Tricritical-like behavior of the nonlinear optical refraction at the nematic-isotropic transition in the E7 thermotropic liquid crystal

We use Z-scan technique to investigate the nonlinear optical response of the thermotropic liquid crystal E7 in the neighborhood of the nematic-isotropic phase transition. The analysis of the data for the nonlinear optical birefringence is compatible with an effective critical exponent of the order parameter, beta = 0.28 +/- 0.03, which is close to the classical value, beta = 0.25, for a tricritical point. The nonlinear optical absorption in the nematic range depends on the geometrical configuration of the nematic director with respect to the polarization beam, and vanishes in the isotropic phase.

Ward identities in Lifshitz-like field theories

In this work, we develop a normal product algorithm suitable to the study of anisotropic field theories in flat space, apply it to construct the symmetries generators and describe how their possible anomalies may be found. In particular, we discuss the dilatation anomaly in a scalar model with critical exponent z = 2 in six spatial dimensions.

Ternäre polymerhaltige Lösungen - Phasenverhalten und Grenzflächenspannung

Im Rahmen dieser Arbeit wurden experimentelle und theoretische Untersuchungen zum Phasen- und Grenzflächenverhalten von ternären Systemen des Typs Lösungsmittel/Fällungsmittel/Polymer durchgeführt. Diese Art der Mischungen ist vor allem für die Planung und Durchführung der Membranherstellung von Bedeutung, bei der die genaue Kenntnis des Phasendiagramms und der Grenzflächenspannung unabdingbar ist. Als Polymere dienten Polystyrol sowie Polydimethylsiloxan. Im Fall des Polystyrols kam Butanon-2 als Lösungsmittel zum Einsatz, wobei drei niedrigmolekulare lineare Alkohole als Fällungsmittel verwendet wurden. Für Polydimethylsiloxan eignen sich Toluol als Lösungsmittel und Ethanol als Fällungsmittel. Durch Lichtstreumessungen, Dampfdruckbestimmungen mittels Headspace-Gaschromatographie (VLE-Gleichgewichte) sowie Quellungsgleichgewichten lassen sich die thermodynamischen Eigenschaften der binären Subsysteme charakterisieren. Auf Grundlage der Flory-Huggins-Theorie kann das experimentell bestimmte Phasenverhalten (LLE-Gleichgewichte) in guter Übereinstimmung nach der Methode der Direktminimierung der Gibbs'schen Energie modelliert werden. Zieht man die Ergebnisse der Aktivitätsbestimmung von Dreikomponenten-Mischungen mit in Betracht, so ergeben sich systematische Abweichungen zwischen Experiment und Theorie. Sie können auf die Notwendigkeit ternärer Wechselwirkungsparameter zurückgeführt werden, die ebenfalls durch Modellierung zugänglich sind.Durch die aus den VLE- und LLE-Untersuchungen gewonnenen Ergebnissen kann die sog. Hump-Energie berechnet werden, die ein Maß für die Entmischungstendenz darstellt. Diese Größe eignet sich gut zur Beschreibung von Grenzflächenphänomenen mittels Skalengesetzen. Die für binäre Systeme gefundenen theoretisch fundierten Skalenparameter gelten jedoch nur teilweise. Ein neues Skalengesetz lässt erstmals eine Beschreibung über die gesamte Mischungslücke zu, wobei ein Parameter durch eine gemessene Grenzflächenspannung (zwischen Fällungsmittel/Polymer) ersetzt werden kann.

