Fractional excitations in the Luttinger liquid

We reconsider the spectrum of the Luttinger liquid (LL) usually understood in terms of phonons (density fluctuations), and within the context of bosonization we give an alternative representation in terms of fractional states. This allows to make contact with Bethe Ansatz which predicts similar fractional states. As an example we study the spinon operator in the absence of spin rotational invariance and derive it from first principles: we find that it is not a semion in general; a trial Jastrow wavefunction is also given for that spinon state. Our construction of the new spectroscopy based on fractional states leads to several new physical insights: in the low-energy limit, we find that the $S_{z}=0$ continuum of gapless spin chains is due to pairs of fractional quasiparticle-quasihole states which are the 1D counterpart of the Laughlin FQHE quasiparticles. The holon operator for the Luttinger liquid with spin is also derived. In the presence of a magnetic field, spin-charge separation is not realized any longer in a LL: the holon and the spinon are then replaced by new fractional states which we are able to describe.Comment: Revised version to appear in Physical Review B. 27 pages, 5 figures. Expands cond-mat/9905020 (Eur.Phys.Journ.B 9, 573 (1999)

Preduals of semigroup algebras

For a locally compact group G, the measure convolution algebra M(G) carries a natural coproduct. In previous work, we showed that the canonical predual C 0(G) of M(G) is the unique predual which makes both the product and the coproduct on M(G) weak*-continuous. Given a discrete semigroup S, the convolution algebra ℓ 1(S) also carries a coproduct. In this paper we examine preduals for ℓ 1(S) making both the product and the coproduct weak*-continuous. Under certain conditions on S, we show that ℓ 1(S) has a unique such predual. Such S include the free semigroup on finitely many generators. In general, however, this need not be the case even for quite simple semigroups and we construct uncountably many such preduals on ℓ 1(S) when S is either ℤ+×ℤ or (ℕ,⋅)

Mixing and non-mixing local minima of the entropy contrast for blind source separation

In this paper, both non-mixing and mixing local minima of the entropy are analyzed from the viewpoint of blind source separation (BSS); they correspond respectively to acceptable and spurious solutions of the BSS problem. The contribution of this work is twofold. First, a Taylor development is used to show that the \textit{exact} output entropy cost function has a non-mixing minimum when this output is proportional to \textit{any} of the non-Gaussian sources, and not only when the output is proportional to the lowest entropic source. Second, in order to prove that mixing entropy minima exist when the source densities are strongly multimodal, an entropy approximator is proposed. The latter has the major advantage that an error bound can be provided. Even if this approximator (and the associated bound) is used here in the BSS context, it can be applied for estimating the entropy of any random variable with multimodal density.Comment: 11 pages, 6 figures, To appear in IEEE Transactions on Information Theor

Implications of factorization for the determination of hadronic form factors in D_s^+ \ra \phi transition

Using factorization we determine the allowed domains of the ratios of form factors, $x = A_2(0)/A_1(0)$ and $y = V(0)/A_1(0)$, from the experimentally measured ratio R_h \equiv \Gamma(D_s^+ \ra \phi \rho^+)/\Gamma(D_s^+ \ra \phi \pi^+) assuming three different scenarios for the $q^2$-dependence of the form factors. We find that the allowed domains overlap with those obtained by using the experimentally measured ratio R_{s\ell} = \Gamma(D^+_s \ra \phi \ell^+ \nu_{\ell})/\Gamma(D^+_s \ra \phi \pi^+) provided that the phenomenological parameter $a_1$ is $1.23$. Such a comparison presents a genuine test of factorization. We calculate the longitudinal polarization fraction, \Gamma_L/\Gamma \equiv \Gamma(D_s^+ \ra \phi_L \rho^+_L)/\Gamma(D_s^+ \ra \phi \rho^+), in the three scenarios for the $q^2$-dependence of the form factors and emphasize the importance of measuring $\Gamma_L/\Gamma$. Finally we discuss the $q^2$-distribution of the semileptonic decay and find that it is rather insensitive to the scenarios for the $q^2$-dependence of the form factors, and unless very accurate data can be obtained it is unlikely to discriminate between the different scenarios. Useful information on the value of $x$ might be obtained by the magnitude of the $q^2$-distribution near $q^2 = 0$. However the most precise information on $x$ and $y$ would come from the knowledge of the longitudinal and left-right transverse polarizations of the final vector mesons in hadronic and/or semileptonic decays.Comment: Latex 10 pages( 4 figures), PAR/LPTHE/94-3

Character Levels and Character Bounds. II

This paper is a continuation of [GLT], which develops a level theory and establishes strong character bounds for finite simple groups of linear and unitary type in the case that the centralizer of the element has small order compared to $|G|$ in a logarithmic sense. We strengthen the results of [GLT] and extend them to all groups of classical type