Fakultät für Physik - Digitale Hochschulschriften der LMU - Teil 04/05

This thesis is concerned with two main topics: first, the advancement of the density matrix renormalization group (DMRG) and, second, its applications. In the first project of this thesis we exploit the common mathematical structure of the numerical renormalization group and the DMRG, namely, matrix product states (MPS), to implement an efficient numerical treatment of a two-lead, multi-level Anderson impurity model. By adopting a star-like geometry, where each species (spin and lead) of conduction electrons is described by its own so-called Wilson chain, instead of a single Wilson chain we achieve a very significant reduction in the numerical resources required to obtain reliable results.

Moreover, we show that it is possible to find an "optimal" chain basis, in which chain degrees of freedom of different Wilson chains become effectively decoupled from each other further out on the Wilson chains. This basis turns out to also diagonalize the model's chain-to-chain scattering matrix.

In the second project we show that Chebychev expansions offer numerically efficient representations for calculating spectral functions of one-dimensional lattice models using MPS methods. The main features of this Chebychev matrix product state (CheMPS) approach are: (i) it achieves uniform resolution over the spectral function's entire spectral width; (ii) it offers a well-controlled broadening scheme; (iii) it is based on using MPS tools to recursively calculate a succession of Chebychev vectors, (iv) whose entanglement entropies were found to remain bounded with increasing recursion order for all cases analyzed here. We present CheMPS results for the structure factor of spin-1/2 antiferromagnetic Heisenberg chains and perform a detailed finite-size analysis. Making comparisons to benchmark methods, we find that CheMPS yields results comparable in quality to those of correction vector DMRG, at dramatically reduced numerical cost and agrees well with Bethe Ansatz results for an infinite system, within the limitations expected for numerics on finite systems.

Following these technologically focused projects we study the so-called Kondo cloud by means of the DMRG in the third project.

The Kondo cloud describes the effect of spatially extended spin-spin correlations of a magnetic moment and the conduction electrons which screen the magnetic moment through the Kondo effect at low temperatures. We focus on the question whether the Kondo screening length, typically assumed to be proportional to the inverse Kondo temperature, can be extracted from the spin-spin correlations. We investigate how perturbations which destroy the Kondo effect, like an applied gate potential or a magnetic field, affect the formation of the screening cloud.

In a forth project we address the impact of Quantum (anti-)Zeno physics resulting from repeated single-site resolved observations on the many-body dynamics. We use time-dependent DMRG to obtain the time evolution of the full many-body wave function that is then periodically projected in order to simulate realizations of stroboscopic measurements. For the example of a 1-D lattice of spin-polarized fermions with nearest-neighbor interactions, we find regimes for which many-particle configurations are stabilized and destabilized depending on the interaction strength and the time between observations.

Cells are the fundamental units of which all life forms are composed. To understand the elementary organization of life, it is therefore meaningful to start the investigation on the single cell level.Modern microscopy permits the examination of both subcellular processes and collective microbial behavior. In this microscopic regime, fluctuations are of eminent importance. As this noise is an inherent property of such systems, life evolved robust systems, which work effectively in spite of severe fluctuations. Moreover, life also makes use of these fluctuations for its benefit.

For modeling purposes in this field of research, concepts from statistical mechanics and from the analysis of stochastic processes can be applied to account for the fluctuations.

This work is roughly divided into two parts, which also address the background, concepts and literature of the corresponding topics.

The first part is concerned with the modeling of intracellular processes, for which noise is important. In this context two publications, which arose from collaborations with experimental biophysicists, are discussed:

In gene therapy external genetic material is injected into cells to remedy deficient behavior. To characterize this process, fluorophore encoding plasmids were administered to eukaryotic cells by means of two chemical transfection methods. The distribution of expression levels is explained by several strongly stochastic steps during transfection and subsequent quasi-deterministic gene expression.

The second collaboration addresses the switching kinetics between different phenotypes in bacteria. In the case at hand, the emergence of "competence" in B. subtilis is studied. This ability (to take up genetic material from the extracellular medium) is strongly regulated by a network of interacting genes. While the different phenotypes are associated with stable fixed points of non-linear differential equations, switching between phenotypes relies on fluctuations in the small number of mRNA molecules.

The second part of this work elaborates on collective, stochastic growth of many cells in an expanding colony. The corresponding manuscript analyzes a theoretical model with methods from statistical mechanics.

Microbial colony growth is sometimes seen as a model system for range expansion or colonization processes. Inspired by experiments, a stochastic surface growth process, in the form of a generalized Eden model, is set up and analyzed. The model explicitly takes into account selection between two strains, irreversible mutations, and the roughness of the propagating colony front. The asymmetric character of mutations implies the existence of an absorbing state, where only the mutant strain is at the front of the expanding population. Hence, the model combines two interesting branches of non-equilibrium statistical mechanics: phase transitions to absorbing states and dynamic surface roughening. As usual for these processes, one can define critical exponents, which describe the divergence of observables near the phase transition, and admit organization of models into universality classes. It turns out that the coupling between roughening dynamics and population dynamics induces qualitative different behavior. As a consequence, the model cannot be assigned to any universality class currently known.

