Efficient representation of fully many-body localized systems using tensor networks

We propose a tensor network encoding the set of all eigenstates of a fully many-body localized system in one dimension. Our construction, conceptually based on the ansatz introduced in Phys. Rev. B 94, 041116(R) (2016), is built from two layers of unitary matrices which act on blocks of $\ell$ contiguous sites. We argue this yields an exponential reduction in computational time and memory requirement as compared to all previous approaches for finding a representation of the complete eigenspectrum of large many-body localized systems with a given accuracy. Concretely, we optimize the unitaries by minimizing the magnitude of the commutator of the approximate integrals of motion and the Hamiltonian, which can be done in a local fashion. This further reduces the computational complexity of the tensor networks arising in the minimization process compared to previous work. We test the accuracy of our method by comparing the approximate energy spectrum to exact diagonalization results for the random field Heisenberg model on 16 sites. We find that the technique is highly accurate deep in the localized regime and maintains a surprising degree of accuracy in predicting certain local quantities even in the vicinity of the predicted dynamical phase transition. To demonstrate the power of our technique, we study a system of 72 sites and we are able to see clear signatures of the phase transition. Our work opens a new avenue to study properties of the many-body localization transition in large systems.Comment: Version 2, 16 pages, 16 figures. Larger systems and greater efficienc

Towards a more realistic population of bright spiral galaxies in cosmological simulations

We present an update to the multiphase SPH galaxy formation code by Scannapieco et al. We include a more elaborate treatment of the production of metals, cooling rates based on individual element abundances, and a scheme for the turbulent diffusion of metals. Our SN feedback model now transfers energy to the ISM in kinetic and thermal form, and we include a prescription for the effects of radiation pressure from massive young stars on the ISM. We calibrate our new code on the well studied Aquarius haloes and then use it to simulate a sample of 16 galaxies with halo masses between 1x10^11 and 3x10^12 M_sun. In general, the stellar masses of the sample agree well with the stellar mass to halo mass relation inferred from abundance matching techniques for redshifts z=0-4. There is however a tendency to overproduce stars at z>4 and to underproduce them at z<0.5 in the least massive haloes. Overly high SFRs at z<1 for the most massive haloes are likely connected to the lack of AGN feedback in our model. The simulated sample also shows reasonable agreement with observed star formation rates, sizes, gas fractions and gas-phase metallicities at z=0-3. Remaining discrepancies can be connected to deviations from predictions for star formation histories from abundance matching. At z=0, the model galaxies show realistic morphologies, stellar surface density profiles, circular velocity curves and stellar metallicities, but overly flat metallicity gradients. 15 out of 16 of our galaxies contain disk components with kinematic disk fraction ranging between 15 and 65 %. The disk fraction depends on the time of the last destructive merger or misaligned infall event. Considering the remaining shortcomings of our simulations we conclude that even higher kinematic disk fractions may be possible for LambdaCDM haloes with quiet merger histories, such as the Aquarius haloes.Comment: 26 pages, 20 figures, accepted for publication in MNRA

Statistical and Dynamical Downscaling of Numerical Climate Simulations : Enhancement and Evaluation for East Asia

The overall aim of this thesis is to present methods, which improve evaluating dynamical downscaling approaches or enhance statistical downscaling schemes. These methods are illustrated along examples of both approaches for the East Asian region. The transfer of information from a large scale to a smaller scale is referred to as downscaling. Two different approaches are employed in climate science for downscaling purposes, i.e. textit{dynamical downscaling} and textit{statistical downscaling}. In order to give a better description of the downscaled data, this thesis presents methods, which help evaluating and interpreting the data and results of further studies in a better way, for both approaches. Dynamical Downscaling is based on a spatially limited atmospheric general circulation model, a so-called regional climate model (RCM). At the boundaries of the RCM lateral boundary conditions (LBC) are provided by a climate simulation performed with a global general circulation model (GCM). This thesis proposes methods for evaluating RCM simulations. First, a qualitative evaluation, that investigates whether single atmospheric dynamics are resolved by the RCM, is presented. Second, a newly developed evaluation method, that investigates by cross-spectral analysis on which temporal scales a RCM is able to generate variability independently from the GCM defining the LBC, is introduced. To this end, cross-spectra are estimated point-to-point between the RCM and a bi-linearly interpolated version of the GCM defining the LBC. Both methods are illustrated along RCM simulations performed for a domain covering East Asia. The RCM COSMO-CLM has been adapted for this purpose, and was driven by climate simulations performed with ECHAM5 and the re-analysis ERA-40 at its boundaries. The qualitative evaluation shows that both summer monsoon and winter monsoon dynamics are resolved by COSMO-CLM. The cross-spectral analysis suggests that the potential of COSMO-CLM to generate variability independently from the GCM depends on both dynamical features, i.e. monsoons and inter-tropical convergence zone, and on numerical parameters, i.e. horizontal resolution and domain extension. Statistical downscaling is based on statistical transfer functions between the output of large scale climate simulations and observations on the local scale. While an abundance of statistical methods for this kind of purpose are available, it is crucial from case to case to find physically meaningful predictors, which allow further interpretations of the results. Deriving and applying such predictors is demonstrated along a statistical downscaling study for precipitation properties in the Poyang catchment in Eastern China. The dichotomous variable, if 24~h accumulated rainfall exceeds a certain threshold, is taken from local rain gauges for summer. Empirical orthogonal functions (EOF) are calculated for relative vorticity at 850~hPa and vertical velocity at 500~hPa taken from ERA-40 re-analysis data. Both information are linked by logistic regression. The most dominant EOF-predictor can be associated with meso-$alpha$-scale disturbances, which are part of the summer monsoon dynamics in this region. Downscaled data is often requested for further studies in climate science, but also in other disciplines. Thus, developing evaluation methods for assessing the quality of RCM simulations, and deriving physically interpretable predictors for statistical downscaling schemes are crucial enhancements for the downscaling procedure

Regularized Newton Methods for X-ray Phase Contrast and General Imaging Problems

Like many other advanced imaging methods, x-ray phase contrast imaging and tomography require mathematical inversion of the observed data to obtain real-space information. While an accurate forward model describing the generally nonlinear image formation from a given object to the observations is often available, explicit inversion formulas are typically not known. Moreover, the measured data might be insufficient for stable image reconstruction, in which case it has to be complemented by suitable a priori information. In this work, regularized Newton methods are presented as a general framework for the solution of such ill-posed nonlinear imaging problems. For a proof of principle, the approach is applied to x-ray phase contrast imaging in the near-field propagation regime. Simultaneous recovery of the phase- and amplitude from a single near-field diffraction pattern without homogeneity constraints is demonstrated for the first time. The presented methods further permit all-at-once phase contrast tomography, i.e. simultaneous phase retrieval and tomographic inversion. We demonstrate the potential of this approach by three-dimensional imaging of a colloidal crystal at 95 nm isotropic resolution.Comment: (C)2016 Optical Society of America. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibite