Tomonaga-Luttinger liquid theory for one-dimensional attractive Fermi gases

Abstract

The one-dimensional (1D) Yang-Gaudin model-an integrable $δ$-function interacting Fermi gas, serves as a paradigm in quantum many-body physics, encompassing phenomena from spin-charge separation to the Luther-Emery liquid. However, a consistent description of the Luther-Emery liquid and the bosonization of Fulde-Ferrell-Larkin-Ovchinnikov (FFLO)-like pairing states in the 1D attractive Fermi gas remains elusive. In this work, we develop a universal Tomonaga-Luttinger liquid (TLL) theory to describe the FFLO state across both weak and strong coupling regimes. We rigorously derive the low-energy effective Hamiltonian using bosonization, revealing the emergence of a two-component Luttinger liquid: one exhibiting spin-charge coupling in the weakly attractive regime, and another featuring charge-charge separation in the strongly attractive regime. For the weakly attractive regime, we further derive the renormalization-group equations for the sine-Gordon term in the spin sector and show that this term undergoes a relevant-irrelevant phase transition driven by the magnetic field. For the strongly attractive regime, we analyze the dynamical correlation functions of the FFLO pairing state based on the derived effective Hamiltonian. Finally, we propose an experimental scheme using ultracold atoms to verify the Luther-Emery liquid behavior and the subtle phenomena of spin-charge coupling and charge-charge separation.

Figures (3)

• Figure 1: Energy spectra of the paired and unpaired fermions, which are numerically calculated from the thermodynamic Bethe ansatz equations under the setting $\mu=0,\, H=1,\ , \gamma=20$. The thermodynamic Bethe ansatz equations are given in Xi-Wen Guan:2023. In the unpaired fermions sector($E_u$), there only exist fermions with spin-up. In the the bound pair sector($E_b$), there exist equal numbers of fermions with spin-up and spin-down. At the low-energy scale, we can linearize the spectra at the Fermi points $k_b$ and $k_u$ respectively.
• Figure 2: Pair correlation function in momentum space. The result given by CFT(red danished line) agrees well with our TLL results Eq.\ref{['pair_correlation_function_momentum_space']}(blue solid line) for different values of polarization $p=0.2,0.5,0.8$ and the interaction strength $\gamma=10$. Here the particle density $n=0.1$.
• Figure 3: Exact low-energy excitations of the unpaired fermions and the bound pairs. (a) Schematic illustration of the particle-hole excitation of the unpaired fermions: moving one particle within the two Fermi points to outside the Fermi sea; the bound pairs: moving one pair of paired fermions within the two Fermi points to outside the Fermi sea without pair-breaking. (b) Particle-hole excitation spectra of the unpaired fermions(yellow green) and the bound pairs(blue) calculated from the Bethe ansatz equations, their spectra manifest a novel separation in their sound velocities, which is regarded as the charge-charge separation. This figure is calculated by solving the thermodynamic Bethe ansatz equations numerically under the setting of the interaction strength $\gamma =52$ and polarization $p=0.35$. The thermodynamic Bethe ansatz equations are given in Xi-Wen Guan:2023.