
satisfy the star--triangle relation. By means of these intertwining vectors,
the factorized L-operators for the elliptic -operator are also constructed.
The vertex--IRF correspondence and the factorized L-operators for Belavin's
-matrix are reproduced from those of the elliptic -operator.Comment: 25 pages, amslatex, no figure
Diagonalization of the XXZ Hamiltonian by Vertex Operators
We diagonalize the anti-ferroelectric XXZ-Hamiltonian directly in the
thermodynamic limit, where the model becomes invariant under the action of
affine U_q( sl(2) ).
Our method is based on the representation theory of quantum affine algebras,
the related vertex operators and KZ equation, and thereby bypasses the usual
process of starting from a finite lattice, taking the thermodynamic limit and
filling the Dirac sea. From recent results on the algebraic structure of the
corner transfer matrix of the model, we obtain the vacuum vector of the
Hamiltonian. The rest of the eigenvectors are obtained by applying the vertex
operators, which act as particle creation operators in the space of
eigenvectors.
We check the agreement of our results with those obtained using the Bethe
Ansatz in a number of cases, and with others obtained in the scaling limit ---
the -invariant Thirring model.Comment: 65 page
Enhancement of pair correlation in a one-dimensional hybridization model
We propose an integrable model of one-dimensional (1D) interacting electrons
coupled with the local orbitals arrayed periodically in the chain. Since the
local orbitals are introduced in a way that double occupation is forbidden, the
model keeps the main feature of the periodic Anderson model with an interacting
host. For the attractive interaction, it is found that the local orbitals
enhance the effective mass of the Cooper-pair-like singlets and also the pair
correlation in the ground state. However, the persistent current is depressed
in this case. For the repulsive interaction case, the Hamiltonian is
non-Hermitian but allows Cooper pair solutions with small momenta, which are
induced by the hybridization between the extended state and the local orbitals.Comment: 11 page revtex, no figur
Exact solution of new integrable nineteen-vertex models and quantum spin-1 chains
New exactly solvable nineteen vertex models and related quantum spin-1 chains
are solved. Partition functions, excitation energies, correlation lengths, and
critical exponents are calculated. It is argued that one of the non-critical
Hamiltonians is a realization of an integrable Haldane system. The finite-size
spectra of the critical Hamiltonians deviate in their structure from standard
predictions by conformal invariance.Comment: 16 pages, to appear in Z. Phys. B, preprint Cologne-94-474
Shock waves in two-dimensional granular flow: effects of rough walls and polydispersity
We have studied the two-dimensional flow of balls in a small angle funnel,
when either the side walls are rough or the balls are polydisperse. As in
earlier work on monodisperse flows in smooth funnels, we observe the formation
of kinematic shock waves/density waves. We find that for rough walls the flows
are more disordered than for smooth walls and that shock waves generally
propagate more slowly. For rough wall funnel flow, we show that the shock
velocity and frequency obey simple scaling laws. These scaling laws are
consistent with those found for smooth wall flow, but here they are cleaner
since there are fewer packing-site effects and we study a wider range of
parameters. For pipe flow (parallel side walls), rough walls support many shock
waves, while smooth walls exhibit fewer or no shock waves. For funnel flows of
balls with varying sizes, we find that flows with weak polydispersity behave
qualitatively similar to monodisperse flows. For strong polydispersity, scaling
breaks down and the shock waves consist of extended areas where the funnel is
blocked completely.Comment: 11 pages, 15 figures; accepted for PR
Adsorption of Reactive Particles on a Random Catalytic Chain: An Exact Solution
We study equilibrium properties of a catalytically-activated annihilation reaction taking place on a one-dimensional chain of length () in which some segments (placed at random, with mean concentration
) possess special, catalytic properties. Annihilation reaction takes place,
as soon as any two particles land onto two vacant sites at the extremities
of the catalytic segment, or when any particle lands onto a vacant site on
a catalytic segment while the site at the other extremity of this segment is
already occupied by another particle. Non-catalytic segments are inert with
respect to reaction and here two adsorbed particles harmlessly coexist. For
both "annealed" and "quenched" disorder in placement of the catalytic segments,
we calculate exactly the disorder-average pressure per site. Explicit
asymptotic formulae for the particle mean density and the compressibility are
also presented.Comment: AMSTeX, 27 pages + 4 figure
An alternative order parameter for the 4-state Potts model
We have investigated the dynamic critical behavior of the two-dimensional
4-state Potts model using an alternative order parameter first used by
Vanderzande [J. Phys. A: Math. Gen. \textbf{20}, L549 (1987)] in the study of
the Z(5) model. We have estimated the global persistence exponent by
following the time evolution of the probability that the considered
order parameter does not change its sign up to time . We have also obtained
the critical exponents , , , and using this alternative
definition of the order parameter and our results are in complete agreement
with available values found in literature.Comment: 6 pages, 6 figure
Quantum and Classical Integrable Systems
The key concept discussed in these lectures is the relation between the
Hamiltonians of a quantum integrable system and the Casimir elements in the
underlying hidden symmetry algebra. (In typical applications the latter is
either the universal enveloping algebra of an affine Lie algebra, or its
q-deformation.) A similar relation also holds in the classical case. We discuss
different guises of this very important relation and its implication for the
description of the spectrum and the eigenfunctions of the quantum system.
Parallels between the classical and the quantum cases are thoroughly discussed.Comment: 59 pages, LaTeX2.09 with AMS symbols. Lectures at the CIMPA Winter
School on Nonlinear Systems, Pondicherry, January 199
Decoupling of the S=1/2 antiferromagnetic zig-zag ladder with anisotropy
The spin-1/2 antiferromagnetic zig-zag ladder is studied by exact
diagonalization of small systems in the regime of weak inter-chain coupling. A
gapless phase with quasi long-range spiral correlations has been predicted to
occur in this regime if easy-plane (XY) anisotropy is present. We find in
general that the finite zig-zag ladder shows three phases: a gapless collinear
phase, a dimer phase and a spiral phase. We study the level crossings of the
spectrum,the dimer correlation function, the structure factor and the spin
stiffness within these phases, as well as at the transition points. As the
inter-chain coupling decreases we observe a transition in the anisotropic XY
case from a phase with a gap to a gapless phase that is best described by two
decoupled antiferromagnetic chains. The isotropic and the anisotropic XY cases
are found to be qualitatively the same, however, in the regime of weak
inter-chain coupling for the small systems studied here. We attribute this to a
finite-size effect in the isotropic zig-zag case that results from
exponentially diverging antiferromagnetic correlations in the weak-coupling
limit.Comment: to appear in Physical Review
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