Thermodynamics and excitations of the one-dimensional Hubbard model

TL;DR

This paper reviews the exact solution of the one-dimensional Hubbard model, focusing on string solutions, and compares two thermodynamic approaches, providing detailed analysis of excitations and their dispersion relations.

Contribution

It offers a comprehensive analysis of $k-\Lambda$ string solutions and compares the Yang-Yang and quantum transfer matrix methods for thermodynamics.

Findings

Agreement between Yang-Yang and quantum transfer matrix approaches.

Detailed dispersion curves for elementary excitations at zero magnetic field.

Clarification of $k-\Lambda$ string properties and their role in the model.

Abstract

We review fundamental issues arising in the exact solution of the one-dimensional Hubbard model. We perform a careful analysis of the Lieb-Wu equations, paying particular attention to so-called `string solutions'. Two kinds of string solutions occur: $\Lambda$ strings, related to spin degrees of freedom and $k-\Lambda$ strings, describing spinless bound states of electrons. Whereas $\Lambda$ strings were thoroughly studied in the literature, less is known about $k-\Lambda$ strings. We carry out a thorough analytical and numerical analysis of $k-\Lambda$ strings. We further review two different approaches to the thermodynamics of the Hubbard model, the Yang-Yang approach and the quantum transfer matrix approach, respectively. The Yang-Yang approach is based on strings, the quantum transfer matrix approach is not. We compare the results of both methods and show that they agree. Finally, we obtain the dispersion curves of all elementary excitations at zero magnetic field for the less than half-filled band by considering the zero temperature limit of the Yang-Yang approach.