Critical properties of the metal-insulator transition in anisotropic systems

We study the three-dimensional Anderson model of localization with anisotropic hopping, i.e., weakly coupled chains and weakly coupled planes. In our extensive numerical study we identify and characterize the metal-insulator transition by means of the transfer-matrix method. The values of the critical disorder $W_c$ obtained are consistent with results of previous studies, including multifractal analysis of the wave functions and energy level statistics. $W_c$ decreases from its isotropic value with a power law as a function of anisotropy. Using high accuracy data for large system sizes we estimate the critical exponent as $\nu=1.62\pm0.07$. This is in agreement with its value in the isotropic case and in other models of the orthogonal universality class.Comment: 17 pages, 7 figures, requires svjour.csl and svepj.clo (included), submitted to EPJ

An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces

We present a rigorous scheme that makes it possible to compute eigenvalues of the Laplace operator on hyperbolic surfaces within a given precision. The method is based on an adaptation of the method of particular solutions to the case of locally symmetric spaces and on explicit estimates for the approximation of eigenfunctions on hyperbolic surfaces by certain basis functions. It can be applied to check whether or not there is an eigenvalue in an \epsilon-neighborhood of a given number \lambda>0. This makes it possible to find all the eigenvalues in a specified interval, up to a given precision with rigorous error estimates. The method converges exponentially fast with the number of basis functions used. Combining the knowledge of the eigenvalues with the Selberg trace formula we are able to compute values and derivatives of the spectral zeta function again with error bounds. As an example we calculate the spectral determinant and the Casimir energy of the Bolza surface and other surfaces.Comment: 48 pages, 8 figures, LaTeX, some more typos corrected, more Figures added, some explanations are more detailed now, Fenchel-Nielsen coordinates and numbers for the surface with symmetry group of order 10 corrected, datafiles are now available as ancillary file

Analysis of Schr\"odinger operators with inverse square potentials I: regularity results in 3D

Let $V$ be a potential on \RR^3 that is smooth everywhere except at a discrete set \maS of points, where it has singularities of the form $Z/\rho^2$, with $\rho(x) = |x - p|$ for $x$ close to $p$ and $Z$ continuous on \RR^3 with $Z(p) > -1/4$ for p \in \maS. Also assume that $\rho$ and $Z$ are smooth outside \maS and $Z$ is smooth in polar coordinates around each singular point. We either assume that $V$ is periodic or that the set \maS is finite and $V$ extends to a smooth function on the radial compactification of \RR^3 that is bounded outside a compact set containing \maS. In the periodic case, we let $\Lambda$ be the periodicity lattice and define \TT := \RR^3/ \Lambda. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schr\"odinger-type operator $H = -\Delta + V$ acting on L^2(\TT), as well as for the induced \vt k--Hamiltonians \Hk obtained by restricting the action of $H$ to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper.Comment: 15 pages, to appear in Bull. Math. Soc. Sci. Math. Roumanie, vol. 55 (103), no. 2/201

