This dissertation presents new results on two problems concerning the dynamics of certainclasses of interacting many-body systems in quantum mechanics. Chapter 1 is an introduction,\nwhich includes mathematical preliminaries and a summary of the results.\nIn Chapter 2 we introduce the notion of transmission time to study the dynamics of disordered\nquantum spin chains and prove results relating its behavior to many-body localization properties.\nWe also study two versions of the so-called Local Integrals of Motion (LIOM) representation of\nspin chain Hamiltonians and their relation to dynamical many-body localization. We prove that\nuniform-in-time dynamical localization expressed by a zero-velocity Lieb-Robinson bound implies\nthe existence of a LIOM representation of the dynamics as well as a weak converse of this statement.\nWe also prove that for a class of spin chains satisfying a form of exponential dynamical localization,\nsparse perturbations result in a dynamics in which transmission times diverge at least as a power\nlaw of distance, with a power for which we provide lower bound that diverges with increasing\nsparseness of the perturbation.\nIn Chapter 3 we introduce a class of UV-regularized two-body interactions for fermions in Rd\nand prove a Lieb-Robinson estimate for the dynamics of this class of many-body systems. As a\nstep toward this result, we also prove a propagation bound of Lieb-Robinson type for Schrodinger\noperators. We apply the propagation bound to prove the existence of infnite-volume dynamics as\na strongly continuous group of automorphisms on the CAR algebra.
We review several aspects of Many‐Body Localization‐like properties exhibited by the disordered XY chains: localization properties of the energy eigenstates and thermal states, propagation bounds of Lieb‐Robinson type, decay of correlation functions, absence of particle transport, bounds on the bipartite entanglement, and bounded entanglement growth under the dynamics. We also prove new results on the absence of energy transport and Fock space localization. All these properties are made accessible to mathematical analysis due to the exact mapping of the XY chain to a system of quasi‐free fermions given by the Jordan‐Wigner transformation. Motivated by these results we discuss conjectured properties of more general disordered quantum spin and other systems as possible directions for future mathematical research.
Many-body localization of a disorder interacting boson system in one dimension is studied numerically at the filling factor being one-half, in terms of level statistics, local compressibility, correlation function and entanglement entropies. The von Neumann entanglement entropy is decoupled into a particle number entropy and a configuration entropy. An anomalous volume-law behavior is found for the configuration entanglement entropy to confirm a recent experimental observation [A. Lukin, M. Rispoli, R. Schittko, et al., Science 364, 256 (2019)] for sufficient strong disorder, while the particle number entropy fulfills an area-law corresponding to the total entropy for disordered spin chain. The localization length are extracted from a two-body correlation function for many-body localization states and corresponding time-evolutions states as well. A phase diagrams is established with consisting of an ergodic thermalized region and a many-body-localization region in a parameter space of the disorder strength and the energy density. Two regions are separated by a many-body mobility edge deducted from the standard deviation of the particle-number entanglement entropy, which appears consistent with that based on the localization length. Slow dynamics characterized by a logarithmic time-dependence is explicitly shown for both the particle number entropy and the configuration entropy in an intermediate regime of their time-evolutions, which does not show up in the Anderson localization case, i.e. non-interacting disorder systems.
We investigate the transition from the many-body localized phase to the ergodic thermalized phase at an infinite temperature in an
XY spin chain with
L spins, which experiences power-law decaying interactions in the form of ${V}_{ij}\ensuremath{\propto}1/{\left|i\ensuremath{-}j\right|}^{\ensuremath{\alpha}}$ (
i,j=1,...,L) and a random transverse field. By performing large-scale exact diagonalization for the chain size up to
L=18, we systematically analyze the energy gap statistics, the half-chain entanglement entropy, and the uncertainty of the entanglement entropy of the system at different interaction exponents $\ensuremath{\alpha}$. The finite-size critical scaling allows us to determine the critical disorder strength
Wc and the critical exponent $\ensuremath{\nu}$ at the many-body-localization phase transition, as a function of the interaction exponent $\ensuremath{\alpha}$ in the limit $L\ensuremath{\rightarrow}\ensuremath{\infty}$. We find that both
Wc and $\ensuremath{\nu}$ diverge when $\ensuremath{\alpha}$ decreases to a critical power of ${\ensuremath{\alpha}}_{c}\ensuremath{\simeq}1.16\ifmmode\pm\else\textpm\fi{}0.17$, indicating the absence of many-body localization for $\ensuremath{\alpha}<{\ensuremath{\alpha}}_{c}$. Our result differs from the analytical prediction of ${\ensuremath{\alpha}}_{c}=3/2$ found in a previous work based on a scaling argument [A. L. Burin, Phys. Rev. B 92, 104428 (2015)], but is close to the recent numerical result ${\ensuremath{\alpha}}_{c}\ensuremath{\approx}1$ obtained from quantum dynamics simulation [A. Safavi-Naini, M. L. Wall, O. L. Acevedo, A. M. Rey, and R. M. Nandkishore, Phys. Rev. A 99, 033610 (2019)].
The collective behavior exhibited by a large number of microscopic quantum particles is at the heart of some of the most striking phenomena in condensed matter physics such as Bose-Einstein condensation and superconductivity. Physicists and mathematicians have made great progress in understanding when and how these collective phenomena emerge through the interplay of particle statistics, particle interaction and the value of thermodynamic parameters like the temperature or the chemical potential. Due to the extreme complexity of realistic many-body systems, it is natural to introduce appropriate simplifications to render their analysis feasible. Three examples of such simplifications which have proven themselves as viable starting points for a fruitful and mathematically rigorous analysis of many-body systems are the following: (a) the study of integrable models; (b) the derivation of effective theories, valid on a macroscopic scale, from more fundamental microscopic theories under appropriate coarse-graining; and (c) the use of quantum information theory to understand general connections between correlation, entanglement and particle statistics. In this thesis, we present mathematically rigorous results that were obtained in these three directions. (1) We prove anomalous quantum many-body transport in XY quantum spin chains for certain choices of the external magnetic field. The anomalous transport is described via new kinds of anomalous Lieb-Robinson bounds, including one of power-law type. We note that the XY spin chain is integrable as it can be mapped to free fermions via the non-local Jordan-Wigner transformation. (2) We derive effective macroscopic theories of Ginzburg-Landau type from the microscopic BCS theory of superconductivity in certain circumstances. We study the case of a multi-component order parameter for translation-invariant systems and the condensation of fermion pairs at zero temperature in a domain with a hard boundary. (3) We use techniques from quantum information-theory to derive bounds on the entropy of fermionic reduced density matrices, a measure of the entanglement inherent to a fermionic quantum state.
We study a simple and tractable model of many-body localization. The main idea is to take a renormalization group perspective in which local entanglement is removed to reach a product state. The model is built from a random local unitary which implements a real space renormalization procedure and a fixed point Hamiltonian with random exponentially decaying interactions. We prove that every energy eigenstate is localized, that energy is not transported, and argue that despite being fine tuned, the model is stable to perturbations. We also show that every energy eigenstate obeys an area law for entanglement entropy and we consider the dynamics of entanglement entropy under perturbations. In the case of extensive pertubations we recover a logarithmic growth of entanglement observed in recent numerical simulations.