[Paper Review] Fermi liquids and Luttinger liquids

H. J. Schulz, Gianaurelio Cuniberti+1 more|arXiv (Cornell University)|Jul 28, 1998

Cold Atom Physics and Bose-Einstein Condensates4 citations

TL;DR

This paper provides a comprehensive overview of Fermi liquids and Luttinger liquids using both phenomenological and microscopic approaches, emphasizing their distinct behaviors via the bosonization formalism. It establishes key differences in low-dimensional quantum systems, particularly in transport and disorder effects, and applies the framework to spin chains and ladders, offering a foundational reference for strongly correlated electron systems in one dimension.

ABSTRACT

In these lecture notes, the basic physics of Fermi liquids and Luttinger liquids is presented. Fermi liquids are discussed both from a phenomenological viewpoint, in relation to microscopic approaches, and as renormalization group fixed points. Luttinger liquids are introduced using the bosonization formalism, and their essential differences with Fermi liquids are pointed out. Applications to transport effects, the effect of disorder, quantum spin chains, and spin ladders, both insulating and metallic, are given.

Motivation & Objective

• To provide a pedagogical introduction to the theoretical foundations of Fermi liquids and Luttinger liquids in strongly correlated electron systems.
• To clarify the fundamental differences between Fermi liquids and Luttinger liquids, especially in one-dimensional systems.
• To present the bosonization formalism as a central tool for analyzing Luttinger liquids and their low-energy physics.
• To apply the theoretical framework to concrete physical systems, including quantum spin chains and spin ladders (metallic and insulating).
• To connect phenomenological descriptions with renormalization group fixed points and microscopic models, enhancing theoretical understanding of low-dimensional quantum matter.

Proposed method

• Employing the bosonization formalism to map interacting fermions in one dimension to non-interacting bosons, enabling exact treatment of low-energy excitations.
• Using the renormalization group to analyze fixed points of Fermi liquids and Luttinger liquids, highlighting their stability and universality.
• Applying the Luttinger liquid theory to study transport properties, such as conductance and current response, in one-dimensional systems.
• Incorporating disorder effects through the use of scaling theory and renormalization group analysis to assess localization and metal-insulator transitions.
• Analyzing spin chains and spin ladders using field-theoretic methods, distinguishing between metallic and insulating phases via the Luttinger parameter and spin gap.
• Comparing results from effective field theories with exact solutions in specific models, such as the Heisenberg chain and t-J model.

Experimental results

Research questions

• RQ1 How do the low-energy properties of Fermi liquids differ from those of Luttinger liquids in one-dimensional systems?
• RQ2 What is the role of the bosonization formalism in describing the collective excitations of Luttinger liquids?
• RQ3 How do disorder and interactions affect transport in one-dimensional quantum wires?
• RQ4 What determines the metal-insulator transition in spin-ladder systems, and how does it relate to Luttinger liquid behavior?
• RQ5 In what ways do Luttinger liquids emerge as fixed points in the renormalization group flow of interacting fermion systems?

Key findings

• Luttinger liquids exhibit power-law correlations and absence of quasiparticle peaks in the spectral function, contrasting with the Fermi liquid's delta-function-like quasiparticle peaks.
• The bosonization approach successfully captures the universal low-energy behavior of one-dimensional interacting fermions, including the Luttinger liquid parameter K.
• Disorder in Luttinger liquids leads to localization, with conductance scaling to zero at low temperatures, indicating a breakdown of metallic behavior.
• Spin chains and ladders show distinct phases depending on the Luttinger parameter: gapless for K > 1/2 and gapped for K < 1/2, consistent with field theory predictions.
• Fermi liquids are stable fixed points under renormalization group flow, while Luttinger liquids represent a distinct universality class in one dimension.
• The theory correctly predicts the absence of Fermi surface in Luttinger liquids and the presence of collective charge and spin modes with different velocities.