Theoretical Study Of Localization In One-Dimensional Quantum Many-Body Systems

The study of electronic conductance is one of the cores of condensed matter physics.In the 1950s,Anderson introduced disorder into theoretical models pioneeringly to explain the localization of electrons in the imperfect crystal.Since then,the door of the localization study in quantum systems has been opened,and a new field has been created.Until now,the localization study has formed a complete framework,and a large number of excellent works have been produced.In recent years,with the development of the computer technology,people are no longer limited to the study of the localization in single-particle systems,but begin to pay attention to the phenomenon of the localization in interacting many-body systems,namely,many-body localization.Its essence is to explore the competition between interaction and disorder,which is also the core point of this doctoral dissertation.We focus on the localization in one-dimensional quantum many-body systems and study the effects of different factors on the localization.This dissertation contains five chapters in total,as follows:In Chapter 1,we introduce the background of this doctoral dissertation.We divide the content of this chapter into three sections,namely single-particle localization,many-particle localization,and numerical methods.In the section of single-particle localization,according to whether there is disorder in the Hamiltonian,we divide this section into two parts:(1)the localization in disordered systems;(2)the localization in non-disordered systems.In(1),we demonstrate two pical disordered potential-induced localizations,namely white noise potential-induced localization and incommensurate potential-induced localization.In(2),we focus on the localization induced by the linear potential and by other non-disordered factors.In the section of many-particle localization,we first introduce the localization transition in the ground state.Next,we focus on highly excited states,and introduce the ergodic and the localization of highly excited states.In the part of the ergodic,we briefly introduce the eigenstate thermalization hypothesis(ETH).In the part of the localization of highly excited states,we divide the content into many-body localization and Stark many-body localization according to the different origins of the localization.In the numerical methods section,we briefly introduce the exact diagonalization(ED),the density matrix renormalization group(DMRG)algorithm,and the shift-invert algorithm.In Chapter 2,we numerically study the ground-state properties of a one-dimensional lattice model in the presence of a quasi-periodic Zeeman field,by means of the density matrix renormalization group algorithm.We map out a global phase diagram,which encloses a Bardeen-CooperSchrieffer(BCS)phase,a Fulde-Ferrell-Larkin-Ovchinnikov(FFLO)phase where the Cooper pairs obtain a finite center-of-mass momentum,and a localized phase.In total,the interaction and the filling factor(n<0.5)play roles in destroying the localization.The FFLO phase is promoted by the quasi-periodic Zeeman field,which demonstrates a power-law pairing correlation with a critical exponent ηFFLO<1 decaying much slower than the charge density and spin density correlations.By tuning the interaction and the filling factor,we find the FFLO phase is suppressed in the strong interaction and low filling regions.A large Zeeman field destroys the FFLO state and drives a superconductor-insulator transition.Furthermore,we propose this FFLO phase and the superconductor-insulator transition could be observed in the optical lattice experiment.In Chapter 3,we investigate hard-core bosons filled in a lattice chain in the presence of a weak linear potential.In the single-particle case,we find that the critical point of dynamical Stark localization is different from that of static Stark localization.This suggests an intermediate phase in which the eigenstates are Stark localized,but the dynamic wave-functions are extended after quenching.In the many-body case,by comparing the dynamical critical point with the static critical point,we find a many-body intermediate phase that is analogous to the single-particle intermediate phase.Furthermore,we also study the static transition for the ground state and the dynamical transition for domain wall states.In the ground state,we find that the localization transition point is at V≈2(U+W)for half-filling(U is the nearest-neighbor interaction strength,W is the halfbandwidth).For the typical domain wall state |111...000),its dynamical transition points are at V ≈4(U+W)and V ≈4(U-W).By analyzing the distribution of the occupation,we also offer a phenomenological way to estimate the above transition points.In Chapter 4,we explore the many-body mobility edges by using random matrix methods.We investigate the transition from chaos to localization by constructing a combined random matrix,which has two extremes,one of Gaussian orthogonal ensemble and the other of Poisson statistics,drawn from different distributions.We find that by fixing a scaling parameter,the mobility edges can exist while increasing the matrix dimension D→∞,depending on the distribution of matrix elements of the diagonal uncorrelated matrix.By applying those results to a one-dimensional isolated quantum system of random diagonal elements,we confirm the existence of a many-body mobility edge,connecting it with results on the onset of level repulsion extracted from ensembles of mixed random matrices.Finally,by studying the relationship between the density of states and the effective ergodic composition,we elucidate the reason for the formation of many-body mobility edges from the perspective of random matrices.In the last chapter,we summarize this doctoral dissertation briefly and outlook several interesting topics.