Quarks, Gluons and Frustrated Antiferromagnets

The Contractor Renormalization Group method (CORE) is used to establish the equivalence of various Hamiltonian free fermion theories and a class of generalized frustrated antiferromagnets. In particular, after a detailed discussion of a simple example, it is argued that a generalized frustrated SU(3) antiferromagnet whose single-site states have the quantum numbers of mesons and baryons is equivalent to a theory of free massless quarks. Furthermore, it is argued that for slight modification of the couplings which define the frustrated antiferromagnet Hamiltonian, the theory becomes a theory of quarks interacting with color gauge-fields.Comment: 21 pages, Late

Validation des écoulements par l'analyse discriminante

L'étude des écoulements souterrains et du drainage est généralement réalisée par l'établissement des cartes et des profils piézométriques (en effectuant des campagnes de relevés piézométriques dans les forages et les piézomètres captant la nappe). On essaye ici de confirmer les résultats obtenus à partir de l'interprétation de la piézométrie par le traitement statistique multidimensionnel des données hydrogéochimiques des sources, et plus précisément l'analyse factorielle discriminante, et de caractériser l'importance et la dominance de l'un des paramètres responsables de la minéralisation des eaux karstiques : les phénomènes climatiques (saisons), et les caractéristiques du système (lithologie, structure du drainage, gradients). L'application à la nappe des calcaires du Bathonien-Bajocien du plateau de Besançon (Jura externe, France) a permis de démontrer que la variabilité chimique des eaux aux exutoires karstiques est conditionnée plus par les caractéristiques hydrogéologiques de l'aquifère (unités hydrogéologiques) que par les évènements hydroclimatiques (effet "campagne" ou période d'étude), et de resituer les forages par rapport aux sources testées ; l'association d'un groupe de forages avec certaines sources est à relier avec des circulations rapides en zone libre, ce cas est celui du regroupement de la source du Trébignon et les puits de Chailluz (situés dans la zone libre de l'aquifère) qui présentent le même comportement hydrochimiques. Alors que d'autres forages sont groupés avec le fonctionnement de la nappe en zone captive, ce cas correspond à celui de l'affectation des puits de Thise (implantés dans la partie captive de l'aquifère) à la source de Roche- lez-Beaupré. Ces résultats corroborent ceux obtenus par l'étude piézométrique.The plateau of Besançon (Eastern France) spreads between the large valley of the Ognon to the north and the enclosed valley of the Doubs to the south. Stratigraphically, the principal formations belong to the middle Jurassic era and the region is marked by a brittle tectonic characterized by NE-SW faults. Bajocian and Bathonian limestones constitute a karstic aquifer (200 m in thickness), which is free for much of its extent but becomes confined under Oxfordian marls towards the Doubs valley. The hydrogeological and structural boundaries of the drainage basin are complex, as are its hydrodynamic functioning and the hydrochemical responses observed at the outlets. The reservoir is drained principally by the Mouillère, Trébignon and Roche-lez-Beaupré springs, these outlets being situated mainly in the Doubs valley. The region is largely covered by dense vegetation, essentially forests, which constitute a groundwater recharge zone, and by the conglomeration of Besançon. One of the most karstified regions in France, the site includes two groups of wells that supply the town of Besançon with drinking water: the first group is situated on the unconfined zone of the aquifer (Chailluz), and the second is located on the confined part of the karstic aquifer (Thise). The wells are equipped with piezometers to monitor water levels continuously.The rainfall distribution in this karstic region is homogeneous. During the 1991- 1992 hydrological year, the observed precipitation was 1144 mm at Besançon, with an evapotranspiration constituting about 50 % of the measured precipitation. On the Besançon plateau, the water mineralization of the different karstic springs is related to the nature of the underlying geological media (lithology, importance of karstification, etc.), to the water residence time and to seasonal variables (characteristics of the rainfall episode, hydroclimatic events, etc.). The water chemistry is relatively stable during dry intervals, but exhibits high fluctuations