Statistical mechanics of polymer models: rigorous results and applications

Monomer-dimer models are amongst the models in statistical mechanics which found application in many areas of science, ranging from biology to social sciences. This model describes a many-body system in which monoatomic and diatomic particles subject to hard-core interactions get deposited on a graph. In our work we provide an extension of this model to higher-order particles. The aim of our work is threefold: first we study the thermodynamic properties of the newly introduced model. We solve analytically some regular cases and find that, differently from the original, our extension admits phase transitions. Then we tackle the inverse problem, both from an analytical and numerical perspective. Finally we propose an application to aggregation phenomena in virtual messaging services.

The interplay of localization and interactions in quantum many-body systems

Disorder and interactions both play crucial roles in quantum transport. Decades ago, Mott showed that electron-electron interactions can lead to insulating behavior in materials that conventional band theory predicts to be conducting. Soon thereafter, Anderson demonstrated that disorder can localize a quantum particle through the wave interference phenomenon of Anderson localization. Although interactions and disorder both separately induce insulating behavior, the interplay of these two ingredients is subtle and often leads to surprising behavior at the periphery of our current understanding. Modern experiments probe these phenomena in a variety of contexts (e.g. disordered superconductors, cold atoms, photonic waveguides, etc.); thus, theoretical and numerical advancements are urgently needed. In this thesis, we report progress on understanding two contexts in which the interplay of disorder and interactions is especially important.

The first is the so-called “dirty” or random boson problem. In the past decade, a strong-disorder renormalization group (SDRG) treatment by Altman, Kafri, Polkovnikov, and Refael has raised the possibility of a new unstable fixed point governing the superfluid-insulator transition in the one-dimensional dirty boson problem. This new critical behavior may take over from the weak-disorder criticality of Giamarchi and Schulz when disorder is sufficiently strong. We analytically determine the scaling of the superfluid susceptibility at the strong-disorder fixed point and connect our analysis to recent Monte Carlo simulations by Hrahsheh and Vojta. We then shift our attention to two dimensions and use a numerical implementation of the SDRG to locate the fixed point governing the superfluid-insulator transition there. We identify several universal properties of this transition, which are fully independent of the microscopic features of the disorder.

The second focus of this thesis is the interplay of localization and interactions in systems with high energy density (i.e., far from the usual low energy limit of condensed matter physics). Recent theoretical and numerical work indicates that localization can survive in this regime, provided that interactions are sufficiently weak. Stronger interactions can destroy localization, leading to a so-called many-body localization transition. This dynamical phase transition is relevant to questions of thermalization in isolated quantum systems: it separates a many-body localized phase, in which localization prevents transport and thermalization, from a conducting (“ergodic”) phase in which the usual assumptions of quantum statistical mechanics hold. Here, we present evidence that many-body localization also occurs in quasiperiodic systems that lack true disorder.

Inverse problem for the wave equation : partial data and novel boundary sources

An inverse problem for the wave equation is a mathematical formulation of the problem to convert measurements of sound waves to information about the wave speed governing the propagation of the waves. This doctoral thesis extends the theory on the inverse problems for the wave equation in cases with partial measurement data and also considers detection of discontinuous interfaces in the wave speed. A possible application of the theory is obstetric sonography in which ultrasound measurements are transformed into an image of the fetus in its mother's uterus. The wave speed inside the body can not be directly observed but sound waves can be produced outside the body and their echoes from the body can be recorded. The present work contains five research articles. In the first and the fifth articles we show that it is possible to determine the wave speed uniquely by using far apart sound sources and receivers. This extends a previously known result which requires the sound waves to be produced and recorded in the same place. Our result is motivated by a possible application to reflection seismology which seeks to create an image of the Earth s crust from recording of echoes stimulated for example by explosions. For this purpose, the receivers can not typically lie near the powerful sound sources. In the second article we present a sound source that allows us to recover many essential features of the wave speed from the echo produced by the source. Moreover, these features are known to determine the wave speed under certain geometric assumptions. Previously known results permitted the same features to be recovered only by sequential measurement of echoes produced by multiple different sources. The reduced number of measurements could increase the number possible applications of acoustic probing. In the third and fourth articles we develop an acoustic probing method to locate discontinuous interfaces in the wave speed. These interfaces typically correspond to interfaces between different materials and their locations are of interest in many applications. There are many previous approaches to this problem but none of them exploits sound sources varying freely in time. Our use of more variable sources could allow more robust implementation of the probing.

