Green's functions on finite lattices and their connection to the infinite lattice limit

It is shown that the Green's function on a finite lattice in arbitrary space dimension can be obtained from that of an infinite lattice by means of translation operator. Explicit examples are given for one- and two-dimensional lattices

A glimpse of a Luttinger liquid

The concept of a Luttinger liquid has recently been established as a fundamental paradigm vital to our understanding of the properties of one-dimensional quantum systems, leading to a number of theoretical breakthroughs. Now theoretical predictions have been put to test by the comprehensive experimental study.Comment: Unedited version of N&V article in Nature materials 4, 273 (2005

The Magic Angle "Mystery" in Electron Energy Loss Spectroscopy: Relativistic and Dielectric Corrections

Recently it has been demonstrated that a careful treatment of both longitudinal and transverse matrix elements in electron energy loss spectra can explain the mystery of relativistic effects on the {\it magic angle}. Here we show that there is an additional correction of order $(Z\alpha)^2$ where $Z$ is the atomic number and $\alpha$ the fine structure constant, which is not necessarily small for heavy elements. Moreover, we suggest that macroscopic electrodynamic effects can give further corrections which can break the sample-independence of the magic angle.Comment: 10 pages (double column), 6 figure

Exact ground states for the four-electron problem in a two-dimensional finite Hubbard square system

We present exact explicit analytical results describing the exact ground state of four electrons in a two dimensional square Hubbard cluster containing 16 sites taken with periodic boundary conditions. The presented procedure, which works for arbitrary even particle number and lattice sites, is based on explicitly given symmetry adapted base vectors constructed in r-space. The Hamiltonian acting on these states generates a closed system of 85 linear equations providing by its minimum eigenvalue the exact ground state of the system. The presented results, described with the aim to generate further creative developments, not only show how the ground state can be exactly obtained and what kind of contributions enter in its construction, but emphasize further characteristics of the spectrum. On this line i) possible explications are found regarding why weak coupling expansions often provide a good approximation for the Hubbard model at intermediate couplings, or ii) explicitly given low lying energy states of the kinetic energy, avoiding double occupancy, suggest new roots for pairing mechanism attracting decrease in the kinetic energy, as emphasized by kinetic energy driven superconductivity theories.Comment: 37 pages, 18 figure

Sine-Gordon Model - Renormalization Group Solutions and Applications

The sine-Gordon model is discussed and analyzed within the framework of the renormalization group theory. A perturbative renormalization group procedure is carried out through a decomposition of the sine-Gordon field in slow and fast modes. An effective slow modes's theory is derived and re-scaled to obtain the model's flow equations. The resulting Kosterlitz-Thouless phase diagram is obtained and discussed in detail. The theory's gap is estimated in terms of the sine-Gordon model paramaters. The mapping between the sine-Gordon model and models for interacting electrons in one dimension, such as the g-ology model and Hubbard model, is discussed and the previous renormalization group results, obtained for the sine-Gordon model, are thus borrowed to describe different aspects of Luttinger liquid systems, such as the nature of its excitations and phase transitions. The calculations are carried out in a thorough and pedagogical manner, aiming the reader with no previous experience with the sine-Gordon model or the renormalization group approach.Comment: 44 pages, 7 figure

Kondo effect in an integer-spin quantum dot

The Kondo effect is a key many-body phenomenon in condensed matter physics. It concerns the interaction between a localised spin and free electrons. Discovered in metals containing small amounts of magnetic impurities, it is now a fundamental mechanism in a wide class of correlated electron systems. Control over single, localised spins has become relevant also in fabricated structures due to the rapid developments in nano-electronics. Experiments have already demonstrated artificial realisations of isolated magnetic impurities at metallic surfaces, nanometer-scale magnets, controlled transitions between two-electron singlet and triplet states, and a tunable Kondo effect in semiconductor quantum dots. Here, we report an unexpected Kondo effect realised in a few-electron quantum dot containing singlet and triplet spin states whose energy difference can be tuned with a magnetic field. This effect occurs for an even number of electrons at the degeneracy between singlet and triplet states. The characteristic energy scale is found to be much larger than for the ordinary spin-1/2 case.Comment: 12 page

