Tomonaga–Luttinger liquid

Tomonaga–Luttinger liquid
Name	Tomonaga–Luttinger liquid
Theorists	Sin-Itiro Tomonaga, Joel Luttinger
Year	1950, 1963
Related concepts	Fermi liquid theory, Bosonization, Conformal field theory
Systems	Carbon nanotubes, Quantum wires, Edge states

Tomonaga–Luttinger liquid. It is a theoretical model describing the low-energy excitations of interacting electrons in one-dimensional conductors, fundamentally distinct from the Fermi liquid theory applicable in higher dimensions. The concept originated from the work of Sin-Itiro Tomonaga in 1950 and was later extended by Joel Luttinger in 1963, with the modern synthesis formalized in the 1980s. This model predicts exotic collective behavior where individual quasiparticles are absent, replaced by separate excitations of charge and spin.

Introduction

The breakdown of Fermi liquid theory in one spatial dimension necessitated a new theoretical framework, leading to the development of this model. Pioneering work by Sin-Itiro Tomonaga on a soluble model for interacting bosons laid the groundwork, which Joel Luttinger later adapted using a linearized dispersion approximation. The full implications were realized decades later by physicists including Daniel C. Mattis, Elliott H. Lieb, and F. Duncan M. Haldane, who connected it to Conformal field theory. Its necessity arises because even weak interactions in one-dimensional systems drastically alter the nature of the ground state and its excitations.

Theoretical model

The theoretical description typically begins with a model of spin-1/2 fermions on a line, such as the Hubbard model or a Thirring model, with short-range interactions. A key technique is Bosonization, which maps the interacting fermionic system onto a theory of non-interacting bosons representing collective density waves. The linearization of the dispersion near the Fermi points is crucial, as employed in the original Tomonaga model. This approach reveals that the elementary excitations are independent spinons and holons, a phenomenon known as Spin-charge separation.

Physical properties

The model exhibits several non-Fermi liquid properties. The single-particle spectral function lacks a sharp quasiparticle peak, instead showing power-law singularities. Various correlation functions, such as those for charge density waves and superconducting pairing, decay with distance as power laws with non-universal exponents dependent on interaction strength. The distinct velocities for spin and charge excitations, a hallmark of Spin-charge separation, are directly measurable. Furthermore, the system displays a universal conductance quantized in units of $2e^2/h$, as predicted by the Landauer formula.

Experimental realizations

Experimental verification has been achieved in several engineered one-dimensional systems. Semiconducting quantum wires fabricated using techniques like Molecular-beam epitaxy on substrates such as Gallium arsenide have provided key evidence. Measurements on single-walled carbon nanotubes have shown clear signatures of Spin-charge separation via Raman spectroscopy. The edge states of certain quantum Hall systems, described by the Chiral Luttinger liquid theory, also exhibit its properties. More recently, ultracold atoms confined in optical lattices created by teams at institutions like MIT and the Max Planck Institute have served as tunable simulators.

Relation to other models

It is intimately connected to several major theoretical frameworks in condensed matter physics. It serves as a prime example of a Conformal field theory in two dimensions, sharing algebraic structures with models like the XXZ spin chain. In the limit of strong repulsion, it maps onto the Tomonaga model of interacting bosons. The Hubbard model at half-filling exhibits a transition to a Mott insulator described by this framework. Its chiral version is essential for understanding the edge states of the Fractional quantum Hall effect, as formalized by X. G. Wen.

Applications

The framework provides the essential theory for understanding transport in nanoscale one-dimensional conductors. It is critical for modeling the behavior of carbon nanotube-based electronics and interconnects in potential future technologies. The theory underpins the analysis of edge state transport in devices exhibiting the Integer quantum Hall effect and Fractional quantum Hall effect. It also informs the study of spin ladders and certain quasi-one-dimensional materials like the Bechgaard salts. Furthermore, concepts from this model have influenced the analysis of out-of-equilibrium dynamics in quantum impurity problems, such as the Kondo model.

Category:Condensed matter physics Category:Quantum mechanics Category:Theoretical physics