Pinning-Effekte kolumnarer Schwerionenspuren in Kuprat-Supraleiter-Dünnschichten

In this work the flux line dynamics in High-Temperature Superconductor (HTSC) thin films in the presence of columnar defects was studied using electronic transport measurements. The columnar defects which are correlated pinning centers for vortices were generated by irradiation with swift heavy ions at the Gesellschaft für Schwerionenforschung (GSI) in Darmstadt. In the first part, the vortex dynamics is discussed within the framework of the Bose-glass model. This approach describes the continuous transition from a vortex liquid to a Bose-glass phase which is characterized by the localization of the flux lines at the columnar defects. The critical behavior of the characteristic length and time scales for temperatures in the vicinity of this phase transition were probed by scaling properties of experimentally obtained current-voltage characteristics. In contrast to the predicted universal properties of the critical behavior the scaling analysis shows a strong dependence of the dynamic critical exponent on the experimentally accessible electric field range. In addition, the predicted divergence of the activation energy in the limit of low current densities was experimentally not confirmed.The dynamic behavior of flux lines in spatially resolved irradiation geometries is reported in the second part. Weak pinning channels with widths between 10 µm and 100 µm were generated in a strong pinning environment with the use of metal masks and the GSI microprobe, respectively. Measurements of the anisotropic transport properties of these structures show a striking resemblance to the results in YBCO single crystals with unidirected twin boundaries which were interpreted as a guided vortex motion effect. The use of two additional test bridges allowed to determine in parallel the resistivities of the irradiated and unirradiated parts as well as the respective current-voltage characteristics. These measurements provided the input parameters for a numerical simulation of the potential distribution in the spatially resolved irradiation geometry. The results are interpreted within a model that describes the hydrodynamic interaction between a Bose-glass phase and a vortex liquid. The interface between weakly pinned flux lines in the unirradiated channels and strongly pinned vortices leads to a nonuniform vortex velocity profile and therefore a variation of the local electric field. The length scale of these interactions was estimated for the first time in measuring the local variation of the electric field profile in a Bose-glass contact.Finally, a method for the determination of the true temperature in HTSC thin films at high dissipation levels is described. In this regime of electronic transport the occurrence of a flux flow instability is accompanied by heating effects in the vortex system. The heat propagation properties of the film/substrate system are deduced from the time dependent voltage response to a short high current density pulse of rectangular shape. The influence of heavy ion irradiation on the heat resistance at the film/substrate interface is studied.

Computer simulation studies of vector spin glasses

Die vorliegende Doktorarbeit befasst sich mit klassischen Vektor-Spingläsern eine Art von ungeordneten Magneten - auf verschiedenen Gittertypen. Da siernbedeutsam für eine experimentelle Realisierung sind, ist ein theoretisches Verständnis von Spinglas-Modellen mit wenigen Spinkomponenten und niedriger Gitterdimension von großer Bedeutung. Da sich dies jedoch als sehr schwierigrnerweist, sind neue, aussichtsreiche Ansätze nötig. Diese Arbeit betrachtet daher den Limesrnunendlich vieler Spindimensionen. Darin entstehen mehrere Vereinfachungen im Vergleichrnzu Modellen niedriger Spindimension, so dass für dieses bedeutsame Problem Eigenschaften sowohl bei Temperatur Null als auch bei endlichen Temperaturenrnüberwiegend mit numerischen Methoden ermittelt werden. Sowohl hyperkubische Gitter als auch ein vielseitiges 1d-Modell werden betrachtet. Letzteres erlaubt es, unterschiedliche Universalitätsklassen durch bloßes Abstimmen eines einzigen Parameters zu untersuchen. "Finite-size scaling''-Formen, kritische Exponenten, Quotienten kritischer Exponenten und andere kritische Größen werden nahegelegt und mit numerischen Ergebnissen verglichen. Eine detaillierte Beschreibung der Herleitungen aller numerisch ausgewerteter Gleichungen wird ebenso angegeben. Bei Temperatur Null wird eine gründliche Untersuchung der Grundzustände und Defektenergien gemacht. Eine Reihe interessanter Größen wird analysiert und insbesondere die untere kritische Dimension bestimmt. Bei endlicher Temperatur sind der Ordnungsparameter und die Spinglas-Suszeptibilität über die numerisch berechnete Korrelationsmatrix zugänglich. Das Spinglas-Modell im Limes unendlich vieler Spinkomponenten kann man als Ausgangspunkt zur Untersuchung der natürlicheren Modelle mit niedriger Spindimension betrachten. Wünschenswert wäre natürlich ein Modell, das die Vorteile des ersten mit den Eigenschaften des zweiten verbände. Daher wird in Modell mit Anisotropie vorgeschlagen und getestet, mit welchem versucht wird, dieses Ziel zu erreichen. Es wird auf reizvolle Wege hingewiesen, das Modell zu nutzen und eine tiefergehende Beschäftigung anzuregen. Zuletzt werden sogenannte "real-space" Renormierungsgruppenrechnungen sowohl analytisch als auch numerisch für endlich-dimensionale Vektor-Spingläser mit endlicher Anzahl von Spinkomponenten durchgeführt. Dies wird mit einer zuvor bestimmten neuen Migdal-Kadanoff Rekursionsrelation geschehen. Neben anderen Größen wird die untere kritische Dimension bestimmt.

