Fractional Excitations,Quantum Magnetism And Dynamical Correlations In One-Dimensional Many-Body Systems

Quantum many-body systems are described by Hilbert spaces with exponentially large dimensions which cannot be easily reduced in the presence of interparticle in-teractions.This poses significant challenges to theoretical investigation of quantum many-body phenomena.In order to understand their low-energy behaviors,effective theories have been developed and proven to be great successes.One celebrated example in one spatial dimension(1D)is the Tomonaga-Luttinger liquid(TLL)theory,which explains exotic collective excitations in 1D systems.However,it is not applicable in the quantum critical region where fascinating universal scaling behaviors dictated by quan-tum critical points emerge.Fortunately,by studying 1D quantum integrable systems via the Bethe ansatz method,one can not only verify universal characteristics of Luttinger liquid but also discuss quantum criticality,both in a rigorous way.This thesis mainly focuses on thermodynamics,quantum criticality,low-energy excitations,and quantum liquid behavior in three quite different 1D integrable models-the Lieb-Liniger model,1D spin-1/2 Heisenberg chain,and SU(2)Fermi gases with repulsive interaction.Based on exact solutions,we present universal characteristics of Luttinger liquid and scaling behaviors in quantum critical region for 1D quantum many-body systems.Moreover,combining our theoretical results with experiments on solid state materials and cold atoms,we discuss in detail measurement and observation of thermodynamic properties of 1D spin-1/2 chains and propose an experimental scheme for testing spin-charge separation.The main contents of this thesis are as follows:1.Thermodynamics and Quantum Criticality of 1D Quantum Integrable SystemsWe solve the thermodynamic Bethe ansatz(TBA)Equations of the Lieb-Liniger model,1D spin-1/2 Heisenberg chain,and repulsive Fermi gases in the vicinity of quantum critical points,and obtain scaling functions and thermodynamic quantities analytically.These exact results reveal the fact that near quantum critical points,ther-modynamic quantities of these three models show universal behaviors.This indicates that they belong to the same universality class dictated by the free-fermion type quantum critical point.We also analyze and discuss related experimental results of the solid state material CuPzN.2.Low-energy Excitations and Quantum Liquids in 1D Quantum Integrable SystemsBy studying the Bethe ansatz(BA)Equations and Thermodynamic Bethe ansatz(TBA)Equations of the Lieb-Liniger model,1D spin-1/2 Heisenberg chain,and repul-sive Fermi gases,we obtain particle-hole excitations in charge sector and two-spinon excitation spectrum in spin sector,which verifies the Luttinger Liquid behaviors in low energy region.We also derive a linear dependence of specific heat on temperature for these three models,and prove additivity rules of macroscopic thermodynamic properties for repulsive Fermi gases.Relating Wilson ratio to Luttinger parameter,we study the universal quantum liquid behavior of these three models.We further point out that 1D repulsive SU(2)Fermi gases exhibit spin-charge separation phenomenon.3.Spin-Charge Separation in 1D Repulsive Fermi GasesSo far there is still no convincing evidence of the spin-charge separation in one dimension,from either theoretical or experimental perspective.We study a 1D repulsive Fermi gas in strong coupling limit and find that its TBA equations of the spin sector is the same as that of a Heisenberg spin chain.In light of this,we for the first time obtain its spin and charge dynamical structure factors.It is shown that with an increasing coupling,the sound velocity of spin quasi-particle decreases while the the sound velocity of charge quasi-particle increases.Based on this result,we further propose using Bragg spectroscopy to experimentally verify the spin-charge separation in one dimension.