Correlation Function Of Single Spin Flipped One Dimensional Fermi Gas

The study of one-dimensional quantum gas has been one of the hot spots in the field of cold atomic physics for many years.Quantum gas can be classified according to the statistical property into Fermi,Bose and Bose Fermi mixtures system,described by either the continuous model or the lattice model.This paper deals with the details of the derivation of correlation functions in continuous model,numerical calculation and analysis of correlation functions of an attractively interacting Fermi gas with single spin reversed.With a brief introduction to one dimensional model,we accurately calculate all kinds of corresponding correlation functions in different continuous models.Starting from the Bose system,one-body correlation function,second-order correlation function and high-order local correlation function in the model are analytical derived by using the symmetry of wave functions and commutation relations between the operators.Then,the method in the Bose system is applied to calculate the correlation functions in the spinless Fermi system.Next,the spin degree is added to the spinless Fermi system,which becomes Gaudin-Yang model.The model studied in this paper is a special Gaudin-Yang model in which one particle reverses its spin and interacts with other particles attractively.This model can be also called single spin flipped Fermi gas.The spin-down particle can be regarded as an impurity compared with the rest of particles.Then correlation functions are analytical calculated by the same method as the first two models.Then the Bethe Ansatz(BA)method is used to deduce first quantized wave function corresponding to the system.BA equations corresponding to the model are solved numerically.By doing all these,the quasi-momentum and the system's spin rapidity corresponding to each particle are obtained.And then they are brought into the expression of the wave function and we apply the Monte Carlo method to calculate the multiple integrals of the function expression,to get the numerical results of the correlation functions.We analyze the one-body correlation function and the second-order correlation function for different interaction strengths.By comparison,the shape of one-body correlation function and the second-order correlation function between up-spin particles change with the strength of interactions: it becomes narrower at the tip,but the trend remains unchanged.The second-order correlation function between spin up and spin down particles shows that at the position of the spin down particles,the probability of finding spin-up particles is large,and it becomes larger with the enhancement of interaction.This shows that the attraction interaction only affects on the correlation between up-spin and down-spin.