Ultra-sensitive pressure dependence of bandgap of rutile-GeO2 revealed by many body perturbation theory

The reported values of bandgap of rutile GeO2 calculated by the standard density functional theory within local-density approximation (LDA)/generalized gradient approximation (GGA) show a wide variation (similar to 2 eV), whose origin remains unresolved. Here, we investigate the reasons for this variation by studying the electronic structure of rutile-GeO2 using many-body perturbation theory within the GW framework. The bandgap as well as valence bandwidth at Gamma-point of rutile phase shows a strong dependence on volume change, which is independent of bandgap underestimation problem of LDA/GGA. This strong dependence originates from a change in hybridization among O-p and Ge-(s and p) orbitals. Furthermore, the parabolic nature of first conduction band along X-Gamma-M direction changes towards a linear dispersion with volume expansion. (C) 2015 AIP Publishing LLC.

Many-body theory of positron-atom interactions

A many-body theory approach is developed for the problem of positron-atom scattering and annihilation. Strong electron- positron correlations are included nonperturbatively through the calculation of the electron-positron vertex function. It corresponds to the sum of an infinite series of ladder diagrams, and describes the physical effect of virtual positronium formation. The vertex function is used to calculate the positron-atom correlation potential and nonlocal corrections to the electron-positron annihilation vertex. Numerically, we make use of B-spline basis sets, which ensures rapid convergence of the sums over intermediate states. We have also devised an extrapolation procedure that allows one to achieve convergence with respect to the number of intermediate- state orbital angular momenta included in the calculations. As a test, the present formalism is applied to positron scattering and annihilation on hydrogen, where it is exact. Our results agree with those of accurate variational calculations. We also examine in detail the properties of the large correlation corrections to the annihilation vertex.

Many-body theory calculations of positron binding to negative ions

A many-body theory approach developed by the authors [Phys. Rev. A 70, 032720 (2004)] is applied to positron bound states and annihilation rates in atomic systems. Within the formalism, full account of virtual positronium (Ps) formation is made by summing the electron-positron ladder diagram series, thus enabling the theory to include all important many-body correlation effects in the positron problem. Numerical calculations have been performed for positron bound states with the hydrogen and halogen negative ions, also known as Ps hydride and Ps halides. The Ps binding energies of 1.118, 2.718, 2.245, 1.873 and 1.393 eV and annihilation rates of 2.544, 2.482, 1.984, 1.913 and 1.809 ns^{-1}, have been obtained for PsH, PsF, PsCl, PsBr and PsI, respectively.

Geometric and Combinatorial Aspects of NonEquilibrium Statistical Mechanics

Non-Equilibrium Statistical Mechanics is a broad subject. Grossly speaking, it deals with systems which have not yet relaxed to an equilibrium state, or else with systems which are in a steady non-equilibrium state, or with more general situations. They are characterized by external forcing and internal fluxes, resulting in a net production of entropy which quantifies dissipation and the extent by which, by the Second Law of Thermodynamics, time-reversal invariance is broken. In this thesis we discuss some of the mathematical structures involved with generic discrete-state-space non-equilibrium systems, that we depict with networks in all analogous to electrical networks. We define suitable observables and derive their linear regime relationships, we discuss a duality between external and internal observables that reverses the role of the system and of the environment, we show that network observables serve as constraints for a derivation of the minimum entropy production principle. We dwell on deep combinatorial aspects regarding linear response determinants, which are related to spanning tree polynomials in graph theory, and we give a geometrical interpretation of observables in terms of Wilson loops of a connection and gauge degrees of freedom. We specialize the formalism to continuous-time Markov chains, we give a physical interpretation for observables in terms of locally detailed balanced rates, we prove many variants of the fluctuation theorem, and show that a well-known expression for the entropy production due to Schnakenberg descends from considerations of gauge invariance, where the gauge symmetry is related to the freedom in the choice of a prior probability distribution. As an additional topic of geometrical flavor related to continuous-time Markov chains, we discuss the Fisher-Rao geometry of nonequilibrium decay modes, showing that the Fisher matrix contains information about many aspects of non-equilibrium behavior, including non-equilibrium phase transitions and superposition of modes. We establish a sort of statistical equivalence principle and discuss the behavior of the Fisher matrix under time-reversal. To conclude, we propose that geometry and combinatorics might greatly increase our understanding of nonequilibrium phenomena.