during flood periods. Outlets were sampled every 7 to 10 days from January 1991 to August 1992, in order to follow variations in water physico-chemistry. The sampling frequencies covered diverse water-level situations in the aquifer (low-water period, high-water period, drying-up period, etc.), and allowed the application of some multidimensional statistical tools.The piezometric profile and lithologic correlations between the boreholes of Chailluz and Thise show that the highest hydraulic gradient is calculated between the Chailluz boreholes and the Trébignon spring. The hydraulic gradient is low in the confined part of the aquifer (between the Thise boreholes and the spring of Roche-lez- Beaupré). Two zones can be identified: 1. a free zone distinguished by flow and drainage relationships between the Chailluz boreholes and the Trébignon spring; 2. a confined zone linking the Thise boreholes and the Roche-lez-Beaupré spring, which corresponds to a common water circulation. To confirm the results obtained from the piezometric study, we have used multidimensional statistical techniques to process the hydrochemical data from the karstic springs.Factorial discriminant analysis is a multivariate method that allows the testing of hypotheses. It belongs to the field of decisional statistics, and is based on the comparison of multidimensional intragroup variances to intergroup variance. These methods can show the presence of certain relationships between a qualitative explained criterion and a group of quantitative explanatory characters, and they allow one to describe these latter relationships. Cross discriminant analysis can be used to evaluate the pertinence of a given hypothesis and to identify the dominant factor(s). The introduction of a qualitative variable within a population allows the division of this population into different groups, with each individual assigned to one group. Discrimination of the groups consists of maximizing the variance between their centres of gravity (a principal components analysis is performed on the matrix of the centres of gravity of the defined groups a priori); one can then clarify the properties that distinguish the different groups. If the individual is close to the centre of gravity of its group, it is "well classified". In the case where the distance to the centre of gravity of its group is superior to that to the centre of gravity of another group, the individual is "poorly classified" and it will be reassigned to this other group.Another advantage of these methods is the use of supplementary individuals, which are not involved in the definition of the groups. Supplementary individuals are reassigned to the groups to which they are statistically linked. In karstic hydrogeology, cross factorial discriminant analysis of a population enables one to validate the classification criteria based on the analysis of flows and concentrations. These units of classification deal with the hydrokinematic phases (seasonal variable) or with the intrinsic systems (geographic variable). Therefore, in our case we have crossed two qualitative factors, "system" (Trébignon, Roche-lez-Beaupré) and "season" (spring, summer, autumn, winter) by undertaking two different analyses on the same series of data.The application of these techniques to the limestone aquifer of the Bathonian- Bajocian shows that intrinsic factors predominate in determining the mineralization of the outlet water. On the other hand, this method enables one to associate borehole chemistry to tested springs: the association of a group of boreholes with particular springs is explained by the rapid circulation and drainage in the unconfined zone of the karstic media. This is the case of the association of the Chailluz boreholes (situated in the unconfined zone of the aquifer) with the Trébignon spring, which behave hydrochemically in the same manner. Other boreholes are grouped with the confined aquifer; this case corresponds to the assignment of the Thise boreholes (located in the confined zone of the aquifer) to the Roche-lez-Beaupré spring. These results corroborate those obtained from the piezometric study. Factorial discriminant analysis allowed the verification of the appropriation of the predefined groups to the discriminant factors: hydrogeologic character (lithology, structure of the drainage, gradients, etc.) and hydroclimatic phenomena (season, hydrometeorological episodes, etc.). The choice of the property groups based on the geographic and geologic criteria proved pertinent