The aim of this thesis is to study the environmental dependence of populations of low-mass X-ray binaries (LMXBs) in early-type galaxies. The dissertation is mainly based on archival data of the Chandra observatory, complemented by the infrared data of the Spitzer observatory as well as by the results of the Hubble Space Telescope and ground based optical observations. The dissertation investigates the dependence of LMXB populations on the stellar density, velocity dispersion and the age of the stellar population.

To investigate the LMXB dependence on the stellar density and velocity dispersion, we studied different sub populations of low-mass X-ray binaries -- dynamically formed systems in globular clusters and in the nucleus of M31, and presumably primordial X-ray binaries in the fields of galaxies. We found that the luminosity distributions of globular cluster and field LMXBs differ throughout the entire luminosity range. The fraction of faint (Lx < E37 erg/s) sources in globular clusters is ~ 4 times smaller than in the field population. This may present a challenge for the models suggesting that the entire LMXB population was formed dynamically in globular clusters and then expelled to the field due to dynamical interactions or as globular clusters dissolve. The luminosity function of dynamically formed sources in the nucleus of M31 is similar to that of globular cluster sources at the faint end but differs at the bright end, that the M31 nucleus hosting significantly fewer bright sources than globular clusters (and field population). The difference between their luminosity distributions is likely caused by the factor of ~ 10-20 difference in stellar velocities in globular clusters and galactic nuclei, which leads to different dynamical formation channels.

In order to investigate the LMXB dependence on the stellar age we collected a sample of 20 nearby massive E/S0 galaxies covering a broad range of stellar ages from ~ 1 to > 10 Gyrs. We found statistically significant correlation of the specific frequency of LMXBs (number per unit stellar mass) with the age of the host galaxy and its globular cluster content. Overall, older galaxies tend to have more X-ray binaries per unit stellar mass than the younger ones. This can be explained as a combined effect of two factors. On one hand, there appears to be an intrinsic correlation of the specific number of low-mass X-ray binaries in the field with the age of the stellar population. On the other hand, the older galaxies tend to have richer globular cluster population and therefore a larger number of dynamically formed X-ray sources associated with them. We also found a significant difference in the luminosity functions of low-mass X-ray

binaries in younger and older galaxies, with the former having a flatter luminosity distribution and hosting more bright sources. This behavior is unrelated to the difference between luminosity distributions of the globular cluster and field sources and reflects the intrinsic age dependence of the LMXB populations.

The Antennae galaxies (NGC 4038/39) are the nearest and best-studied major

merger of two gas-rich spirals in the local Universe. They are

named after the characteristic pair of tidal tails that protrude out of their

main galactic disks. Due to their proximity the

Antennae are extremely well sampled by modern high-resolution observations

over an enormous wavelength range, from radio to X-ray. This allows for

a comprehensive multiwavelength approach to the present-day morphology

and dynamical history of the system. The goal of this Thesis is to test the available high-quality multiwavelength data against new high-resolution merger simulations as a key to improve our understanding of several

merger-induced physical processes, such as starbursts and the

formation of young star clusters. These processes are expected to have

been even more important in the formation and early evolution of

galaxies.

First of all, accurate initial conditions for the interaction orbit

and the initial galaxy models need to be constrained. To this end,

we perform an extended parameter search in order to obtain a suitable

match to the Antennae galaxies.

Using these new initial conditions, we are able to present the

first high-resolution numerical simulations that successfully

match the detailed large- and small-scale morphology and kinematics of the Antennae

galaxies. Moreover, the spatial

distribution of star-forming regions as well as the

total star formation rate are reproduced in excellent quantitative

agreement, in particular, if we adopt a very weak stellar feedback.

We find that the Antennae are currently in a special

phase of their evolution, shortly after the second pericenter. They

will merge soon within the next ~50 Myr.

In addition, we compare the star formation histories of all

published hydrodynamical Antennae simulations with the observed

cluster age distribution. The latter is well approximated by a

power-law dN/dt ~ t^gamma, declining as gamma =

1. Under the assumption that clusters form at rates proportional to

the total star formation, it is found that the variations in the

simulated formation histories alone cannot account for most of the

steep decline in the observed age distribution. This provides strong

evidence for efficient, prolonged cluster disruption in the Antennae,

similar to more quiescent galaxies.

Finally, we address the question of whether the Antennae will

evolve into a typical elliptical galaxy. We find that

the virialized merger remnant resembles an oblate, fast-rotating early-type

with surface brightness profile well fitted by a Sersic function of

index n ~ 5. For high metallicities (Z > Z_sun) the

stellar remnant of the Antennae may add to a population of present-day

ellipticals after secular evolution of another ~2.5 -3 Gyr.