Two applications of the gauge/gravity duality

The anti de Sitter/conformal field theory correspondence, or AdS/CFT for short, is an equivalence between a string theory with gravity defined on a space and a quantum field theory without gravity on the boundary of this space. The correspondence makes it possible to compute observables in strongly interacting field theories. We will focus on the weak form of correspondence, which explains how classical supergravity in five dimensions is related to four dimensional strongly interacting field theory. This thesis is divided into two parts. The first part introduces the correspondence, and the second part contains three original research publications. The publications present two different applications of the correspondence. The first application is to quantum chromodynamics. In particular we derive results on the spatial string tension at finite temperature and thermodynamics of quantum chromodynamics. The second application is to condensed matter physics, namely quantum Hall transitions. We derive results for frequency independent and dependent conductivities in quantum Hall transitions, and find a universal behavior in our solutions.Tuntemamme fysiikka selittää hyvin hiukkasfysiikan ja painovoiman ilmiöitä - erikseen. Näitä kuvaavien teorioiden yhdistäminen on ollut yksi suurimmista ongelmista teoreettisessa fysiikassa viimeisen sadan vuoden ajan. Vuonna 1998 tiedeyhteisö löysi mittagravitaatiodualiteetin, joka yhdistää tietyn gravitaatioteorian ja kenttäteorian. Mittagravitaatiodualiteetti on yhteys tyypin IIB säieteorian ja supersymmetrisen kvanttikenttäteorian välillä. Erikoisen mielenkiintoiseksi dualiteetin tekee se, että säieteoria sisältää gravitaation ja se on määritelty viidessä ulottuvuudessa, kenttäteoria puolestaan elää tämän avaruuden reunalla neljässä ulottuvuudessa ilman gravitaatiota. Dualiteetti perustuu holografiaperiaatteeseen. Periaatteen mukaan gravitaatioteorian vapausasteet voidaan kuvata tämän teorian reuna-avaruudessa. Tämä herättää luonnollisen kysymyksen: kuinka pieneen tilaan voimme koodata kaiken tiedon, voimmeko kuvata koko maailman hiekanjyväsessä? Dualiteettia voidaan hyödyntää kvanttikromodynamiikan syvempään ymmärrykseen. Erityisesti häiriöteorian ulottumattomissa olevat laskut ovat olleet ongelmallisia fyysikoille. Dualiteetti tarjoaa luonnollisen tavan tarkastella tällaisia, vahvan kytkennän, teorioita. Väitöskirjan yksi päätulos kvanttikromodynamiikasta liittyy kvarkkien potentiaalin määrittämiseen. Saamamme tulokset yhtyvät hyvin tunnettujen tulosten kanssa, joita on aiemmin esitetty kirjallisuudessa eri menetelmien avulla laskettuina. Toinen tulos tarkastelee kvanttikromodynamiikan yleistä termodynamiikkaa ja kuinka voimme ymmärtää sitä dualiteetin avulla. Dualiteettia voidaan soveltaa myös kiinteän olomuodon fysiikkaan. Yksi erityinen esimerkki on kvantittunut Hallin ilmiö, jota väitöskirjassa tarkastellaan. Väitöstyössä laskettiin sähkönjohtavuus siirtymissä eri tilojen välillä. Tämä vertautuu hyvin kirjallisuudessa aiemmin esitettyihin tuloksiin

AC transport at holographic quantum hall transitions

26 pages including 7 figures. (v2): correction to section 3, added reference. (v3): clarified comparison to previous work, (v4): a couple of comments and references addedWe compute AC electrical transport at quantum Hall critical points, as modeled by intersecting branes and gauge/gravity duality. We compare our results with a previous field theory computation by Sachdev, and find unexpectedly good agreement. We also give general results for DC Hall and longitudinal conductivities valid for a wide class of quantum Hall transitions, as well as (semi)analytical results for AC quantities in special limits. Our results exhibit a surprising degree of universality; for example, we find that the high frequency behavior, including subleading behavior, is identical for our entire class of theories.Peer reviewe

Analysis of Schrodinger operators with inverse square potentials II: FEM and approximation of eigenfunctions in the periodic case

In this article, we consider the problem of optimal approximation of eigenfunctions of Schrödinger operators with isolated inverse square potentials and of solutions to equations involving such operators. It is known in this situation that the finite element method performs poorly with standard meshes. We construct an alter- native class of graded meshes, and prove and numerically test optimal approximation results for the finite element method using these meshes. Our numerical tests are in good agreement with our theoretical results

Analysis of Schrodinger operators with inverse square potentials I: regularity results in 3D

Let V be a potential on R3 that is smooth everywhere except at a discrete set S of points, where it has singularities of the form Z/ 2, with (x) = |x − p| for x close to p and Z continuous on R3 with Z(p) > −1/4 for p 2 S. Also assume that and Z are smooth outside S and Z is smooth in polar coordinates around each singular point. We either assume that V is periodic or that the set S is finite and V extends to a smooth function on the radial compactification of R3 that is bounded outside a compact set containing S. In the periodic case, we let be the periodicity lattice and define T := R3/ . We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schr¨odinger-type operator H = − + V acting on L2(T), as well as for the induced k–Hamiltonians Hk obtained by restricting the action of H to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper

