[Paper Review] Interacting fermions in one dimension: The Tomonaga-Luttinger model

K. Schönhammer|arXiv (Cornell University)|Oct 30, 1997

Cold Atom Physics and Bose-Einstein Condensates12 references12 citations

TL;DR

This paper presents a detailed theoretical analysis of the Tomonaga-Luttinger model for interacting fermions in one dimension, leveraging fermion-boson duality to map the system onto coupled harmonic oscillators. Using bosonization techniques, it derives exact expressions for the momentum distribution and spectral functions, establishing the foundational framework of Luttinger liquid behavior in one-dimensional quantum systems.

ABSTRACT

The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low energy spectrum of noniteracting fermions is identical to the one of a harmonic chain. This fermion-boson transmutation allows to describe interacting fermions as a system of coupled oscillators. Tomonaga's model of interacting nonrelativistic fermions on a ring is presented and first discussed in low order perturbation theory. After introducing the concept of two independent species of right and left moving fermions the exact solution of the Tomonaga-Luttinger model is discussed in detail. The momentum distribution and spectral functions are calculated using the method of the bosonization of the field operator. The general Luttinger liquid phenomenology is shortly discussed.

Motivation & Objective

• To provide a comprehensive theoretical description of interacting fermions in one dimension using the Tomonaga-Luttinger model.
• To demonstrate how the low-energy physics of noninteracting fermions maps onto a harmonic chain via fermion-boson transmutation.
• To derive exact expressions for the momentum distribution and spectral functions using bosonization of the field operator.
• To establish the phenomenology of the Luttinger liquid phase in one-dimensional systems.
• To clarify the role of right- and left-moving fermionic excitations in the exact solution of the model.

Proposed method

• Employing the Tomonaga model of nonrelativistic fermions on a ring with long-range interactions.
• Introducing separate degrees of freedom for right- and left-moving fermions to decouple the system into independent modes.
• Applying the method of bosonization to transform the fermionic field operators into bosonic fields, enabling exact solution techniques.
• Using the bosonized Hamiltonian to calculate the momentum distribution function and spectral functions via correlation functions.
• Analyzing the system in the low-energy limit to extract universal Luttinger liquid properties.
• Validating results through perturbative checks in low order and confirming consistency with exact solutions.

Experimental results

Research questions

• RQ1 How can the low-energy behavior of interacting fermions in one dimension be exactly solved using bosonization?
• RQ2 What are the momentum distribution and spectral functions in the Tomonaga-Luttinger model, and how do they deviate from noninteracting Fermi gas behavior?
• RQ3 How do right- and left-moving fermionic modes contribute to the exact solution of the model?
• RQ4 What universal features of the Luttinger liquid emerge from the exact solution of the Tomonaga-Luttinger model?
• RQ5 In what way does fermion-boson duality simplify the description of one-dimensional interacting fermions?

Key findings

• The momentum distribution function in the Tomonaga-Luttinger model exhibits power-law behavior, characteristic of Luttinger liquid physics, rather than a sharp Fermi surface.
• The spectral function shows a singular behavior at the Fermi momentum, consistent with the absence of quasiparticle peaks in one-dimensional interacting fermions.
• The exact solution confirms the existence of two independent modes—right- and left-moving fermions—whose coupling leads to collective charge and spin density waves.
• Bosonization provides an exact mapping of the interacting fermion system onto a system of coupled harmonic oscillators, enabling analytical tractability.
• The model exhibits universal scaling behavior in correlation functions, independent of microscopic details, confirming the robustness of Luttinger liquid phenomenology.
• Minor corrections in the final version (v3) confirm the consistency and accuracy of the derived expressions for the momentum distribution and spectral functions.