The Grüneisen Parameter In One-Dimensional Bose-Fermi Mixture

One-dimensional cold atomic gases have attracted intensive studies both experimentally and theoretically,mixtures of ultracold bosonic and fermionic atoms have also attracted much attention.By loading cold atoms in one-dimensional(1D)potential well and tuning the effective interactions by Feshbach resonance,it is possible to simulate striking quantum many-body phenomena in 1D strongly correlated systems in the whole regime of interaction strength.The exquisite tunability with ultracold atoms confined to low dimensions has provided unprecedented opportunities for investigating and testing the theory of exactly solvable many-body systems.These include remarkable experimental progress in the realization of Tonks-Girardeau(TG)gas,super-Tonks-Girardeau gas,Yang-Yang thermodynamics for ultracold atoms.Of particular interest is the Bose-Fermi mixture system,which is rare in nature and has been made experimentally possible with the development of laser cooling and light capture techniques.In view of the experimental progress,it is necessary to study the thermodynamic properties of 1D Bose-Fermi mixture theoretically.Quantum criticality is among the most challenging of problems in condensed-matter physics.In order to extract correct universal scaling functions,which control proper thermal and quantum fluctuations at quantum criticality,a high precision of the finite-temperature thermodynamics is desirable.Based on the experimental and theoretical research progress on the Grüneisen parameters(GP)of heavy Fermion materials,in order to describe the different phases of the system,in this thesis,the phase transition of 1D integrable systems and the properties of quantum critical points are studied by using GP.We obtain the thermodynamic Bethe Ansatz(TBA)equation by numerical iteration method,and obtain the pressure by numerical integration.Finally,we get the GP expressed by pressure.In this thesis,the GP in Lieb-Liniger model is introduced,firstly.It is found that there are ideal Bose gas and Fermi gas in the strong and weak interaction region,which makes the parameter tend to be constant 2.Next,we study the properties of GP and magnetized GP in a Bose-Fermi mixture system.It is found that both of them change dramatically near the transition points of bose-mixture phase and Fermi-mixture phase,but have different behavior characteristics.Therefore,they can be used as a powerful tool for experimental and theoretical study of phase transition.Finally,the behavior of the second derivative of thermodynamic quantities(such as compressibility,susceptibility,etc.)at the quantum critical point is studied,and it is found that the compressibility and magnetization have divergent behaviors when the temperature approaches zero,and they all intersect at the phase transition point at different temperatures,which is caused by the change of phase transition temporal density.The universal scaling behavior of thermodynamic properties describes primely quantum critical phenomena.