Quantum Phase Transition In One Dimensional Spin Chain And Bose-einstein Condensates In Degenerate Rings

Quantum phase transition plays an important role in condensed matter physics.The ground states of a physical system will experience dramastic changes when they cross phase transition points.They are ususlly described by Ginzburg-Landau symmetry bro-ken theory based on order parameters and long-rangle correlations.In the resent years,new quantities of studying phase transition have been proposed,such as,topological order parameters,quantum entanglement,geometric phase(GP),fidelity susceptibility(FS),and so on.When a physical system evolves adiabatically along a closed path in its parameter space,its wavefunction will accumulate a GP besides the dynamical phase.The GP can reveal topological properties of wavefunctions,whose first order deriva-tive with respect to the driven parameters would be divergent when phase transitions take place due to energy gap closing.It has universal scaling behaviors around the phase transition points.FS characterizes how quickly the wavefunction changes with the driv-en parameters.Since the wavefunctions in different phases can be qiute different,FS also can be divergent and possesses some scaling relations when a phase transtion hap-pens.In earlier literature,numerical methods were used to get the divergent behaviors of GP and FS.We find their exact scaling behaviors can be determined by the singular function expansion method.These exact results can greatly enrich our understanding of GP and FS as well as their roles in characterizing quantum phase transitons.Bose-Einstein condensation(BEC)is an important topic in quantum statistic me-chanics that plays a significant role in understanding the superfluidity in liquid He and in BCS theory.Since the realization of BEC in ultracold atoms,a large amount of studies on degenerate quantum gas have been conducted.Ultracold atomic system-s enjoy great flexibiity and controllability and the interactions between atoms can be tuned through Feshbach resonance.The realization of synthetic gauge fields in ultra-cold atoms has open a new avenue for the study of quantum Hall effect,topological insulator and topological superconductor.Expecially,realization of spin-orbit coupling(SOC)in both bosons and fermions has paved the way for simulating SOC of electrons,studying topological phase transitions and searching novel BEC.SOC realised in 2016 also has extended the investigation of exotic matter states.SOC can change the energy bands of single particle and make the ground state to be degenerate.In free space,the degerate space is a ring in momentum space for two dimensional SOC and has infinite degeneracy.Then the BEC is a plane wave or standing wave determined by interactions.We find new BEC mechanisms by studying a bilayer bosonic system with two dimen-sional SOC and interlayer tunneling.Time-reversal symmetry and inversion symmetry play an important role here.In this artical,we introduce the investigation of quantum phase transition in two spin chain models and the BEC in degenerate rings.The results are as follows:.We illustrate the basic idea of singular function expansion using the exactly solv-able XY model and then apply this method to a little complex but still exactly solvable extended Ising model.We acquire divergent behaviros of GP and FS around critical points,which can further our understanding of their relationships with quantum phase transitions.For XY model,the exact scaling relations with length N of the spin chain for both GP and FS are obtained as well as their scaling behaviors around critical points.The coefficient of divergent term is determined by the way of energy gap closing.On the other hand,we also get the intimate relations of these coefficients and the constant term.As to the extended Ising model,when the energy gap is not closed at specials points,the scaling laws of GP and FS with spin chain length N break down.However,their scaling laws around critical points remaine.In this way,the intimate relations between these coefficients and the constant term also break down.·We use the mean-field theory to describe the quantum phase transition of bilayer spin-orbit coupling bosonic gas,obtain the groundstate phase diagram by using both imaginary-time evolution and variational methods,and analyse the crucial role of symmetry in determining ground states.When the degenerate space of single particle ground states consists of two identical rings,the spin-momentum locking could be relaxed,so gound state can still be a plane wave with certain polarization or a standing wave even when the density interaction of different spins dominates.It can only be a standing wave in the single layer system.What is more,we also find equal weight plane wave,zero-momentum states with full polarization or equal weight.When the degenerate space is two different rings,we find plane wave,standing wave,stripe states with two or four wave vectors if the two rings are not so close.Otherwise,we get BEC with excited states.