Universal Window for Two Dimensional Critical Exponents

Two dimensional condensed matter is realised in increasingly diverse forms that are accessible to experiment and of potential technological value. The properties of these systems are influenced by many length scales and reflect both generic physics and chemical detail. To unify their physical description is therefore a complex and important challenge. Here we investigate the distribution of experimentally estimated critical exponents, $\beta$, that characterize the evolution of the order parameter through the ordering transition. The distribution is found to be bimodal and bounded within a window $\sim 0.1 \le \beta \le 0.25$, facts that are only in partial agreement with the established theory of critical phenomena. In particular, the bounded nature of the distribution is impossible to reconcile with existing theory for one of the major universality classes of two dimensional behaviour - the XY model with four fold crystal field - which predicts a spectrum of non-universal exponents bounded only from below. Through a combination of numerical and renormalization group arguments we resolve the contradiction between theory and experiment and demonstrate how the "universal window" for critical exponents observed in experiment arises from a competition between marginal operators.Comment: 26 pages, 5 figures and 6 tables. Uses longtable packag

Hirnantia sagittifera

The brachiopod Hirnantia sagittifera (M’Coy) and trilobite morphs of the genus Mucronaspis from the topmost Ordovician Porkuni Stage of the central East Baltic are described and compared with those from the Hirnantian Stage of other regions. These important Hirnantian taxa occur in the Livonian Tongue of the Central Baltoscandian Facies Belt of the Baltic Basin, where the Porkuni Stage is represented by the non-graptolitic Kuldiga and Saldus formations. Hirnantia sagittifera appears in the lowermost part of the Porkuni Stage and is rather widely distributed in the basin in spite of its rare finds in each locality. Our study of trilobites of the genus Mucronaspis has enabled us to observe morphological changes in its exoskeleton in time and to identify a succession of five morphotypes (morphs). In some characteristics these morphs are similar to those of different alleged species of the genus Mucronaspis (M. olini, M. danai, M. ganabina, M. mucronata) but they cannot be definitely assigned to any of these species due to some variances. However, here for the first time a stratigraphically ordered collection is presented, which deserves attention in revising the taxonomy of highly variable Mucronaspis. The described brachiopods and trilobites occur mainly in the strata correlated with the Normalograptus extraordinarius graptolite Biozone. However, the uppermost finds of both taxa come from strata correlated with the N. persculptus Biozone