This thesis explores the effects singularities have on stationary and dynamical properties of many-body quantum systems. In papers I and II we find that the ground state suffers a Z2 symmetry breaking phase transition (PT) when a single impurity is added to a Bose-Einstein condensate (BEC) in a double well (bosonic Josephson junction). The PT occurs for a certain value of the BEC-impurity interaction energy, Λc . A result of the PT is the mean-field dynamics undergo chaotic motion in phase space once the symmetry is broken. We determine the critical scaling exponents that characterize the divergence of the correlation length and fidelity susceptibility at the PT, finding that the BEC-impurity system belongs to the same universality class as the Dicke and Lipkin-Meshkov-Glick models (which also describe symmetry breaking PTs in systems of bosons). In paper III we study the dynamics of a generic two-mode quantum field following a quench where one of the terms in the Hamiltonian is flashed on and off. This model is relevant to BECs in double wells as well as other simple many-particle systems found in quantum optics and optomechanics. We find that when plotted in Fock-space plus time, the semiclassical wave function develops prominent cusp-shaped structures after the quench. These structures are singular in the classical limit and we identify them as catastrophes (as described by the Thom-Arnold catastrophe theory) and show that they arise from the coalescence of classical (mean-field) trajectories in a path integral description. Furthermore, close to the cusp the wave function obeys a remarkable set of scaling relations signifying these structures as examples of universality in quantum dynamics. Within the cusp we find a network of vortex-antivortex pairs which are phase singularities caused by interference. When the mean-field Hamiltonian displays a Z2 symmetry breaking PT modelled by the Landau theory of PTs we calculate scaling exponents describing how the separation distance between the members of each pair diverges as the PT is approached. We also find that the cusp becomes infinitely stretched out at the PT due to critical slowing down. In paper IV we investigate in greater detail the morphology of the vortex network found within cusp catastrophes in many-body wave functions following a quench. In contrast to the cusp catastrophes studied so far in the literature, these structures live in Fock space which is fundamentally granular. As such, these cusps represent a new iii type of catastrophe, which we term a ‘quantum catastrophe’. The granularity of Fock space introduces a new length scale, the quantum length lq = N −1 which effectively removes the vortex cores. Nevertheless, a subset of the vortices persist as phase singularities as can be shown by integrating the phase of the wave function around circuits in Fock-space plus time. Whether or not the vortices survive in a quantum catastrophe is governed by the separation of the vortex-antivortex pairs lv ∝ N −3/4 in comparison to lq , i.e. they survive if lv lq . When particle numbers are reached such that lq ≈ lv the vortices annihilate in pairs. / Thesis / Doctor of Philosophy (PhD)

Strong quantum correlations between many particles in low dimensions lead to emergence of interesting phases of matter. These phases are often studied through the properties of the many-body eigenstates of an interacting quantum many-body system. The folklore example of topological order in the ground states is the fractional quantum Hall (FQH) effect. With the current developments in the field of ultracold atoms in optical lattices, realizing FQH physics on a lattice and being able to create and braid anyons is much awaited from the view point of fault tolerant quantum computing. This thesis contributes to the field of FQH effect and anyons in a lattice setting. Conformal field theory has been useful to build interesting lattice FQH models which are few-body and non-local. We provide a general scheme of truncation to arrive at tractable local models whose ground states have the desired topological properties. FQH models are known to host anyons, but, it is a hard task when it comes to braiding them on small sized lattices with edges. To get around this problem, we demonstrate that one can squeeze the anyons and braid them successfully within a smaller area by crawling them like snakes on modest sized open lattices. As a numerically cheap approach to detect topological quantum phase transitions, we again resort to anyons that are only well defined in a topological phase. We create defects and study a simple quantity such as the charge of the defect to test whether the phase supports anyons or not. On the other hand, with the advent of many-body localization (MBL) and quantum many-body scars, interesting eigenstate phases which were otherwise only known to occur in ground states have been identified even at finite energy densities in the many-body spectra of generic systems. This thesis also contributes to the field of non-equilibrium physics by portraying models that display interesting non-ergodic phases and also quantum many-body scars. For instance, we show that an emergent symmetry in a disordered model can be used as a tool to escape MBL in a single eigenstate while not preventing the rest of the states from localizing. This can lead to an interesting situation of weakly broken MBL phase where a non-MBL state lives in the spectrum of MBL like states. We also demonstrate the emergence of a non-ergodic, but also a non-mbl phase in a non-local model with SU(2) symmetry. We provide two constructions of rather different models with quantum many-body scars with chiral and non-chiral topological order.

