Correlation Functions And Polaron-Molecule Crossover In One Dimensional Attractive Fermi Gas

In the past few decades,the ultracold Fermi gas has been used as a superb experimental platform for understanding many-body physics due to its high manipulability.In particular,the experiments in threedimensional(3D)tunable Fermi liquid of ultracold atoms give an opportunity to realize Fermi polaron,a new form of quasi-particle induced by the mobile impurity,and the impurity systems attract more and more attention.However,most of the studies on the Fermi polaron problem,for example,the polaron-molecule transition in 3D fermionic medium more or less involve with approximations based on variational methods,or numerical techniques such as renormalization group or quantum Monte-Carlo.In contrast,many 1D systems can be treated in exact ways,for instance,Bethe ansatz,Bose-Fermi mapping,and non-perturbative quantum field theory,etc.,which offers a test bench to understand the quasi-particle physics phenomena like quantum criticality,quantum correlation,and momentum distribution.In this work we mainly study the correlation functions in one dimensional attractive Fermi polaron system of spin-1/2 Fermi gas with one flipped spin as an impurity immersed in the fully polarized spin-up media.By means of exact Bethe ansatz method and string momentum assumption,we calculate the analytical results of ground state correlation functions in both free fermion and strong attraction limit.The tendency of correlation functions from weak to strong interaction is computed numerically using Monte Carlo Method.In the weak attraction regime,correlation functions behave polaron-like while they become sharp as the interaction growth.The correlation functions in the infinite attractive limit can be decomposed into the free fermion term with the number of particles reduced by one,and the molecular term indicated by the emergence of a sharp peak or dip in the central area of the correlation.Thus the emergence of peak or dip in correlation functions may be treated as symbol of the system undergoes a polaron-molecule crossover.Furthermore,we study the correlation functions of excited states,which can be classified into gas states,with real momentum,and bound states,with string momentum.In the strong attraction limit,gas states are also called STG(super Tonks-Girardeau)gas,in which case there is no peak or dip found.Tan contact is found to be proportional to the cubic of the interaction strength and the total energy is verified to satisfy the Tan adiabatic theorem in the strong attraction regime.The analytical calculation on single impurity problem not only offers a benchmark to the 1D polaron problem,but also sheds light on exploring equilibrium and non-equilibrium many-body physics such as quantum phase transitions.