Within this Thesis, we present an improved numerical

model for the interacting Antennae galaxies that may serve as a

test-bed for further investigations of this archetypal

merger. As next steps, we plan to investigate the extended hot gas

component found in X-ray observations of the Antennae, and test our

present results in a code comparison project.

We discuss exclusive charmless hadronic B decays. In particular we perform a phenomenological analysis of the decay into 2 longitudinal vector mesons and present analytical integral-expressions for the decay into proton and antiproton. The light nucleon form factor is part of the decay amplitude into protons and is calculated and regularized explicitly at subleading power in the large energy limit.

Femtosecond stimulated Raman microscopy (FSRM) is an upcoming technique in vibrational microscopy that combines optical imaging with the chemical sensitivity of Raman scattering. It allows for the quantitative, space resolved representation of microscopic samples. In FSRM, chemical contrast relies on femtosecond stimulated Raman scattering (FSRS), a nonlinear Raman process providing signal levels, that are strongly amplified with respect to spontaneous Raman scattering. Therefore, FSRM benefits from shortened exposure times and improved signal/noise-ratios. Even though FSRS is an optically nonlinear effect, its spectral signature resembles the (polarised) spontaneous Raman spectra. Autofluorescence of the sample does not contribute to the FSRS signature. FSRM turns out to be suitable for chemical imaging of biological systems, as cells and tissues. Since their signal strengths are quite low, data with high signal/noise-ratios is essential for the specific

analysis of the sample. These requirements can be provided by FSRM. Moreover, the linear dependency of the FSRS signal on the sample concentration facilitates the quantitative

analysis of its molecular composition. Given that tissue shows a wide heterogeneity on the molecular level, spectrally broad information, which is indeed provided by FSRM, is indispensable for chemically sensitive imaging. The intention of the present work was to establish FSRM as a tool for biological and medical imaging. FSRM is a scanning technique that relies on the nonlinear interaction of two ultrashort laser pulses with a Raman active medium. One of them is a spectrally narrow and intense picosecond pulse (Raman pump pulse), the other a spectrally broad and weak femtosecond pulse (Raman probe pulse). Both pulses are coupled collinearly into a scanning microscope where stimulated Raman scattering occurs in the sample positioned at the focus. This interaction leads to a spectral modification of the Raman probe pulse, which is recorded by the aid of a multi-channel detector. Its FSRS spectrum can be obtained by referencing the femtosecond pulse in presence of the Raman pump pulse to the one in absence. By raster-scanning the sample space-resolved FSRS spectra are retrieved. In the first FSRM setup a laser/amplifier system served as laser source. Its peak intensities are not compatible with the requirements of biological systems. Therefore a new light source has been developed for FSRM. It is based on a laser oscillator running at high repetition rate, which provides ultrashort femtosecond pulses with a spectral width over more than 3000 cm-1. These serve directly as Raman probe pulses. The Raman pump pulse is generated out of the laser spectrum by means of a ytterbium based fiber amplifier.

Stimulated Raman microscopy is considered as a disturbance-free, nonlinear microscopy technique. In the context of this work, a further nonlinear contribution

to the FSRS signal is reported for the first time. It shows up as a strong oscillation of the baseline in the spectrum and deteriorates the signal/noise-ratio as well as the spectral selectivity. This background can be identified as a spectral interference between the Raman probe pulse and an additional electric field, which is generated by a four-wave-mixing process between the Raman pump and Raman probe pulse. Properties and methods to suppress the spectral interferences are presented.

In the work for this thesis, a split-mirror-setup was designed and build, which was used to split the XUV laser-pulse of FELs (Free Electron Laser) into two identical pulses from which one can be delayed. With this setup the laser pulses of FLASH, Hamburg and SCSS, Harima(Japan) where characterized temporally, to determine the temporal pulse-structure for subsequent experiments. The intermolecular dynamics of the homonuclear diatomic molecules nitrogen and oxygen were examined and the experimental results were reproduced by classical simulations. In the measurement with oxygen for an energy band of the coincident singly charged ions, an ionization probability was found that depends on the delay between the two XUV-pulses. This can most probably be explained by the autoionization of an excited singly charged molecular state. Subsequently the investigation of the two photon double ionization (TPDI) of deuterium is presented. In the single pulse experiments simulations within the Born-Oppenheimer approximation made it possible to distinguish between the direct and sequential TPDI. In the pump-probe experiments light was shed onto the dynamics of the TPDI. In addition, experiments with strong few-cycle near-infrared (NIR) pulses are presented that examined the carrier envelope phase (CEP) dependence of the non-sequential double ionization of argon. Implementing single-shot CEP-tagging in conjunction with coincidence spectroscopy allowed to achieve unprecedented accuracy in measuring correlated electron dynamics.