Smoothed universal correlations in the two-dimensional Anderson model

We report on calculations of smoothed spectral correlations in the two-dimensional Anderson model for weak disorder. As pointed out in (M. Wilkinson, J. Phys. A: Math. Gen. 21, 1173 (1988)), an analysis of the smoothing dependence of the correlation functions provides a sensitive means of establishing consistency with random matrix theory. We use a semiclassical approach to describe these fluctuations and offer a detailed comparison between numerical and analytical calculations for an exhaustive set of two-point correlation functions. We consider parametric correlation functions with an external Aharonov-Bohm flux as a parameter and discuss two cases, namely broken time-reversal invariance and partial breaking of time-reversal invariance. Three types of correlation functions are considered: density-of-states, velocity and matrix element correlation functions. For the values of smoothing parameter close to the mean level spacing the semiclassical expressions and the numerical results agree quite well in the whole range of the magnetic flux.Comment: 12 pages, 14 figures submitted to Phys. Rev.

The CCP4 suite : integrative software for macromolecular crystallography

The Collaborative Computational Project No. 4 (CCP4) is a UK-led international collective with a mission to develop, test, distribute and promote software for macromolecular crystallography. The CCP4 suite is a multiplatform collection of programs brought together by familiar execution routines, a set of common libraries and graphical interfaces. The CCP4 suite has experienced several considerable changes since its last reference article, involving new infrastructure, original programs and graphical interfaces. This article, which is intended as a general literature citation for the use of the CCP4 software suite in structure determination, will guide the reader through such transformations, offering a general overview of the new features and outlining future developments. As such, it aims to highlight the individual programs that comprise the suite and to provide the latest references to them for perusal by crystallographers around the world

Rare events and other deviations from universality in disordered conductors

Gegenstand dieser Arbeit ist die Untersuchung von statistischen Eigenschaften der ungeordneten Metallen im Rahmen des Anderson-Modells der Lokalisierung. Betrachtet wird ein Elektron auf einem Gitter mit "Nächste-Nachbarn-Hüpfen" und zufälligen potentiellen Gitterplatzenergien. Wegen der Zufälligkeit zeigen die Elektroneigenschaften, zum Beispiel die Eigenenergien und -zustände, irreguläre Fluktuationen, deren Statistik von der Amplitude der Potentialenergie abhängt. Mit steigender Amplitude wird das Elektron immer mehr lokalisiert, was schliesslich zum Metall-Isolator-Übergang führt. In dieser Arbeit wird die Statistik insbesondere im metallischen Bereich untersucht, und dadurch der Einfluss der Lokalisierung an den Eigenschaften des Systems betrachtet. Zuerst wird die Statistik der Matrixelemente des Dipoloperators untersucht. Die numerischen Ergebnisse für das Anderson-Modell werden mit Vorhersagen der semiklassischen Näherung verglichen. Dann wird der spektrale Strukturfaktor betrachtet, der als Fourier-Transformation der zwei-Punkt Zustandsdichtekorrelationsfunktion definiert wird. Dabei werden besonders die nichtuniversellen Abweichungen von den Vorhersagen der Zufallsmatrixtheorie untersucht. Die Abweichungen werden numerisch ermittelt, und danach mit den analytischen Vorhersagen verglichen. Die Statistik der Wellenfunktionen zeigt ebenfalls Abweichungen von der Zufallsmatrixtheorie. Die Abweichungen sind am größten für Statistik der großen Wellenfunktionsamplituden, die sogenannte seltene Ereignisse darstellen. Die analytischen Vorhersagen für diese Statistik sind teilweise widersprüchlich, und deshalb ist es interessant, sie auch numerisch zu untersuchen