Gaussian field theories, random Cantor sets and multifractality

The computation of multifractal scaling properties associated with a critical field theory involves non-local operators and remains an open problem using conventional techniques of field theory. We propose a new description of Gaussian field theories in terms of random Cantor sets and show how universal multifractal scaling exponents can be calculated. We use this approach to characterize the multifractal critical wave function of Dirac fermions interacting with a random vector potential in two spatial dimensions. We show that the multifractal scaling exponents are self-averaging.Comment: Extensive modifications of previous version; exact results replace numerical calculation

Apport de la géologie, de l’hydrogéologie et des isotopes de l’environnement à la connaissance des «nappes en creux» du Grand Yaéré (Nord Cameroun)

La carte piézométrique de la nappe du Logone-Chari-Tchad met en évidence des anomalies piézométriques interprétées comme des « nappes en creux ». Les informations de l’hydrogéologie et des isotopes de l’environnement conduisent à remettre en question les grandes profondeurs des niveaux statiques observées par certains auteurs dans ces dépressions piézométriques. Les données hydrogéologiques démontrent que dans la zone déprimée de la surface piézométrique, l’aquifère est de type bicouche. Par ailleurs, la distribution des teneurs en isotopes stables (oxygène-18 et deutérium) et en tritium confirme le cloisonnement des aquifères :La relation δ 2H vs. δ 18O montre que les effets d’enrichissement par évaporation lors de la recharge des nappes ne sont très marqués que dans les eaux des nappes superficielles dont les niveaux statiques ne dépassent pas 20 m de profondeur. Les dépressions fermées dont les points les plus bas atteignent 60 m sous la surface du sol s’interprètent difficilement dans l’hypothèse d’une reprise évaporatoire.Il résulte de cette étude que l’absence de dépendance nette entre les niveaux piézométriques superficiels et les niveaux profonds place le problème des anomalies piézométriques du Grand Yaéré dans un contexte totalement différent de celui des anciennes interprétations qui s’appuyaient sur l’hypothèse d’une nappe libre généralisée monocouche. À l’avenir, la construction de la carte piézométrique de la nappe du Logone-Chari-Tchad devra tenir compte de la structure des deux nappes superposées.Piezometric depressions, common in sub-Saharan Africa, are major hydrogeological anomalies manifested by closed curves, pronounced hollows and dips attaining several tens of meters below the regional water table level. The Logone-Chari-Chad piezometric map reveals piezometric anomalies that have been interpreted as depressed aquifers. The depth of the water table is 60 m in the Tagawa-Am Talia axis, 40 m between Louba-Louba and Andirni and 30 m around Yagoua. Factors linked to evaporation are generally thought to be responsible for these depressed zones.The objective of this study (based on the saturated zone) was: 1) to place the Logone-Chari-Chad piezometric anomalies in their hydrogeological settings, and 2) to evaluate the use of environmental isotopes to explain their formation processes. To achieve our goal, 27 water supply points (8 boreholes and 19 wells) were selected from the borders and centre of the Logone-Chari-Chad depression. Samples were collected between 1989 and 1991. Measurements performed in the field involved static water levels, whereas the laboratory analyses 18O, 2H and 3H were performed at the International Atomic Energy Agency (IAEA) laboratory in Vienna, within the framework of the project RAF/8/012 funded by IAEA.The new geological and hydrogeological data demonstrate that in the depressed zone of the piezometric surface, the aquifer has two layers. In contrast, the Logone-Chari-Chad piezometric map was previously drawn considering the aquifer as a single-layer. From a hydrochemical point of view, the groundwater in the Logone-Chari-Chad aquifer is stratified: calcium bicarbonate type water was found at the surface (shallow groundwater), whereas sodium carbonate type water was found at depth (deeper groundwater).Seasonal piezometric fluctuations of 1.5 to 3 m have been observed in the shallow groundwater. In the deeper groundwater, they range from 0.20 to 0.30 m. The difference in the values of water table fluctuation leads not only to variations in the mode of groundwater circulation, but also to variations in the hydrodynamic properties of aquifers, such as transmissivity.The distribution in stable isotope contents (18O, 2H and 3H) confirmed the compartmentalization of aquifers. The correlation between 3H and 18O showed that there are two water types, with different recharge modes and episodes. On the border of the depression, shallow groundwater pinches out on the semi-permeable substratum, resulting in a tritium content greater than 4 UT. In the depression axis, there is deeper groundwater with a tritium content below 4 UT.The relationship between 2H and 18O shows that the enrichment effects of evaporation at the time of recharge are very pronounced only in the shallow groundwater, where the static level does not exceed 20 m below the soil surface. The closed piezometric depressions, whose deepest point attains 60 m below the soil surface, cannot be explained by the presence of intense evaporation. The variation in tritium content with respect to the static level shows that in the depressed zone, the first 20 m are characterized by a tritium content greater than 4 UT, whereas at depths of 30 m or more, tritium contents are lower than 4 UT.The absence of dependence between shallow and deep piezometric levels invalidates the interpretation of great water depths proposed in previous studies of the piezometric depression of the Logone-Chari-Chad water table. Thus, the hypothesis that the Logone-Chari-Chad is a single-layer system should be abandoned. The future construction of the piezometric map of the Logone-Chari-Chad water table should take into account the structure and lithology of the two superimposed layers

Excitation spectrum of the homogeneous spin liquid

We discuss the excitation spectrum of a disordered, isotropic and translationally invariant spin state in the 2D Heisenberg antiferromagnet. The starting point is the nearest-neighbor RVB state which plays the role of the vacuum of the theory, in a similar sense as the Neel state is the vacuum for antiferromagnetic spin wave theory. We discuss the elementary excitations of this state and show that these are not Fermionic spin-1/2 `spinons' but spin-1 excited dimers which must be modeled by bond Bosons. We derive an effective Hamiltonian describing the excited dimers which is formally analogous to spin wave theory. Condensation of the bond-Bosons at zero temperature into the state with momentum (pi,pi) is shown to be equivalent to antiferromagnetic ordering. The latter is a key ingredient for a microscopic interpretation of Zhang's SO(5) theory of cuprate superconductivityComment: RevTex-file, 16 PRB pages with 13 embedded eps figures. Hardcopies of figures (or the entire manuscript) can be obtained by e-mail request to: [email protected]

