Low-Dimensional Spin Systems: Hidden Symmetries, Conformal Field Theories and Numerical Checks

We review here some general properties of antiferromagnetic Heisenberg spin chains, emphasizing and discussing the role of hidden symmetries in the classification of the various phases of the models. We present also some recent results that have been obtained with a combined use of Conformal Field Theory and of numerical Density Matrix Renormalization Group techniques.Comment: To be published in the proceedings of the XIII Conference on "Symmetries in Physics", held in Bregenz (Voralberg, Austria), 21-24/7/2003. Plain LaTeX2e, 4 EPS figure

Effective mapping of spin-1 chains onto integrable fermionic models. A study of string and Neel correlation functions

We derive the dominant contribution to the large-distance decay of correlation functions for a spin chain model that exhibits both Haldane and Neel phases in its ground state phase diagram. The analytic results are obtained by means of an approximate mapping between a spin-1 anisotropic Hamiltonian onto a fermionic model of noninteracting Bogolioubov quasiparticles related in turn to the XY spin-1/2 chain in a transverse field. This approach allows us to express the spin-1 string operators in terms of fermionic operators so that the dominant contribution to the string correlators at large distances can be computed using the technique of Toeplitz determinants. As expected, we find long-range string order both in the longitudinal and in the transverse channel in the Haldane phase, while in the Neel phase only the longitudinal order survives. In this way, the long-range string order can be explicitly related to the components of the magnetization of the XY model. Moreover, apart from the critical line, where the decay is algebraic, we find that in the gapped phases the decay is governed by an exponential tail multiplied by algebraic factors. As regards the usual two points correlation functions, we show that the longitudinal one behaves in a 'dual' fashion with respect to the transverse string correlator, namely both the asymptotic values and the decay laws exchange when the transition line is crossed. For the transverse spin-spin correlator, we find a finite characteristic length which is an unexpected feature at the critical point. We also comment briefly the entanglement features of the original system versus those of the effective model. The goodness of the approximation and the analytical predictions are checked versus density-matrix renormalization group calculations.Comment: 28 pages, plain LaTeX, 2 EPS figure

The resolution of whole Earth seismic tomographic models

We evaluate the resolution of whole Earth structure achieved by compressional wave traveltime data from the International Seismological Centre (ISC); the measure of resolution we employ, provided by the direct calculation of the model resolution matrix, is more rigorous than the traditional (and computationally cheaper) one obtained through synthetic/checkerboard tests. Our work completes the introductive study of Boschi (2003), where only mantle models derived with a very simple regularization scheme were considered. Here, we expand Boschi's database with measurements of compressional waves reflected by, or refracted through, the Earth's core-mantle boundary (CMB) and core. In analogy with the work of Soldati et al. (2003), we treat CMB topography and heterogeneous outer core structure as free parameters of our inversions; analysing model resolution matrices, we attempt to explain the known discrepancy between deep Earth structure mapped by seismic waves reflected and refracted by the cor

Using the Post—Widder formula to compute the Earth's viscoelastic Love numbers

The post-glacial or post-seismic relaxation of a Maxwell viscoelastic earth, 1-D or slightly laterally heterogeneous, can be calculated in a normal-mode approach, based on an application of the propagator technique. This semi-analytical approach, widely documented in the literature, allows to compute the response of an earth model whose rheological parameters vary quite strongly with depth, at least as accurately and efficiently as by 1-D numerical integration (Runge-Kutta). Its main drawback resides in the need to identify the roots of a secular polynomial, introduced after reformulating the problem in the Laplace domain, and required to transform the solution back to the time domain. Root finding becomes increasingly difficult, and ultimately unaffordable, as the complexity of rheological profiles grows: the secular polynomial gradually gets more ill behaved, and a larger number of more and more closely spaced roots is to be found. Here, we apply the propagator method to solve the Earth's viscoelastic momentum equation, like in the above-mentioned normal-mode framework, but bypass root finding, using, instead, the Post-Widder formula to transform the solution, found again in the Laplace domain, back to the time domain. We test our method against earlier normal-mode results, and prove its effectiveness in modelling the relaxation of earth models with extremely complex rheological profile

Gauge/string duality and scalar glueball mass ratios

It has been shown by Polchinski and Strassler that the scaling of high energy QCD scattering amplitudes can be obtained from string theory. They considered an AdS slice as an approximation for the dual space of a confining gauge theory. Here we use this approximation to estimate in a very simple way the ratios of scalar glueball masses imposing Dirichlet boundary conditions on the string dilaton field. These ratios are in good agreement with the results in the literature. We also find that they do not depend on the size of the slice.Comment: 5 pages, no figures. References updated. Version published in JHE

Stable particles in anisotropic spin-1 chains

Motivated by field-theoretic predictions we investigate the stable excitations that exist in two characteristic gapped phases of a spin-1 model with Ising-like and single-ion anisotropies. The sine-Gordon theory indicates a region close to the phase boundary where a stable breather exists besides the stable particles, that form the Haldane triplet at the Heisenberg isotropic point. The numerical data, obtained by means of the Density Matrix Renormalization Group, confirm this picture in the so-called large-D phase for which we give also a quantitative analysis of the bound states using standard perturbation theory. However, the situation turns out to be considerably more intricate in the Haldane phase where, to the best of our data, we do not observe stable breathers contrarily to what could be expected from the sine-Gordon model, but rather only the three modes predicted by a novel anisotropic extension of the Non-Linear Sigma Model studied here by means of a saddle-point approximation.Comment: 8 pages, 7 eps figures, svjour clas

Multiple resolution surface wave tomography: the Mediterranean basin

From a large set of fundamental-mode surface wave phase velocity observations, we map the transversely isotropic lateral heterogeneities in the upper-mantle shear velocity structure. We design a multiple resolution inversion procedure, which allows us to parametrize any selected region more finely than the rest of the globe. We choose, as a high-resolution region, the upper mantle underlying the Mediterranean basin. We formulate the inverse problem as in a previous paper by Boschi & Ekström, calculating regional JWKB (Jeffreys-Wentzel-Kramers-Brillouin) surface wave sensitivity kernels for each pixel of a 2°× 2° starting model, including the high-resolution global crustal map Crust 2.0. We find that the available surface wave data can resolve the most important geophysical features of the region of interest, providing a reliable image of intermediate spatial wavelengt