A FL* approach to High Tc cuprates

The FL* liquid is a concept introduced some years ago to describe an electronic Fermi liquid where the particle excitations are not only the standard spin 1/2 charged electrons, but also particle excitations carrying charge but spinless, named holons, and carrying only spin 1/2 but neutral, named spinons. Such kind of excitations have been introduced immediately after the discovery of high Tc superconductivity in the cuprates by Anderson and Kivelson, mainly relying on analogy with one-dimensional models, but the idea that in planar systems they can coexist with bound states of them in the form of an electronic Fermi liquid is quite recent. It has benn shown that to the FL* liquid one can apply two suitable topological versions of the Luttinger theorem on the Fermi volume. In this thesis it is analyzed a possible FL* nature of the low-energy physics of high Tc cuprates, with a verification of the Luttinger theorem, to an approach proposed mainly by Sachdev, based upon a suitabky revised versione of the RVB formalism pioneered by Anderson. A brief comment is added on a gauge approach with semionic statistics of holons and spinons, originally proposed by Laughlin, where an initially overlooked FL* nature seems to emerge. Finally a comparison of some experimental data with results obtained in the FL* approach to cuprates is discussed.The FL* liquid is a concept introduced some years ago to describe an electronic Fermi liquid where the particle excitations are not only the standard spin 1/2 charged electrons, but also particle excitations carrying charge but spinless, named holons, and carrying only spin 1/2 but neutral, named spinons. Such kind of excitations have been introduced immediately after the discovery of high Tc superconductivity in the cuprates by Anderson and Kivelson, mainly relying on analogy with one-dimensional models, but the idea that in planar systems they can coexist with bound states of them in the form of an electronic Fermi liquid is quite recent. It has benn shown that to the FL* liquid one can apply two suitable topological versions of the Luttinger theorem on the Fermi volume. In this thesis it is analyzed a possible FL* nature of the low-energy physics of high Tc cuprates, with a verification of the Luttinger theorem, to an approach proposed mainly by Sachdev, based upon a suitabky revised versione of the RVB formalism pioneered by Anderson. A brief comment is added on a gauge approach with semionic statistics of holons and spinons, originally proposed by Laughlin, where an initially overlooked FL* nature seems to emerge. Finally a comparison of some experimental data with results obtained in the FL* approach to cuprates is discussed

Extracting features from eigenfunctions: higher Cheeger constants and sparse eigenbasis approximation

This thesis investigates links between the eigenvalues and eigenfunctions of the Laplace-Beltrami operator, and the higher Cheeger constants of smooth Riemannian manifolds, possibly with boundary. The higher Cheeger constants give a loose description of the major geometric features of a manifold. We obtain a new lower bound on the negative Laplace-Beltrami eigenvalues in terms of the corresponding higher Cheeger constant. The level sets of Laplace-Beltrami eigenfunctions sometimes reveal sets with small Cheeger ratio, representing well-separated features of the manifold. Some manifolds have their major features entwined across several eigenfunctions, and no single eigenfunction contains all the major features. In this case, there may exist carefully chosen linear combinations of the eigenfunctions, each with large values on a single feature, and small values elsewhere. We can then apply a soft-thresholding operator to these linear combinations to obtain new functions, each supported on a single feature. We show that the Cheeger ratios of the level sets of these functions also give an upper bound on the Laplace-Beltrami eigenvalues. We extend these level set results to nonautonomous dynamical systems, and show that the dynamic Laplacian eigenfunctions reveal sets with small dynamic Cheeger ratios. In a later chapter, we propose a numerical method for identifying features represented in eigenvectors arising from spectral clustering methods when those features are not cleanly represented in a single eigenvector. This method provides explicit candidates for the soft-thresholded linear combinations of eigenfunctions mentioned above. Many data clustering techniques produce collections of orthogonal vectors (e.g. eigenvectors) which contain connectivity information about the dataset. This connectivity information must be disentangled by some secondary procedure. We propose a method for finding an approximate sparse basis for the space spanned by the leading eigenvectors, by applying thresholding to linear combinations of eigenvectors. Our procedure is natural, robust and efficient, and it provides soft-thresholded linear combinations of the inputted eigenfunctions. We develop a new Weyl-inspired eigengap heuristic and heuristics based on the sparse basis vectors, suggesting how many eigenvectors to pass to our method

