The Iterative Solutions Of TBA Equations Quantum Criticality For One-dimensional Quantum Gas

One of the fundamental and meaningful theoretical models in cold atom research area is the one-dimensional interacting quantum gas.The system may be composed of bosonic atoms,fermionic atoms gas or the mixture of them.The typical theoretical models have been established in the 1960s and widely generalized in later decades.These fundamental theories,such as Lieb-Liniger Model,Gaudin-Yang Model and Lai-Yang Model,have contributed in exploring the features of more experimentally realistic one-dimensional quantum gas.Therefore taking an insight into these theories becomes the primary task in our paper.Compared to the Bose gas model,the Fermi gas model will exhibit more fancy and interesting physical phenomenon due to its characteristic intrinsic properties.Especially for the attractive Fermi gas,the emergence of Cooper pairs allows us to classify the atoms in the system into two species,namely the bound paired fermions and unpaired fermions.The energies of these two species fermions will separate and form a gap.Corresponding to this quantum effect,the TBA equations of this model which reflects the quantum thermodynamic fluctuation will become two formulas,in which one stands for quantum behavior of the paired fermion and the other stands for the quantum behavior of the unpaired fermion.Meanwhile,the high order solutions of the TBA equations can be obtained by the iterative method.In addition,the high order relations between these two species fermi chemical potentials and the interaction strength also can be acquired in this iterative progress.The quantum critical phenomena is closely related to the quantum phase transition which will be accompanied with the changing parameters of the system.The quantum critical behavior of one-dimensional integrable model can be determined by exactly solving the BA equations.Therefore in this paper we study the quantum phase transitions of attractive Fermi gas by analytically solving the TBA equations,which can be obtained from the BA equations in Yang-Yang thermodynamic method.