Platforms for the realization and characterization of Tomonaga–Luttinger liquids

The concept of a Tomonaga–Luttinger liquid (TLL) has been established as a fundamental theory for the understanding of 1D quantum systems. Originally formulated as a replacement for the Fermi liquid theory of Landau, which accurately predicts the behaviour of most 3D metals but fails dramatically in 1D, the TLL description applies to an even broader class of 1D systems, including bosons and anyons. After a certain number of theoretical breakthroughs, its descriptive power has now been confirmed experimentally in different experimental platforms. They extend from organic conductors, carbon nanotubes, quantum wires, topological edge states of quantum spin Hall insulators to cold atoms, Josephson junctions, Bose liquids confined within 1D nanocapillaries, and spin chains. In the ground state of such systems, quantum fluctuations become correlated on all length scales, but, counter-intuitively, no long-range order exists. This Review will illustrate the validity of conformal field theory for describing real-world systems, establishing the boundaries for its application, and discuss how the quantum-critical TLL state governs the properties of many-body systems in 1D. The Tomonaga–Luttinger liquid framework can be used to describe 1D quantum systems, spanning fermions, bosons and anyons. In this Review, we discuss the various platforms that can host TLL states, including Josephson junctions, cold atoms and topological materials, and discuss the advances TLL theory can provide in quantum criticality, nonequilibrium dynamics and condensed-matter physics exploration. In 1D systems, the concept of quasiparticles with the same quantum numbers as individual particles fails. Instead, low-energy excitations consist of linearly dispersing collective modes linked to fluctuations in particle and spin densities, leading to the phenomenon of spin–charge separation. The energy and momentum distributions of a 1D system exhibit power law singularities rather than steps at the Fermi energy, indicating a critical state known as the Tomonaga–Luttinger liquid (TLL). The critical exponents of this state depend on the Tomonaga–Luttinger parameters. The initial experimental validation of TLL theory comes from studies on organic conductors, quantum wires, carbon nanotubes and spin chains. Recent advances include experimental probes in ultracold atomic gases and Josephson junction chains, opening new avenues for exploring TLL physics. The TLL framework extends to systems such as 1D quantum antiferromagnets, bosonic systems and edge states in fractional quantum Hall systems (chiral TLLs) and topological insulators (helical TLLs). Nonlinear corrections to dispersion further refine the theory. The interplay between Luttinger liquid behaviour and higher-dimensional systems presents an exciting opportunity to explore many-body effects, such as hinge states of higher-order topological insulators, layer domain walls in van der Waals heterostructures, carbon nanotubes deposited on graphene substrates, or atomic wires on semiconducting surface. It opens the field to new quantum phases or exotic boundary states. In 1D systems, the concept of quasiparticles with the same quantum numbers as individual particles fails. Instead, low-energy excitations consist of linearly dispersing collective modes linked to fluctuations in particle and spin densities, leading to the phenomenon of spin–charge separation. The energy and momentum distributions of a 1D system exhibit power law singularities rather than steps at the Fermi energy, indicating a critical state known as the Tomonaga–Luttinger liquid (TLL). The critical exponents of this state depend on the Tomonaga–Luttinger parameters. The initial experimental validation of TLL theory comes from studies on organic conductors, quantum wires, carbon nanotubes and spin chains. Recent advances include experimental probes in ultracold atomic gases and Josephson junction chains, opening new avenues for exploring TLL physics. The TLL framework extends to systems such as 1D quantum antiferromagnets, bosonic systems and edge states in fractional quantum Hall systems (chiral TLLs) and topological insulators (helical TLLs). Nonlinear corrections to dispersion further refine the theory. The interplay between Luttinger liquid behaviour and higher-dimensional systems presents an exciting opportunity to explore many-body effects, such as hinge states of higher-order topological insulators, layer domain walls in van der Waals heterostructures, carbon nanotubes deposited on graphene substrates, or atomic wires on semiconducting surface. It opens the field to new quantum phases or exotic boundary states.