Criticality in the 2+1-dimensional compact Higgs model and fractionalized insulators

We use a novel method of computing the third moment M_3 of the action of the 2+1-dimensional compact Higgs model in the adjoint representation with q=2 to extract correlation length and specific heat exponents nu and alpha, without invoking hyperscaling. Finite-size scaling analysis of M_3 yields the ratio (1+alpha)/nu and 1/nu separately. We find that alpha and nu vary along the critical line of the theory, which however exhibits a remarkable resilience of Z_2 criticality. We propose this novel universality class to be that of the quantum phase transition from a Mott-Hubbard insulator to a charge-fractionalized insulator in two spatial dimensions.Comment: 4 pages, 3 figures, submitted to Phys. Rev. Let

Spectral Statistics in Chiral-Orthogonal Disordered Systems

We describe the singularities in the averaged density of states and the corresponding statistics of the energy levels in two- (2D) and three-dimensional (3D) chiral symmetric and time-reversal invariant disordered systems, realized in bipartite lattices with real off-diagonal disorder. For off-diagonal disorder of zero mean we obtain a singular density of states in 2D which becomes much less pronounced in 3D, while the level-statistics can be described by semi-Poisson distribution with mostly critical fractal states in 2D and Wigner surmise with mostly delocalized states in 3D. For logarithmic off-diagonal disorder of large strength we find indistinguishable behavior from ordinary disorder with strong localization in any dimension but in addition one-dimensional $1/|E|$ Dyson-like asymptotic spectral singularities. The off-diagonal disorder is also shown to enhance the propagation of two interacting particles similarly to systems with diagonal disorder. Although disordered models with chiral symmetry differ from non-chiral ones due to the presence of spectral singularities, both share the same qualitative localization properties except at the chiral symmetry point E=0 which is critical.Comment: 13 pages, Revtex file, 8 postscript files. It will appear in the special edition of J. Phys. A for Random Matrix Theor

Counting Majorana zero modes in superconductors

A counting formula for computing the number of (Majorana) zero modes bound to topological point defects is evaluated in a gradient expansion for systems with charge-conjugation symmetry. This semi-classical counting of zero modes is applied to some examples that include graphene and a chiral p-wave superconductor in two-dimensional space. In all cases, we explicitly relate the counting of zero modes to Chern numbers.Comment: 21 pages, 3 figure

Conductance scaling at the band center of wide wires with pure non--diagonal disorder

Kubo formula is used to get the scaling behavior of the static conductance distribution of wide wires showing pure non-diagonal disorder. Following recent works that point to unusual phenomena in some circumstances, scaling at the band center of wires of odd widths has been numerically investigated. While the conductance mean shows a decrease that is only proportional to the inverse square root of the wire length, the median of the distribution exponentially decreases as a function of the square root of the length. Actually, the whole distribution decays as the inverse square root of the length except close to G=0 where the distribution accumulates the weight lost at larger conductances. It accurately follows the theoretical prediction once the free parameter is correctly fitted. Moreover, when the number of channels equals the wire length but contacts are kept finite, the conductance distribution is still described by the previous model. It is shown that the common origin of this behavior is a simple Gaussian statistics followed by the logarithm of the E=0 wavefunction weight ratio of a system showing chiral symmetry. A finite value of the two-dimensional conductance mean is obtained in the infinite size limit. Both conductance and the wavefunction statistics distributions are given in this limit. This results are consistent with the 'critical' character of the E=0 wavefunction predicted in the literature.Comment: 10 pages, 9 figures, RevTeX macr

Zero-modes in the random hopping model

If the number of lattice sites is odd, a quantum particle hopping on a bipartite lattice with random hopping between the two sublattices only is guaranteed to have an eigenstate at zero energy. We show that the localization length of this eigenstate depends strongly on the boundaries of the lattice, and can take values anywhere between the mean free path and infinity. The same dependence on boundary conditions is seen in the conductance of such a lattice if it is connected to electron reservoirs via narrow leads. For any nonzero energy, the dependence on boundary conditions is removed for sufficiently large system sizes.Comment: 12 pages, 11 figure