Landau-Zener problem with waiting at the minimum gap and related quench dynamics of a many-body system

We discuss a technique for solving the Landau-Zener (LZ) problem of finding the probability of excitation in a two-level system. The idea of time reversal for the Schrodinger equation is employed to obtain the state reached at the final time and hence the excitation probability. Using this method, which can reproduce the well-known expression for the LZ transition probability, we solve a variant of the LZ problem, which involves waiting at the minimum gap for a time t(w); we find an exact expression for the excitation probability as a function of t(w). We provide numerical results to support our analytical expressions. We then discuss the problem of waiting at the quantum critical point of a many-body system and calculate the residual energy generated by the time-dependent Hamiltonian. Finally, we discuss possible experimental realizations of this work.

Free turbulent shear layer in a point vortex gas as a problem in nonequilibrium statistical mechanics

This paper attempts to unravel any relations that may exist between turbulent shear flows and statistical mechanics through a detailed numerical investigation in the simplest case where both can be well defined. The flow considered for the purpose is the two-dimensional (2D) temporal free shear layer with a velocity difference Delta U across it, statistically homogeneous in the streamwise direction (x) and evolving from a plane vortex sheet in the direction normal to it (y) in a periodic-in-x domain L x +/-infinity. Extensive computer simulations of the flow are carried out through appropriate initial-value problems for a ``vortex gas'' comprising N point vortices of the same strength (gamma = L Delta U/N) and sign. Such a vortex gas is known to provide weak solutions of the Euler equation. More than ten different initial-condition classes are investigated using simulations involving up to 32 000 vortices, with ensemble averages evaluated over up to 10(3) realizations and integration over 10(4)L/Delta U. The temporal evolution of such a system is found to exhibit three distinct regimes. In Regime I the evolution is strongly influenced by the initial condition, sometimes lasting a significant fraction of L/Delta U. Regime III is a long-time domain-dependent evolution towards a statistically stationary state, via ``violent'' and ``slow'' relaxations P.-H. Chavanis, Physica A 391, 3657 (2012)], over flow time scales of order 10(2) and 10(4)L/Delta U, respectively (for N = 400). The final state involves a single structure that stochastically samples the domain, possibly constituting a ``relative equilibrium.'' The vortex distribution within the structure follows a nonisotropic truncated form of the Lundgren-Pointin (L-P) equilibrium distribution (with negatively high temperatures; L-P parameter lambda close to -1). The central finding is that, in the intermediate Regime II, the spreading rate of the layer is universal over the wide range of cases considered here. The value (in terms of momentum thickness) is 0.0166 +/- 0.0002 times Delta U. Regime II, extensively studied in the turbulent shear flow literature as a self-similar ``equilibrium'' state, is, however, a part of the rapid nonequilibrium evolution of the vortex-gas system, which we term ``explosive'' as it lasts less than one L/Delta U. Regime II also exhibits significant values of N-independent two-vortex correlations, indicating that current kinetic theories that neglect correlations or consider them as O(1/N) cannot describe this regime. The evolution of the layer thickness in present simulations in Regimes I and II agree with the experimental observations of spatially evolving (3D Navier-Stokes) shear layers. Further, the vorticity-stream-function relations in Regime III are close to those computed in 2D Navier-Stokes temporal shear layers J. Sommeria, C. Staquet, and R. Robert, J. Fluid Mech. 233, 661 (1991)]. These findings suggest the dominance of what may be called the Kelvin-Biot-Savart mechanism in determining the growth of the free shear layer through large-scale momentum and vorticity dispersal.

Accurate localized resolution of identity approach for linear-scaling hybrid density functionals and for many-body perturbation theory

© 2015 IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.A key component in calculations of exchange and correlation energies is the Coulomb operator, which requires the evaluation of two-electron integrals. For localized basis sets, these four-center integrals are most efficiently evaluated with the resolution of identity (RI) technique, which expands basis-function products in an auxiliary basis. In this work we show the practical applicability of a localized RI-variant ('RI-LVL'), which expands products of basis functions only in the subset of those auxiliary basis functions which are located at the same atoms as the basis functions. We demonstrate the accuracy of RI-LVL for Hartree-Fock calculations, for the PBE0 hybrid density functional, as well as for RPA and MP2 perturbation theory. Molecular test sets used include the S22 set of weakly interacting molecules, the G3 test set, as well as the G2-1 and BH76 test sets, and heavy elements including titanium dioxide, copper and gold clusters. Our RI-LVL implementation paves the way for linear-scaling RI-based hybrid functional calculations for large systems and for all-electron many-body perturbation theory with significantly reduced computational and memory cost.