Nonstandard finite-size effects at discontinuous phase transitions: Degenerate low-temperature states and boundary conditions

In dieser Dissertation wird das Skalenverhalten derÜbergangstemperatur von Systemen an diskontinuierlichen Phasenübergängen aus einem Zwei- Zustands-Modell abgeleitet und erweitert. Es wird erläutert, wie sich das Skalenverhalten für periodische Randbedingungen drastisch verändern kann, sobald der Entartungsgrad der geordneten Phasen von der Teilchenzahl abhängt. Eswerden Modellsysteme in zwei und drei Dimensionen betrachtet, deren Zustandssummen mittels analytischer, kombinatorischer Argumente berechnet werden. Für das kompliziertere, isotrope Plaquettemodell in drei Dimensionen können durch diese Rechnungen Ordnungsparameter definiert werden. Diese werden, zusammen mit dem veränderten Skalenverhalten selbskonsistent durch anspruchsvolle und hochpräzise, sogenannte multikanonische Monte-Carlo Simulationen überprüft und bestätigt

Network and Algebraic Topology of Influenza Evolution

Evolution is a force that has molded human existence since its divergence from chimpanzees about 5.4 million years ago. In that same amount of time, an influenza virus, which replicates every six hours, would have undergone an equivalent number of generations over only a hundred years. The fast replication times of influenza, coupled with its high mutation rate, make the virus a perfect model to study real-time evolution at a mega-Darwin scale, more than a million times faster than human evolution. While recent developments in high-throughput sequencing provide an optimal opportunity to dissect their genetic evolution, a concurrent growth in computational tools is necessary to analyze the large influx of complex genomic data. In my thesis, I present novel computational methods to examine different aspects of influenza evolution. I first focus on seasonal influenza, particularly the problems that hamper public health initiatives to combat the virus. I introduce two new approaches: 1. The q2-coefficient, a method of quantifying pathogen surveillance, and 2. FluGraph, a technique that employs network topology to track the spread of seasonal influenza around the world. The second chapter of my thesis examines how mutations and reassortment combine to alter the course of influenza evolution towards pandemic formation. I highlight inherent deficiencies in the current phylogenetic paradigm for analyzing evolution and offer a novel methodology based on algebraic topology that comprehensively reconstructs both vertical and horizontal evolutionary events. I apply this method to viruses, with emphasis on influenza, but foresee broader application to cancer cells, bacteria, eukaryotes, and other taxa

Real-space simulation of two-dimensional interacting quantum condensed matter systems

The proliferation of quantum fluctuations and long-range entanglement presents an outstanding challenge for the numerical simulation of quantum condensed matter systems with exotic ground states. In this thesis, I tackle two classes of two-dimensional interacting models on the honeycomb lattice: multi-orbital Hubbard models on zig zag transition metal dichalcogenide nanoribbons and generalised Kitaev models on periodic clusters. In the first part of the thesis, I discuss novel results obtained in a comparative study of mean field theory (MFT) and determinant quantum Monte Carlo (DetQMC). MFT reveals the influence of the edge filling on the ground state of the ribbons. The unbiased, numerically exact DetQMC confirms the stability of one of the possible ground states, albeit with quantitative differences, such as the critical Hubbard interaction for the onset of magnetic order. Unfortunately, DetQMC is severely plagued by the sign problem for this model. The variance of its estimators grows exponentially as most of the relevant edge fillings are reached and simulations are rendered unfeasibly expensive from the computational standpoint. Motivated by the difficulties posed by the sign problem, I carry out a survey of general purpose numerical methods. The second part of the thesis addresses quantum spin liquids — which have attracted increasing attention — presenting a toolset of Chebyshev spectral methods developed here, namely: the finite temperature Chebyshev polynomial and the hybrid Lanczos-Chebyshev methods. The first one enables studies of temperature dependence for quantities of experimental interest, such as the specific heat, with a two-fold speed-up with respect to state-of-the-art methods. The second one gives access to spectral functions efficiently and with unparalleled flexibility. I use it to obtain novel results for the spin susceptibility of the Kitaev-Ising model, unravelling dynamical signatures of a liquid–to–liquid transition. Finally, I briefly discuss the integration of the novel Chebyshev toolset with existing open-source software