The critical behavior of one-dimensional interacting Fermi systems is expected to display universality features, called Luttinger liquid behavior. Critical exponents and certain thermodynamic quantities are expected to be related among each others by model-independent formulas. We establish such relations, the proof of which has represented a challenging mathematical problem, for a general model of spinning fermions on a one dimensional lattice; interactions are short ranged and satisfy a positivity condition which makes the model critical at zero temperature. Proofs are reported in two papers: in the present one, we demonstrate that the zero temperature response functions in the thermodynamic limit are Borel summable and have anomalous power-law decay with multiplicative logarithmic corrections. Critical exponents are expressed in terms of convergent expansions and depend on all the model details. All results are valid for the special case of the Hubbard model. 1 Main Results 1.1
A one-dimensional interacting Fermi system with weak-coupling parameters
g1(<0) and
g2 is reexamined in the third-order renormalization-group approach. Our result indicates that the value of the invariant coupling ${g}_{1}^{\ensuremath{'}}$ varies, depending on whether the limit $\ensuremath{\omega}\ensuremath{\rightarrow}0$,
q=0 or $\ensuremath{\omega}=0$, $q\ensuremath{\rightarrow}0$ is taken. For $\ensuremath{\omega}\ensuremath{\rightarrow}0$,
q=0, the invariant coupling ${g}_{1}^{\ensuremath{'}}$ scales to $\frac{{g}_{1}^{\ensuremath{'}}}{\ensuremath{\pi}v}\ensuremath{\simeq}\ensuremath{-}1.24$, and for $\ensuremath{\omega}=0$, $q\ensuremath{\rightarrow}0$, it scales to $\frac{{g}_{1}^{\ensuremath{'}}}{\ensuremath{\pi}v}\ensuremath{\simeq}\ensuremath{-}0.716$. We also argue that the invariant coupling ${g}_{1}^{\ensuremath{'}}$ obtained from neither limit is appropriate for calculating the critical exponents of various response functions.
Abstract We present a unified derivation of the pressure equation of states, thermodynamics and scaling functions for the one-dimensional (1D) strongly attractive Fermi gases with SU(w) symmetry. These physical quantities provide a rigorous understanding on a universality class of quantum criticality characterized by the critical exponents z = 2 and correlation length exponent ν = 1/2. Such a universality class of quantum criticality can occur when the Fermi sea of one branch of charge bound states starts to fill or becomes gapped at zero temperature. The quantum critical cone can be determined by the double peaks in specific heat, which serve to mark two crossover temperatures fanning out from the critical point. Our method opens to further study on quantum phases and phase transitions in strongly interacting fermions with large SU(w) and non-SU(w) symmetries in one dimension.