Masterprojecten Theoretische Fysica 2009

J.S.Caux

Nonlinear dynamics in cold atomic gases

supervisor: J.S. Caux

Cold atoms in optical lattices or solid-state chips are now commonly used as realizations of toy models of strongly interacting condensed matter systems. In particular, one-dimensional gases of interacting bosons are being intensively studied by both experimentalists and theorists. This project aims at studying the dynamical response of such systems from the point of view of exactly solvable models, and to make the link with the recently developed nonlinear Luttinger liquid theory of 1D interacting systems by computing the singularity exponents of certain response functions.

Conserved charges in solvable models for quantum dots

supervisor: J.S. Caux

Single electrons trapped in quantum dots are one of the most promising realizations of qubits for the eventual realization of quantum computers. Due to weak couplings of the electron spin with the nuclear magnetic moments of the underlying material, though, the coherence time of quantum states is limited. This project will aim at studying exactly solvable models for such systems, to unearth the precise structure of their conserved (time-independent) quantities, and to establish which consequences these have on the decoherence rate of actual qubit realizations.

Perturbed quantum spin chains

supervisor: J.S. Caux

Quantum spin chains, commonly studied by both theorists and experimentalists, offer textbook examples of strongly-correlated quantum systems. Although theoretical models often have the property of being exactly solvable, actual realizations always contain 'integrability-breaking' additional couplings. This project aims at understanding examples of such deformed models from the point of view of the exactly solvable ones, and to make the link with the general theory of interacting one-dimensional systems based on conformal field theory and Luttinger liquid theory.

B. Nienhuis

Rigidity of granular matter

supervisor: B. Nienhuis

Here we propose a mixed theoretical and experimental research project in which we wish to answer a seemingly simple question. Consider a typical pile of sand and imagine to remove grains arbitrarily. After the removal of a certain fraction of the grains, the pile will collapse under its own weight. It is not known at what filling fraction this takes place, even in infinitesimal gravity. It would be interesting to know how the physical properties of the pile, such as its response to stress, behave in neighborhood of the point at which the pile collapses. What is the nature of the skeleton of grains that is just sufficient to resist stress. These questions can be approached experimentally by measuring the mechanical properties of a suspension of grains in a density matched solvent, to achieve effectively zero gravity. Theoretically one can do computer simulations of simple models of the system. This requires the development of (relatively simple) computer programs. The simplest models may also admit analytical study, considering that the problem is very close to the famous problems of percolation and rigidity percolation, where advanced analytical results exist. (collaboration with Ilya Gruzberg in Chicago, and Daniel Bonn, WZI - UvA)

Contact and rigidity Percolation

supervisor: B. Nienhuis

A more formal project in the same area is the following: A random mixture of conducting and insulating grains can behave as a conductor or an insulator, depending on the composition. At a sharp point in the volume fraction of the conduction component the macroscopic behavior changes. Likewise a mixture of rigid and soft (buttery) grains, can behave as a solid or a fluid. The models to describe this are very similar, called contact percolation and rigidity percolation respectively. About the first much more is known than about the second. Here we intend to study these models side by side.

Correlation functions for polymers in an interface

supervisor: B. Nienhuis

The behavior of linear polymers in solution depends on the 'quality' of the solvent. In a good solvent they form a random trajectory in space, much like a random walk, but not quite. This state of the polymer is called swollen. The fact that a random walk is permitted to visit the same place any number of times, and that a polymer can not have multiple monomers occupy the same space, makes the statistics of the paths different in a subtle way. The standard model for a polymer is called the self avoiding walk (SAW). In a poor solvent the polymer crumples up relatively densely, thus mimimizing the amount of contact with the solvent: it is in the collapsed phase. When the solvent is made progressively poor, for instance by lowering the temperature, or by admixture a poor solvent into a good solvent, the polymer undergoes a phase transition from the swollen to the collapsed phase, usually called the theta point. It is typically described by an interacting-SAW, or ISAW. All this occurs in three dimensions, but just as well in two, when the polymer is dissolved in the interface between two fluids. In this project the polymer in the swollen state, and at the theta point, in two dimensions. The approach makes use of a transfer matrix, which considers the polymer at an arbitrary (non straight) cut through space, which intersects the SAW at arbitrary places. The behavior of these intersection points, as the cut is partially relaid, satisfies certain difference equations called the quantum Knyzhnik-Zamolodchikov equations. These can be used to extract information about correlation functions of the SAW and ISAW. (some aspects in collaboration with Jesper Jacobsen (Ecole Normale Superieure, Paris) and Philippe di Francesco (Institut Physique Theorique Saclay)