Yeung Man Yi = 多體量子系統問題 / 楊曼儀. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2006. / Includes bibliographical references (leaves [247-249]). / Text in English; abstracts in English and Chinese. / Yeung Man Yi = Duo ti liang zi xi tong wen ti / Yang Manyi. / Abstract --- p.i / Acknowledgment --- p.iii / Chapter I --- Computational Quantum Mechanics and Its Applications / 電算量子力學及其應用 --- p.1 / Chapter 1 --- Theoretical Methodology of Electronic Structures: Ab Initio Molecular Orbital Theory --- p.2 / Chapter 1.1 --- Molecular Hamiltonian --- p.2 / Chapter 1.2 --- Hartree Products --- p.5 / Chapter 1.3 --- Slater Determinants and Pauli Exclusion Principle --- p.6 / Chapter 1.4 --- Expansion of Total Electronic Energy in terms of Integrals over MOs --- p.8 / Chapter 1.5 --- Derivation of the Hartree-Fock Equations --- p.11 / Chapter 1.6 --- Orbital Energies and the Koopmans' Theorem --- p.14 / Chapter 1.7 --- AO Basic Sets --- p.17 / Chapter 1.7.1 --- Slater-Type Orbitals --- p.18 / Chapter 1.7.2 --- Gaussian Functions --- p.18 / Chapter 1.8 --- Self-Consistent Field Calculation --- p.19 / Chapter 1.9 --- Hartree-Fock Limit --- p.20 / Chapter 1.10 --- Electron Correlation --- p.20 / Chapter 1.10.1 --- Configuration Interaction --- p.20 / Chapter 1.10.2 --- Density Functional Theory --- p.21 / Chapter 2 --- Theoretical Investigation of Organic Light Emitting Molecules --- p.29 / Chapter 2.1 --- Introduction --- p.29 / Chapter 2.2 --- Methodology --- p.31 / Chapter 2.2.1 --- Theoretical Methodology --- p.31 / Chapter 2.2.2 --- Computational Methodology --- p.35 / Chapter 2.3 --- ADN series --- p.35 / Chapter 2.3.1 --- Molecular Structure --- p.36 / Chapter 2.3.2 --- Electronic Structure --- p.49 / Chapter 2.3.3 --- Absorption and Emission Energy --- p.55 / Chapter 2.3.4 --- Reorganization Energy --- p.56 / Chapter 2.3.5 --- Mobility --- p.57 / Chapter 2.3.6 --- Summary on ADN series --- p.66 / Chapter 2.4 --- XOT series --- p.67 / Chapter 2.4.1 --- Molecular Structure --- p.68 / Chapter 2.4.2 --- Electronic Structure --- p.89 / Chapter 2.4.3 --- Absorption and Emission Energy --- p.96 / Chapter 2.4.4 --- Reorganization Energy and Mobility --- p.98 / Chapter 2.4.5 --- Summary on XOT series --- p.100 / Chapter 2.5 --- KPA series --- p.102 / Chapter 2.5.1 --- Molecular Structure --- p.102 / Chapter 2.5.2 --- Electronic Structure --- p.123 / Chapter 2.5.3 --- Absorption and Emission Energy --- p.131 / Chapter 2.5.4 --- Reorganization Energy and Mobility --- p.131 / Chapter 2.5.5 --- Summary on KPA series --- p.133 / Chapter 2.6 --- NPA series --- p.136 / Chapter 2.6.1 --- Molecular Structure --- p.136 / Chapter 2.6.2 --- Electronic Structure --- p.160 / Chapter 2.6.3 --- Absorption and Emission Energy --- p.166 / Chapter 2.6.4 --- Reorganization Energy and Mobility --- p.167 / Chapter 2.6.5 --- Summary on NPA series --- p.169 / Chapter II --- Analytical Studies of Polarons and the Electron-Phonon Interaction / 極子與電子一聲子相互作用的解析研究 --- p.172 / Chapter 3 --- Study on Holstein Model Using Variational Approximation --- p.173 / Chapter 3.1 --- Holstein Hamiltonian --- p.173 / Chapter 3.2 --- Variational Transformation --- p.175 / Chapter 3.2.1 --- Lang-Firsov Transformation --- p.175 / Chapter 3.2.2 --- Squeezing Transformation --- p.177 / Chapter 3.3 --- Energy and Static Correlation Functions --- p.179 / Chapter 4 --- Study on Holstein Model Using Coupled-Cluster Method --- p.193 / Chapter 4.1 --- Approximation in the Coupled-Cluster Method --- p.193 / Chapter 4.2 --- Approach 1 --- p.195 / Chapter 4.2.1 --- The Zeroth and the First Levels --- p.195 / Chapter 4.2.2 --- Energies and Static Correlation Functions --- p.196 / Chapter 4.3 --- Approach 2 --- p.206 / Chapter 4.3.1 --- The Zeroth and the First Levels --- p.206 / Chapter 4.3.2 --- Energies and Static Correlation Functions --- p.210 / Chapter 4.4 --- Approach 3 --- p.225 / Chapter 4.4.1 --- The Zeroth and the First Levels --- p.226 / Chapter 4.4.2 --- Energies and Static Correlation Functions --- p.228 / Chapter 4.5 --- Comparison with the Variational Method --- p.243