Holographic renormalization for z = 2 Lifshitz spacetimes from AdS

Lifshitz spacetimes with the critical exponent z = 2 can be obtained by the dimensional reduction of Schrödinger spacetimes with the critical exponent z = 0. The latter spacetimes are asymptotically AdS solutions of AdS gravity coupled to an axion–dilaton system and can be uplifted to solutions of type IIB supergravity. This basic observation is used to perform holographic renormalization for four-dimensional asymptotically z = 2 locally Lifshitz spacetimes by the Scherk–Schwarz dimensional reduction of the corresponding problem of holographic renormalization for five-dimensional asymptotically locally AdS spacetimes coupled to an axion–dilaton system. We can thus define and characterize a four-dimensional asymptotically locally z = 2 Lifshitz spacetime in terms of five-dimensional AdS boundary data. In this setup the four-dimensional structure of the Fefferman–Graham expansion and the structure of the counterterm action, including the scale anomaly, will be discussed. We find that for asymptotically locally z = 2 Lifshitz spacetimes obtained in this way, there are two anomalies each with their own associated nonzero central charge. Both anomalies follow from the Scherk–Schwarz dimensional reduction of the five-dimensional conformal anomaly of AdS gravity coupled to an axion–dilaton system. Together, they make up an action that is of the Horava–Lifshitz type with a nonzero potential term for z = 2 conformal gravity.

Topological lattice actions for the 2d XY model

We consider the 2d XY Model with topological lattice actions, which are invariant against small deformations of the field configuration. These actions constrain the angle between neighbouring spins by an upper bound, or they explicitly suppress vortices (and anti-vortices). Although topological actions do not have a classical limit, they still lead to the universal behaviour of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition — at least up to moderate vortex suppression. In the massive phase, the analytically known Step Scaling Function (SSF) is reproduced in numerical simulations. However, deviations from the expected universal behaviour of the lattice artifacts are observed. In the massless phase, the BKT value of the critical exponent ηc is confirmed. Hence, even though for some topological actions vortices cost zero energy, they still drive the standard BKT transition. In addition we identify a vortex-free transition point, which deviates from the BKT behaviour.

Lifshitz holographic renormalization from anti-de Sitter 1

Lifshitz space–times with critical exponent z = 2 can be obtained by dimensional reduction of Schrödinger space–times with critical exponent z = 0. The latter space–times are asymptotically anti-de Sitter (AdS) solutions of AdS gravity coupled to an axion–dilaton system (or even just a massless scalar field). This basic observation is used to perform holographic renormalization for four-dimensional asymptotically locally Lifshitz space–times by dimensional reduction of the corresponding problem of holographic renormalization for five-dimensional asymptotically AdS space–times coupled to an axion–dilaton system. In this setup the four-dimensional structure of the Lifshitz – Fefferman-Graham expansion and the structure of the counterterm action, including the scale anomaly, will be summarized.

From quantum to classical dynamics: The relativistic O ( N ) model in the framework of the real-time functional renormalization group

We investigate the transition from unitary to dissipative dynamics in the relativistic O(N) vector model with the λ(φ2)2 interaction using the nonperturbative functional renormalization group in the real-time formalism. In thermal equilibrium, the theory is characterized by two scales, the interaction range for coherent scattering of particles and the mean free path determined by the rate of incoherent collisions with excitations in the thermal medium. Their competition determines the renormalization group flow and the effective dynamics of the model. Here we quantify the dynamic properties of the model in terms of the scale-dependent dynamic critical exponent z in the limit of large temperatures and in 2≤d≤4 spatial dimensions. We contrast our results to the behavior expected at vanishing temperature and address the question of the appropriate dynamic universality class for the given microscopic theory.