Supersymmetric models for graphene

supervisor: B. Nienhuis

Graphene, a single layer of graphite has been known for a long time to have interesting behavior at specfic points of the conduction band. Here the energy as a function of the momentum locally has the shape of a perfect cone. This is the same as for relativistic (massless) fermions. Recently it was noted that it is not difficult to isolate a single layer of graphite, and this resulted in a large number of experiments and theoretical studies. The conduction electrons of graphene can be drawn as occasional double bonds between carbon atoms forming a hexagonal array. In the neutral state every carbon atom has two single and two double bonds, as if the electrons taks the space of two atoms, thus excluding neighboring positions to be occupied simultaneously. This behavior corresponds with that in a family of models recently constructed by Fendley and Schoutens, which have a special symmetry called supersymmetry. This symmetry helps one to do calculations on the model. This project it is intended to do such calculations to study the properties of these models. (collaboration with Susanne Reffert, U Tokyo)

Virus capsids

supervisor: B. Nienhuis

Viruses consist of a DNA or RNA molecule packaged in a box consisting of a protein, called a capsid. These capsids are completely regular aggegates of multiple copies of the same protein. In many cases the capsid has icosahedral symmetry and consists of building blocks that have the shape of a hexagon or a pentagon. Obvious examples are the pentagonal dodecahedron, and the buckyball, but there are many more. The purpose of this project is to find out how it is possible that the shapes of the capsids are always so regular. In experimental conditions with only the building proteins present also many irregular capsids are formed. The project consists of designing various models for the building process of the capsids, the transitions between different shapes and a variety of other aspects of these processes, and simulating these models to see what they predict for observables in in vitro and in vivo experiments. (collaboration with Debabrata Panja, Akzo)

Entanglement in many-body systems

supervisor: B. Nienhuis

In a quantum many-body system in the ground state the different degrees of freedom are entanglement. Here we study the entanglement of a (relatively large) subset of the degrees of freedom with another subset, possibly the remaining ones. How the amount of entanglement depends on the sizes involved and on the boundary conditions, gives a lot of information about the nature of the ground state. In particular it is a probe to detect a quantum phase transition. (collaboration with Pasquale Calabrese, Torino)

T. Quella

Current algebras for sigma-models on supergroups and supercosets

supervisor: T. Quella

About ten years ago, Maldacena conjectured a celebrated equivalence between string theory on Anti-de Sitter space- times and supersymmetric gauge theories resembling QCD. However, despite immense efforts many open questions remained, especially regarding a covariant treatment of string propagation on Anti-de Sitter space-times. In this project, the latter question will be addressed from the particular perspective of 2D sigma-models and their underlying algebraic structures such as current algebras and integrability. It is well-known that all string vacua can be formulated in terms of conformally invariant sigma-models. There are some especially symmetric string backgrounds (among which the Anti-de Sitter space-times) which are associated to supergroups and supercosets (see e.g. [1, p.1- 10]). In [2] a new class of current algebras has been proposed for such spaces which generalize the famous Kac-Moody algebra symmetries of WZW models and which should allow to address many hitherto unsolved physical questions related to Maldacena's conjecture. In this project, you will analyze the structure of these new current algebras and try to use them to answer physical questions about string spectra etc. Primary goals: