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  • Review Article
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Platforms for the realization and characterization of Tomonaga–Luttinger liquids

Nature Reviews Physics volume 7, pages 565–580 (2025)Cite this article

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Abstract

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.

Key points

  • 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.

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Fig. 1: Josephson junction chains as pinned Tomonaga Luttinger Liquids (TLLs).
Fig. 2: Tomonaga–Luttinger liquids (TLLs) in spin chains and ladders.
Fig. 3: Tomonaga–Luttinger liquids (TLLs) in the edge states of 2D topological insulators (TIs).
Fig. 4: Experimental demonstrations of Tomonaga–Luttinger liquids (TLLs) as helical edge states in 2D topological insulators (TIs).
Fig. 5: Realizations of inter-edge tunnelling devices to probe Tomonaga–Luttinger liquids (TLLs) at the edges of quantum Hall and quantum spin Hall insulators.
Fig. 6: Spin and charge excitations via Bragg spectroscopy.

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Acknowledgements

R.C. was partly supported by the PNRR MUR project PE0000023 NQSTI (TOPQIN and SPUNTO) and PathFinder project IQARO (Grant agreement ID: 101115190) I.B., R.C., E.O. and T.G. thank the Institut Henri Poincaré (UAR 839 CNRS Sorbonne Université) and the Lab Ex CARMIN (ANR-10 LABX-59-01) for their support. T.G. was supported in part by the Swiss National Science Foundation under grants 200020-188687 and 200020-219400. B.W. acknowledges the support of the National Research Foundation (NRF) Singapore, under the Competitive Research Program ‘Towards On-Chip Topological Quantum Devices’ (NRF-CRP21-2018-0001), with further support from the Singapore Ministry of Education (MOE) Academic Research Fund Tier 3 grant (MOE-MOET32023-0003) ‘Quantum Geometric Advantage’. T.D. acknowledges support from the Centre of Excellence for Engineered Quantum Systems, an Australian Research Council Centre of Excellence, CE110001013. M.K. acknowledges the financial support of the Slovenian Research and Innovation Agency through the programme number P1-0125 and the project number J1-2456. R.G.H. was funded in part by the National Science Foundation (US), grant number 2309362.

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Authors and Affiliations

  1. Laboratoire Charles Fabry, Institut d’Optique Graduate School–CNRS–Université Paris-Saclay, Palaiseau, France

    Isabelle Bouchoule

  2. Physics Department ‘E. R. Caianiello’, CNR-SPIN, Università degli Studi di Salerno, Fisciano, Italy

    Roberta Citro

  3. INFN — Gruppo Collegato di Salerno, Salerno, Italy

    Roberta Citro

  4. School of Physics, UNSW Sydney, Sydney, New South Wales, Australia

    Timothy Duty

  5. Qilimanjaro Quantum Tech, Barcelona, Spain

    Timothy Duty

  6. DQMP, University of Geneva, Geneva, Switzerland

    Thierry Giamarchi

  7. Department of Physics and Astronomy, Rice University, Houston, TX, USA

    Randall G. Hulet

  8. Jožef Stefan Institute, Ljubljana, Slovenia

    Martin Klanjšek

  9. LPENSL (UMR 5672), CNRS–Ens de Lyon, Lyon, France

    Edmond Orignac

  10. School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore, Singapore

    Bent Weber

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Glossary

Dimensional crossover

The transition from 1D behaviour to higher-dimensional physics when multiple 1D structures are coupled, posing challenges and opportunities in condensed-matter physics.

Higher-order topological insulators

Advanced materials exhibiting hinge states wherein TLL concepts may apply, suggesting novel boundary phenomena in higher-dimensional systems.

Lieb–Liniger model

A model of 1D non-relativistic bosons with contact interactions, solved exactly by E. Lieb and W. Liniger.

Mott lobes

Regions in the interaction–chemical potential phase diagram in which the system is in a Mott insulating phase.

Quantum criticality

Describes the behaviour of systems in a continuous phase transition, influenced by quantum fluctuations. Tomonaga–Luttinger liquid (TLL) systems are ideal for studying quantum-critical phenomena in reduced dimensions.

Quantum spin liquids

Exotic phases of matter with long-range entanglement, wherein TLL physics intersects with studies of strongly correlated systems.

Tonks–Girardeau gas

Limit of the Lieb–Liniger gas in which the contact interaction becomes infinite. It can be mapped to non-interacting fermions.

Topological matter

Materials with protected edge states, wherein TLL theory may provide information on the interacting topological phases and bulk–boundary correspondence.

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Bouchoule, I., Citro, R., Duty, T. et al. Platforms for the realization and characterization of Tomonaga–Luttinger liquids. Nat Rev Phys 7, 565–580 (2025). https://doi.org/10.1038/s42254-025-00866-w

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