|Abstract:||Under certain conditions, an interacting system can defy the concept of thermalization, a keystone in our understanding of physical processes. Many-body localization (MBL) is a phase of matter in which thermalization does not apply and ergodicity is broken. This striking behavior, which can appears on closed, interacting, quantum systems subject to strong disorder, has been the focus of a large body of theoretical, numerical, and experimental work in recent years. In this thesis we numerically study several aspects of MBL and the ergodic-MBL transition.
Chapter 1 introduces the concept of thermalization in quantum systems, which relies on the idea of the eigenstate thermalization hypothesis (ETH). I then present localization as a phenomenon that breaks ETH, both in its single-particle as in its many-body versions. I discuss some of the main aspects that are known about MBL, as well as some of its open questions, some of which we tackle in later chapters.
Chapter 2 discusses the main numerical methods used in this thesis for the study of MBL. I provide both theoretical background as well as discuss some more practical matters, such as their advantages and disadvantages, or details on the software developed/used in my studies.
In Chapter 3 I present our work on MBL from the point of view of single-particle orbitals. In this work we access a complete set of approximate integrals of motion of a one-dimensional system by computing the one-particle orbitals (OPO) of highly-excited MBL energy eigenstates, which are obtained through the shift-and-invert matrix product state (SIMPS) algorithm. We then study the properties of the OPOs over large systems, up to L = 64. We find that the OPOs drawn from eigenstates at different energy densities have high overlap and their occupations are correlated with the energy of the eigenstates. Moreover, the standard deviation of the inverse participation ratio of these orbitals is maximal at the nose of the mobility edge. Also, the OPOs decay exponentially in real space, with a correlation length that increases at low disorder. In addition, we find that the probability distribution of the strength of the large-range coupling constants of the number operators generated by the OPOs approach a log-uniform distribution at strong disorder.
In Chapter 4 I present our work on the hybridization of eigenstates in either the MBL or the ergodic phase, as well as at the transition. We do so by adiabatially evolving highly excited eigenstates of the Hamiltonian and measuring their their hybridization process with other eigenstates of the system. This hybridization, which dresses the eigenstate and has the potential of bringing it out of the MBL phase through the transition, is a consequence of the "collisions" of eigenstates in energy, which avoid level crossings every time their energy gap is small. The hybridization of eigenstates with each other involves only local regions of the system in the MBL phase, ignores locality in the ergodic phase, and is range-independent at the transition. This range independence suggests the proliferation of long-range resonances at the transition, as well as the divergence of a localization length.
In Chapter 5 I present our studies on the typical and extreme (atypically strong) correlations across a one-dimensional system in the ergodic-MBL phase diagram. While typical correlations decay exponentially with range in the MBL phase, in the ergodic phase they are constant and independent of the range. Surprisingly, we identify a moderate region of the phase in which typical correlations decay as a stretched exponential with range r, and in particular as exp[-r^(1/2)] at the transition, a decay that is reminiscent of the random singlet phase. Moreover, at the transition the distribution of the logarithm of the correlations show vanishing even excess moments and non-zero range-invariant odd excess moments. This distinct behavior at the transition is in contrast with ergodic and MBL phenomenologies. In addition, we study the extreme correlations in the system. Our results suggest that strong long-range correlations proliferate at the transition, in contrast with a decay with range in the MBL phase and the lack of strong long-range correlations in the ergodic phase. Finally, we analyze the probability that a single bit of information is shared across two halves of a system, which has been proposed as a robust order parameter in the ergodic-MBL phase diagram. We find that this probability is non-zero deep in the MBL phase, but vanishes at moderate disorder, well above the transition, thus not providing a proper order parameter.
Chapter 6 I summarize the work presented with an emphasis on context and perspective.