Universality of one-dimensional Fermi systems, I. Response functions and critical exponents

TL;DR

This paper proves that one-dimensional interacting Fermi systems exhibit universal critical behavior characterized by anomalous power-law decay and model-independent relations among critical exponents, with results valid for the Hubbard model.

Contribution

It establishes the Borel summability of response functions and derives universal relations among critical exponents for a broad class of one-dimensional Fermi models.

Findings

Response functions are Borel summable.

Critical exponents exhibit anomalous power-law decay with logarithmic corrections.

Universal relations among critical exponents are proven.

Abstract

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.