Phenomenology of One-Dimensional Quantum Liquids Beyond the Low-Energy Limit

TL;DR

This paper analyzes the behavior of dynamic response functions in one-dimensional quantum liquids beyond low-energy approximations, relating edge singularities to an effective impurity model and providing explicit exponents for various systems.

Contribution

It introduces a method to connect edge singularities in 1D quantum liquids to an effective Hamiltonian, extending the phenomenological description beyond low energies.

Findings

Derived relations between effective Hamiltonian parameters and edge functions.

Expressed response function exponents in terms of $psilon(k)$ and Luttinger parameters.

Fixed exponents for the Heisenberg spin chain using SU(2) symmetry.

Abstract

We consider zero temperature behavior of dynamic response functions of 1D systems near edges of support in momentum-energy plane $(k, \omega).$ The description of the singularities of dynamic response functions near an edge $\epsilon(k)$ is given by the effective Hamiltonian of a mobile impurity moving in a Luttinger liquid. For Galilean-invariant systems, we relate the parameters of such an effective Hamiltonian to the properties of the function $\epsilon (k).$ This allows us to express the exponents which characterize singular response functions of spinless bosonic or fermionic liquids in terms of $\epsilon(k)$ and Luttinger liquid parameters for any $k.$ For an antiferromagnetic Heisenberg spin-1/2 chain in a zero magnetic field, SU(2) invariance fixes the exponents from purely phenomenological considerations.