Dynamics Of A Vortex Ring Under The Local Induction Approximation And Current Phases In Spin Ladder Model

In recent years,ultra-cold atom systems have spurred intense investigations about the generation and dynamics of vortices,as well as the quantum phase transition and control of the low-dimensional quantum materials.The dynamics of vortex ring(VR)is an interesting physical phenomenon in both classical and quantum fluid fields.Vortex dynamics research has important application value in astrophysics,bridges,buildings,flying devices,superconductivity,superfluid and other aspects.A number of research works have discussed the dynamic process of the VR passing through a coaxial spherical obstacle,but the existing works are all numerical calculation results,and the phenomenon has not been analyzed physically.The first work of this paper is based on the mirror image method and the local induction approximation to analyze the dynamic process of the vortex ring passing through the coaxial spherical obstacle.The spin-ladder model with magnetic flux have aroused research interest,along with the realization of artificial gauge field technology.The second part of this paper focuses on the current phases in the square ladder and the Zigzag ladder models with magnetic flux.The first chapter is an introduction.First,we introduce the basic physical properties of vortex lines and VRs.Secondly,we summarize several sets of numerical results for studying the collision of a VR and a coaxial spherical obstacle both in classical fluids and quantum fluids.In the second part,we introduce the basic physical concepts such as quantum phase transition and frustration effect,as well as the research background of current phases in the ladder systems with magnetic flux.The second chapter mainly introduces our calculation methods.In dealing with the dynamics of the vortex ring,we adopt the local induction approximation(LIA),and the mirror image method is used to deal with the boundary conditions.When dealing with the spin system problem,we use the Lanczos algorithm and the Density Matrix Renormalization Group(DMRG)algorithm.Chapter 3 presents the dynamics of the VR passing through a coaxial spherical obstacle.We draw on the mirror image method in electromagnetism to deal with the effect of the obstacle surface.By considering the LIA method,the velocity of VR is equal to the superposition of the self-induced velocity and that casued by the velocity field of the mirror VR.By calculating the trajectory of the VR and its axial motion speed,we can find that when the vortex ring moves toward the obstacle ball,the radius of the vortex ring will become larger.During the approaching process,the VR would fifirst decelerate,then accelerate,and continue acceleration until it arrives at the center of the obstacle.After the VR passes over the center of the obstacle,the effect of the image VR is reserved symmetrically.The evolutions of radius and velocity of the VR are thus symmetric about the center of the obstacle.When the VR is at different distances from the obstacle ball,the above evolution law is perfectly explained by analyzing the position and velocity field distribution characteristics of its image VR.This analysis result is applicable to both classical fluids and quantum fluids.Due to the limiteation of the LIA approximation,we have ignored factors such as friction and quantum fluctuations of the system.In the fourth chapter,we discuss the current phases in square ladder under the action of magnetic flux.In the no-interaction limit,the ground state of the system in is analyzed by Fourier transform.The results show that:under the staggered flux,the system is a just in the vortex current phase.Then under the uniform flux,there will be a horizontal current phase,a vortex phase and a vertical current phase.The current order and charge order in the finite-scale system are calculated by the DMRG method,and the physical characteristics of each quantum current phase are further characterized.In the vortex phase,square vortices with a certain interval appear.The fifth chapter discusses the current phases in the Zigzag ladder model under the action of magnetic flux.Under the staggered flux,the system is in the vortex phase.Under the uniform flux,the system will have three kinds of current phases:the horizontal current phase,the vortex phase and the vertical current phase.We give the phase diagram for non-interacting case.Then we also consider the interactiong cases by using the DMRG method and discuss the corresponding current patterns.In the uniform hopping Zigzag ladder,with the increase of the uniform magnetic flux,the system changes from the horizontal current phase to the vortex phase,and there is a periodic charge order in the vortex phase.Within the system with strong staggered hopping,the currents between the diagonal bonds are suppressed,then the system directly transforms from the horizontal current phase to the vertical current phase by the increasing the uniform magnetic flux.It means that the staggered hopping is not conducive to the vortex flow within a triangular.The sixth chapter is the summary and outlook of our work.