One of the central tasks for biomolecular modeling is to develop accurate and computationally cheap methods. In this dissertation, we present the development of a brand new polarizable force field—Polarizable Simulations with Second order Interaction Model (POSSIM) involving electrostatic polarization. The POSSIM framework combines accuracy of a polarizable force field and computational efficiency of the second-order approximation of the full-scale induced point dipole polarization formalism. POSSIM force field has been extended to include parameters for small molecules serving as models for peptide and protein side-chains. Parameters have been fitted to permit reproducing many-body energies, gas-phase dimerization energies and geometries and liquid-phase heats of vaporization and densities. Quantum mechanical and experimental data have been used as the target for the fitting. The resulting parameters can be used for simulations of the parameterized molecules themselves or their analogues. In addition to this, these force field parameters have been employed in further development of the POSSIM fast polarizable force field for proteins. The POSSIM framework has been expanded to include a complete polarizable force field for proteins. Most of the parameter fitting was done to high-level quantum mechanical data. Conformational geometries and energies for dipeptides have been reproduced within average errors of ca. 0.5 kcal/mol for energies of the conformers (for the electrostatically neutral residues) and 9.7º for key dihedral angles. We have also validated this force field by simulating an elastin-like polypeptide GVG(VPGVG)3 in aqueous solution. Elastin-like peptides with the (VPGVG)n motif are known to exhibit anomalous behavior of their radius of gyration that increases when temperature is lowered (the so called inverse temperature transition). We have simulated the system with the OPLS-AA and POSSIM force fields and demonstrated that our newly developed polarizable POSSIM parameters permit to capture the experimentally observed decrease of the radius of gyration with increasing temperature, while the fixed-charges OPLS-AA ones do not. Furthermore, our fitting of the force field parameters for the peptides and proteins has been streamlined compared with the previous generation of the complete polarizable force field and relied more on transferability of parameters for non-bonded interactions (including the electrostatic component). The resulting deviations from the quantum mechanical data are similar to those achieved with the previous generation, thus the technique is robust and the parameters are transferable. At the same time, the number of parameters used in this work was noticeably smaller than that of the previous generation of our complete polarizable force field for proteins, thus the transferability of this set can be expected to be greater and the danger of force field fitting artifacts is lower. Therefore, we believe that this force field can be successfully applied in a wide variety of applications to proteins and protein-ligand complexes.

Molecular simulation is applied to understanding the behaviour of alkane liquids with the eventual goal of being able to predict the viscosity of an arbitrary alkane mixture from first principles. Such prediction would have numerous scientific and industrial applications, as alkanes are the largest component of fuels, lubricants, and waxes; furthermore, they form the backbones of a myriad of organic compounds. This dissertation details the creation of a potential, a model for how the atoms and molecules in the simulation interact, based on a systematic approximation of the quantum mechanical potential energy surface using machine learning. This approximation has the advantage of producing forces and energies of nearly quantum mechanical accuracy at a tiny fraction of the usual cost. It enables accurate simulation of the large systems and long timescales required for accurate prediction of properties such as the density and viscosity. The approach is developed and tested on methane, the simplest alkane, and investigations are made into potentials for longer, more complex alkanes. The results show that the approach is promising and should be pursued further to create an accurate machine learning potential for the alkanes. It could even be extended to more complex molecular liquids in the future.

No description available.