Efficient numerical methods for the random-field Ising model: Finite-size scaling, reweighting extrapolation, and computation of response functions

It was recently shown [Phys. Rev. Lett. 110, 227201 (2013)] that the critical behavior of the random-field Ising model in three dimensions is ruled by a single universality class. This conclusion was reached only after a proper taming of the large scaling corrections of the model by applying a combined approach of various techniques, coming from the zero-and positive-temperature toolboxes of statistical physics. In the present contribution we provide a detailed description of this combined scheme, explaining in detail the zero-temperature numerical scheme and developing the generalized fluctuation-dissipation formula that allowed us to compute connected and disconnected correlation functions of the model. We discuss the error evolution of our method and we illustrate the infinite limit-size extrapolation of several observables within phenomenological renormalization. We present an extension of the quotients method that allows us to obtain estimates of the critical exponent a of the specific heat of the model via the scaling of the bond energy and we discuss the self-averaging properties of the system and the algorithmic aspects of the maximum-flow algorithm used.

Dimensional dependence of the Stokes-Einstein relation and its violation

We generalize to higher spatial dimensions the Stokes-Einstein relation (SER) as well as the leading correction to diffusivity in finite systems with periodic boundary conditions, and validate these results with numerical simulations. We then investigate the evolution of the high-density SER violation with dimension in simple hard sphere glass formers. The analysis suggests that this SER violation disappears around dimension du = 8, above which it is not observed. The critical exponent associated with the violation appears to evolve linearly in 8 - d, below d = 8, as predicted by Biroli and Bouchaud [J. Phys.: Condens. Matter 19, 205101 (2007)], but the linear coefficient is not consistent with the prediction. The SER violation with d establishes a new benchmark for theory, and its complete description remains an open problem. © 2013 AIP Publishing LLC.

Privileged Words and Sturmian Words

This dissertation has two almost unrelated themes: privileged words and Sturmian words. Privileged words are a new class of words introduced recently. A word is privileged if it is a complete first return to a shorter privileged word, the shortest privileged words being letters and the empty word. Here we give and prove almost all results on privileged words known to date. On the other hand, the study of Sturmian words is a well-established topic in combinatorics on words. In this dissertation, we focus on questions concerning repetitions in Sturmian words, reproving old results and giving new ones, and on establishing completely new research directions. The study of privileged words presented in this dissertation aims to derive their basic properties and to answer basic questions regarding them. We explore a connection between privileged words and palindromes and seek out answers to questions on context-freeness, computability, and enumeration. It turns out that the language of privileged words is not context-free, but privileged words are recognizable by a linear-time algorithm. A lower bound on the number of binary privileged words of given length is proven. The main interest, however, lies in the privileged complexity functions of the Thue-Morse word and Sturmian words. We derive recurrences for computing the privileged complexity function of the Thue-Morse word, and we prove that Sturmian words are characterized by their privileged complexity function. As a slightly separate topic, we give an overview of a certain method of automated theorem-proving and show how it can be applied to study privileged factors of automatic words. The second part of this dissertation is devoted to Sturmian words. We extensively exploit the interpretation of Sturmian words as irrational rotation words. The essential tools are continued fractions and elementary, but powerful, results of Diophantine approximation theory. With these tools at our disposal, we reprove old results on powers occurring in Sturmian words with emphasis on the fractional index of a Sturmian word. Further, we consider abelian powers and abelian repetitions and characterize the maximum exponents of abelian powers with given period occurring in a Sturmian word in terms of the continued fraction expansion of its slope. We define the notion of abelian critical exponent for Sturmian words and explore its connection to the Lagrange spectrum of irrational numbers. The results obtained are often specialized for the Fibonacci word; for instance, we show that the minimum abelian period of a factor of the Fibonacci word is a Fibonacci number. In addition, we propose a completely new research topic: the square root map. We prove that the square root map preserves the language of any Sturmian word. Moreover, we construct a family of non-Sturmian optimal squareful words whose language the square root map also preserves.This construction yields examples of aperiodic infinite words whose square roots are periodic.

On a semilinear Schrodinger equation with critical Sobolev exponent

We consider the semilinear Schrodinger equation -Deltau+V(x)u= K(x) \u \ (2*-2 u) + g(x; u), u is an element of W-1,W-2 (R-N), where N greater than or equal to4, V, K, g are periodic in x(j) for 1 less than or equal toj less than or equal toN, K>0, g is of subcritical growth and 0 is in a gap of the spectrum of -Delta +V. We show that under suitable hypotheses this equation has a solution u not equal 0. In particular, such a solution exists if K equivalent to 1 and g equivalent to 0.