Higher Powers in Gravitation

We consider the Friedmann-Robertson-Walker cosmologies of theories of gravity that generalise the Einstein-Hilbert action by replacing the Ricci scalar, R, with some function, f(R). The general asymptotic behaviour of these cosmologies is found, at both early and late times, and the effects of adding higher and lower powers of R to the Einstein-Hilbert action is investigated. The assumption that the highest powers of R should dominate the Universe's early history, and that the lowest powers should dominate its future is found to be inaccurate. The behaviour of the general solution is complicated, and while it can be the case that single powers of R dominate the dynamics at late times, it can be either the higher or lower powers that do so. It is also shown that it is often the lowest powers of R that dominate at early times, when approach to a bounce or a Tolman solution are generic possibilities. Various examples are considered, and both vacuum and perfect fluid solutions investigated.Comment: 30 pages, 9 figure

Holographic Kondo Model in Various Dimensions

We study the addition of localised impurities to U(N) Supersymmetric Yang-Mills theories in (p+1)-dimensions by using the gauge/gravity correspondence. From the gravity side, the impurities are introduced by considering probe D(8-p)-branes extendingalong the time and radial directions and wrapping an (7-p)-dimensional submanifold of the internal (8-p)-sphere, so that the degrees of freedom are point-like from the gauge theory perspective. We analyse both the configuration in which the branes generate straight flux tubes -corresponding to actual single impurities - and the one in which connected flux tubes are created- corresponding to dimers. We discuss the thermodynamics of both the configurations and the related phase transition. In particular, the specific heat of the straight flux-tube configuration is negative for p<3, while it is never the case for the connected one. We study the stability of the system by looking at the impurity fluctuations. Finally, we characterise the theory by computing one- and two-point correlators of the gauge theory operators dual to the impurity fluctuations. Because of the underlying generalised conformal structure, such correlators can be expressed in terms of an effective coupling constant (which runs because of its dimensionality) and a generalised conformal dimension.Comment: 56 pages, 3 figures; v2: typos correcte

A danger of low copy numbers for inferring incorrect cooperativity degree

Background: A dose-response curve depicts fraction of bound proteins as a function of unbound ligands. Dose-response curves are used to measure the cooperativity degree of a ligand binding process. Frequently, the Hill function is used to fit the experimental data. The Hill function is parameterized by the value of the dissociation constant, and the Hill coefficient which describes the cooperativity degree. The use of Hill's model and the Hill function have been heavily criticised in this context, predominantly the assumption that all ligands bind at once, which lead to further refinements of the model. In this work, the validity of the Hill function has been studied from an entirely different point of view. In the limit of low copy numbers the dynamics of the system becomes noisy. The goal was to asses the validity of the Hill function in this limit, and to see in which ways the effects of the fluctuations change the form of the dose-response curves. Results: Dose-response curves were computed taking into account effects of fluctuations. The effects of fluctuations were described at the lowest order (the second moment of the particle number distribution) by using previously developed Pair Approach Reaction Noise EStimator (PARNES) method. The stationary state of the system is described by nine equations with nine unknowns. To obtain fluctuation corrected dose-response curves the equations have been investigated numerically. Conclusions: The Hill function cannot describe dose-response curves in a low particle limit. First, dose-response curves are not solely parameterized by the dissociation constant and the Hill coefficient. In general, the shape of a dose-response curve depends on the variables that describe how an experiment (ensemble) is designed. Second, dose-response curves are multi valued in a rather non-trivial way

Strong-coupling expansion and effective hamiltonians

When looking for analytical approaches to treat frustrated quantum magnets, it is often very useful to start from a limit where the ground state is highly degenerate. This chapter discusses several ways of deriving {effective Hamiltonians} around such limits, starting from standard {degenerate perturbation theory} and proceeding to modern approaches more appropriate for the derivation of high-order effective Hamiltonians, such as the perturbative continuous unitary transformations or contractor renormalization. In the course of this exposition, a number of examples taken from the recent literature are discussed, including frustrated ladders and other dimer-based Heisenberg models in a field, as well as the mapping between frustrated Ising models in a transverse field and quantum dimer models.Comment: To appear as a chapter in "Highly Frustrated Magnetism", Eds. C. Lacroix, P. Mendels, F. Mil