O presente trabalho apresenta o estudo das propriedades da condensação de Bose-Einstein e da superfluidez em um sistema bosônico disposto em um arranjo unidimensional de potenciais periódicos em formato de anel. O Hamiltoniano efetivo usual em termos dos operadores de campo é implementado na representação construída em termos das funções de Bloch da primeira banda e o problema é resolvido por meio da sua diagonalização através de métodos numéricos. No limite de hopping pequeno, este modelo é essencialmente equivalente à representação usual do modelo de Bose-Hubbard, mas incorpora efeitos adicionais através das energias de Bloch de partícula independente e dos elementos da matriz de dois corpos na situação em que o hopping é grande [19]. Através da inclusão de rotação no sistema, as energias de partícula independente são forçadas a depender da velocidade angular. Isto implica, correspondentemente, uma dependência da velocidade angular nas funções de onda de partícula independente e nos resultados de muitos corpos obtidos através da diagonalização do Hamiltoniano. Com o objetivo de estudar a superfluidez, o critério de dois fluidos é empregado e através de resultados numéricos obtêm-se a variação da fração de superfluido com o quadrado da velocidade angular. Ainda, considera-se aqui uma expressão perturbativa para o parâmetro inercial do sistema expresso em termos das excitações do sistema sem rotação, o que permite relacionar as energias do sistema com rotação com aquelas do sistema sem rotação. Isto é particularmente interessante para obter a fração de superfluido em termos da informação espectral do sistema sem rotação. Resultados semelhantes podem ser encontrados através da definição de superfluido baseada na resposta do sistema a uma variação de fase, imposta através de condições de contorno torcidas [30, 33], mas com a diferença de que os desenvolvimentos aqui não fazem uso da hipótese do modo condensado. De maneira geral, os resultados numéricos obtidos indicam, que pelo menos para este sistema, as frações de superfluido e condensado são quantidades sem relação direta, sugerindo então que mesmo para sistemas gasosos diluídos a idéia de que a superfluidez é uma consequência da condensação de Bose-Einstein deve ser considerada com mais cuidado. / In this work we study the properties of Bose-Einstein condensation and superfluidity in a finite bosonic system in a 1-dimensional ring with a periodic potential under rotation. The usual field effective Hamiltonian is implemented in a representation constructed in terms of the first band Bloch functions and the problem is solved by numeric diagonalization. In the limit of small hopping, this model is essentially equivalent to the quasi-momentum representation of the usual Bose-Hubbard model but incorporates additional effects via Bloch single particle energies and two-body matrix elements in the case of large hopping [19]. By including rotation in the system we force the single particle energies to be a function of the angular velocity. This implies a corresponding angular velocity dependence of the single particle wavefunctions and many-body diagonalization results. In order to study superfluidity, we consider the two fluid criterion. Numerical results for the superfluid fraction involving the change of in rinsic ground state energy with the square of the angular velocity are obtained. We also consider a perturbative expression for the system inertial parameter expressed in terms of the excitation spectrum of the non rotating system, which enables us to relate the energies in the rotating system to the ones in the system without rotation. This is particularly interesting for obtaining superfluid fraction in terms of spectral information of the non rotating system. Similar results can be found by using the definition of superfluid fraction based on the response of the system to a phase variation imposed by means of twisted boundary conditions [30, 33], but with the difference that our developments do not assume the hypothesis of a condensate mode. Our numerical results indicate that in this system condensate and superfluid fractions are quite unrelated in terms of parameter values, indicating that even for dilute gases the concept that superfluidity is a consequence of Bose-Einstein condensation should be considered more carefully.

Estudamos as excitações de baixa energia, presentes em um gás de bosons homogêneo, de spin nulo, sujeitos a uma interação de dois corpos repulsiva e a temperatura zero, utilizando a aproximação gaussiana, que consiste num caso particular de aproximação de campo médio. As equações dinâmicas resultantes foram linearizadas ao redor da solução estática de Hartree-Fock-Bogoliubov. Obtivemos uma banda contínua e limitada inferiormente, além de um segundo ramo discreto, que define um limite inferior para as excitações e que, ao contrário do resultado proveniente do tratamento de Hartree-Fock-Bogoliubov, possui um comportamento linear sem gap com respeito ao momento da excitação no limite de grandes comprimentos de onda, ou seja, possui uma equação de dispersão do tipo fônon. Discutimos também a forma através da qual é possível gerar desvios do equilíbrio, vinculados aos estados excitados, e concluímos haver restrições sobre os possíveis desvios das grandezas características em campo médio gaussiano, quando tais desvios são gerados por transformações infinitesimais unitárias de um corpo tomadas até primeira ordem. / We study low-lying excitations of a spinless, homogeneous bose gas, with repulsive interaction, at zero temperature, in terms of a gaussian mean field approximation. The dynamical equations of this approximation have been linearized in small displacements from the well known static Hartree-Fock-Bogoliubov solution. We obtain a gapped continous band of excitations above a discrete branch with phonon behavior at large wavelengths. We also discuss the allowed forms of excitations and conclude that restrictions exist for the allowed deviations of the general set of gaussian mean field parameters, when they are generated in first orders by infinitesimal unitary transformations.