Many-body theory of gamma spectra from positron-atom annihilation

A many-body theory approach to the calculation of gamma spectra of positron annihilation on many-electron atoms is developed. We evaluate the first-order correlation correction to the annihilation vertex and perform numerical calculations for the noble gas atoms. Extrapolation with respect to the maximal orbital momentum of the intermediate electron and positron states is used to achieve convergence. The inclusion of correlation corrections improves agreement with experimental gamma spectra.

Electronic properties of interfaces and defects from many-body perturbation theory: Recent developments and applications

We review some recent developments in many body perturbation theory (MBPT) calculations that have enabled the study of interfaces and defects. Starting from the theoretical basis of MBPT, Hedin's equations are presented, leading to the CW and CWI' approximations. We introduce the perturbative approach, that is the one most commonly used for obtaining quasiparticle (QP) energies. The practical strategy presented for dealing with the frequency dependence of the self energy operator is based on either plasmon-pole models (PPM) or the contour deformation technique, with the latter being more accurate. We also discuss the extrapolar method for reducing the number of unoccupied states which need to be included explicity in the calculations. The use of the PAW method in the framework of MBPT is also described. Finally, results which have been obtained using, MBPT for band offsets a interfaces and for defects presented, with companies on the main difficulties and cancels.

Schematic representation of the QP corrections (marked with ) to the band edges (E and E-v) and a defect level (F) for a Si/SiO2 interface (Si and O atoms are represented in blue and red, respectively, in the ball and stick model) with an oxygen vacancy leading to a Si-Si bond (the Si atoms involved in this bond are colored light blue).

Electronic properties of zircon and hafnon from many-body perturbation theory

The electronic properties of zircon and hafnon, two wide-gap high-kappa materials, are investigated using many-body perturbation theory (MBPT) combined with the Wannier interpolation technique. For both materials, the calculated band structures differ from those obtained within density-functional theory and MBPT by (i) a slight displacement of the highest valence-band maximum from the Gamma point and (ii) an opening of the indirect band gap to 7.6 and 8.0 eV for zircon and hafnon, respectively. The introduction of vertex corrections in the many-body self-energy does not modify the results except for a global rigid shift of the many-body corrections.

Density functionals from many-body perturbation theory: The band gap for semiconductors and insulators

Theoretically the Kohn-Sham band gap differs from the exact quasiparticle energy gap by the derivative discontinuity of the exchange-correlation functional. In practice for semiconductors and insulators the band gap calculated within any local or semilocal density approximations underestimates severely the experimental energy gap. On the other hand, calculations with an "exact" exchange potential derived from many-body perturbation theory via the optimized effective potential suggest that improving the exchange-correlation potential approximation can yield a reasonable agreement between the Kohn-Sham band gap and the experimental gap. The results in this work show that this is not the case. In fact, we add to the exact exchange the correlation that corresponds to the dynamical (random phase approximation) screening in the GW approximation. This accurate exchange-correlation potential provides band structures similar to the local density approximation with the corresponding derivative discontinuity that contributes 30%-50% to the energy gap. Our self-consistent results confirm substantially the results for Si and other semiconductors obtained perturbatively [R. W. Godby , Phys. Rev. B 36, 6497 (1987)] and extend the conclusion to LiF and Ar, a wide-gap insulator and a noble-gas solid. (c) 2006 American Institute of Physics.

A Statistical Model for Shadowed Body Centric Communications Channels: Theory and Validation

This paper presents a new statistical signal reception model for shadowed body-centric communications channels. In this model, the potential clustering of multipath components is considered alongside the presence of elective dominant signal components. As typically occurs in body-centric communications channels, the dominant or line-of-sight (LOS) components are shadowed by body matter situated in the path trajectory. This situation may be further exacerbated due to physiological and biomechanical movements of the body. In the proposed model, the resultant dominant component which is formed by the phasor addition of these leading contributions is assumed to follow a lognormal distribution. A wide range of measured and simulated shadowed body-centric channels considering on-body, off-body and body-to-body communications are used to validate the model. During the course of the validation experiments, it was found that, even for environments devoid of multipath or specular reflections generated by the local surroundings, a noticeable resultant dominant component can still exist in body-centric channels where the user's body shadows the direct LOS signal path between the transmitter and the receiver.