This thesis, divided into two parts, is concerned with the analysis of spectral and dynamical characteristics of certain quantum spin systems in the presence of either I) quenched disorder, or II) dynamical noise. In the first part, the quantum random energy model (QREM), a mean-field spin glass model with a many-body localisation transition, is studied. In Chapter 2, we attempt a diagrammatic perturbative analysis of the QREM from the ergodic side, proceeding by analogy to the single-particle theory of weak localisation. Whilst we are able to describe diffusion, the analogy breaks down and a description of the onset of localisation in terms of quantum corrections quickly becomes intractable. Some progress is possible by deriving a quantum kinetic equation, namely the relaxation of the one-spin reduced density matrix is determined, but this affords little insight and extension to two-spin quantities is difficult. We change our approach in Chapter 3, studying instead a stroboscopic version of the model using the formalism of quantum graphs. Here, an analytic evaluation of the form factor in the diagonal approximation is possible, which we find to be consistent with the universal random matrix theory (RMT) result in the ergodic regime. In Chapter 4, we replace the QREM's transverse field with a random kinetic term and present a diagrammatic calculation of the average density of states, exact in the large-N limit, and interpret the result in terms of the addition of freely independent random variables. In the second part, we turn our attention to noisy quantum spins. Chapter 5 is concerned with noninteracting spins coupled to a common stochastic field; correlations arising from the common noise relax only due to the spins' differing precession frequencies. Our key result is a mapping of the equation of motion of n-spin correlators onto the (integrable) non-Hermitian Richardson-Gaudin model, enabling exact calculation of the relaxation rate of correlations. The second problem, addressed in Chapter 6, is that of the dynamics of operator moments in a noisy Heisenberg model; qualitatively different behaviour is found depending on whether or not the noise conserves a component of spin. In the case of nonconserving noise, we report that the evolution of the second moment maps onto the Fredrickson-Andersen model - a kinetically constrained model originally introduced to describe the glass transition. This facilitates a rigorous study of operator spreading in a continuous-time model, providing a complementary viewpoint to recent investigations of random unitary circuits.

Estudamos as excitações de baixa energia, presentes em um gás de bosons homogêneo, de spin nulo, sujeitos a uma interação de dois corpos repulsiva e a temperatura zero, utilizando a aproximação gaussiana, que consiste num caso particular de aproximação de campo médio. As equações dinâmicas resultantes foram linearizadas ao redor da solução estática de Hartree-Fock-Bogoliubov. Obtivemos uma banda contínua e limitada inferiormente, além de um segundo ramo discreto, que define um limite inferior para as excitações e que, ao contrário do resultado proveniente do tratamento de Hartree-Fock-Bogoliubov, possui um comportamento linear sem gap com respeito ao momento da excitação no limite de grandes comprimentos de onda, ou seja, possui uma equação de dispersão do tipo fônon. Discutimos também a forma através da qual é possível gerar desvios do equilíbrio, vinculados aos estados excitados, e concluímos haver restrições sobre os possíveis desvios das grandezas características em campo médio gaussiano, quando tais desvios são gerados por transformações infinitesimais unitárias de um corpo tomadas até primeira ordem. / We study low-lying excitations of a spinless, homogeneous bose gas, with repulsive interaction, at zero temperature, in terms of a gaussian mean field approximation. The dynamical equations of this approximation have been linearized in small displacements from the well known static Hartree-Fock-Bogoliubov solution. We obtain a gapped continous band of excitations above a discrete branch with phonon behavior at large wavelengths. We also discuss the allowed forms of excitations and conclude that restrictions exist for the allowed deviations of the general set of gaussian mean field parameters, when they are generated in first orders by infinitesimal unitary transformations.