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arXiv:2501.12097 · Nature Reviews Physics

Platforms for the realization and characterization of Tomonaga–Luttinger liquids

Isabelle Bouchoule, R. Citro, T. Duty, Thierry Giamarchi, Randall G. Hulet, M. Klanjšek, E. Orignac, Bent Weber

Published Sep 8, 2025• 14 citations
arXivHTMLPDF
DOI
10.1038/s42254-025-00866-w

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Interacting fermions that are confined to 1D can only support collective excitations and are thus governed by the Tomonaga-Luttinger liquid (TLL) theory, in which collective excitations decouple into charge and spin modes instead of particle-like excitations. Over the course of this work we have realized analog quantum simulations of the TLL model using a 6-Li atomic system. By taking advantage of a magnetic Feshbach resonance we readily tune the interactions of the sample so that we can transition from a non-interacting gas to either a strongly repulsive or a strongly attractive system, and probe the dynamics of the system within these regimes. By using low-momentum Bragg spectroscopy we were able to measure and characterize the spin-charge separation of a TLL. Our work also presents the first observation of spin and charge density waves that are separately excited from the ground state through a Bragg perturbation. The observed dynamical response function of the spin and charge excitations remarkably captures both the universal behavior of the Tomonaga-Luttinger liquid, as well as the non-linear effects related to the band curvature effect in the charge sector and the back-scattering interaction effect in the spin sector. Up to now considering higher-order corrections to the linear dispersion in either the spin or charge degrees of freedom had remained out of reach for both theory and experiment. In addition to this, we also realized a spin-incoherent Luttinger liquid and characterized its temperature crossover between the coherent and incoherent regimes. On the attractive interactions regime we observe an inversion of spin-charge separation, and we also have seen preliminary signatures of confinement-induced 1D bound states.

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In the last decade it has become evident that information-theoretic quantities, which quantify a quantum entanglement encoded in a quantum state, are extremely useful to analyze a state of mane-body systems.Celebrated examples are the entanglement entropy and the Shannon entropy, which extract universal information of the underlying field theory of the system.Furthermore, a significant concept, the entanglement spectrum, has been proposed recently.It contains more complete information than the entanglement entropy.They have achieved a considerable success in characterizing exotic quantum phases that are beyond conventional descriptions.Inspired by these developments, we propose a new state, which we name a Rényified state, as follows: for a given quantum state |Ψ⟩ = ∑ i ψ i |i⟩ and the basis states |i⟩, the Rényified state is defined by raising the wave-function coefficients ψ i to the power of n (and normalized).The Shannon entropy can be obtained from the Rényified state.Moreover, the latter contains more information than the former.We study the Rényified state starting from a Tomonaga-Luttinger liquids (TLL), which is an important universality class of one-dimensional quantum systems.A TLL is characterized by a TLL parameter K, and described by the free boson field theory with a central charge c = 1 in the conformal field theory (CFT) context.We will show, using numerical calculations, that the Rényified state behaves as a TLL described by a modified TLL parameter K = K/n for small n.For larger n, the Rényified state behaves in a different way: the longitudinal correlation has long-range order while the transverse one remains algebraic.Based on the numerical results, we apply a replica field theory formulation to an analysis of the Rényified state.It suggests that the Rényified state is also a TLL described by a modified TLL parameter K = K/n, when n < 4K.The field theoretical analysis indicates that the TLL description breaks down at n = 4K, which is related to a phase transition in the Rényi-Shannon entropy.Beyond the transition, n > 4K, it suggests that the Rényified state is no longer a TLL since the longitudinal correlation has long-range order while the transverse one remains i algebraic.This exceptional behavior is unlikely to be realized in the ground state of a Hamiltonian with only short-range interactions.This indicates that the Rényified state beyond the transition belongs to a new class of exotic quantum phase.We explain an origin of the exotic behavior by constructing a particular conformal invariant boundary state of a two-component massless free boson.The relationship between the Rényified state and the Rényi-Shannon entropy is also elucidated in boundary CFT formalism.

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We discuss a platform for the synthetic realization of key physical properties of helical Tomonaga Luttinger liquids (HTLLs) with ultracold fermionic atoms in one-dimensional optical lattices. The HTLL is a strongly correlated metallic state where spin polarization and propagation direction of the itinerant particles are locked to each other. We propose an unconventional one-dimensional Fermi-Hubbard model which, at quarter filling, resembles the HTLL in the long wavelength limit, as we demonstrate with a combination of analytical (bosonization) and numerical (density matrix renormalization group) methods. An experimentally feasible scheme is provided for the realization of this model with ultracold fermionic atoms in optical lattices. Finally, we discuss how the robustness of the HTLL against backscattering and imperfections, well known from its realization at the edge of two-dimensional topological insulators, is reflected in the synthetic one-dimensional scenario proposed here.
Interacting fermions that are confined to 1D can only support collective excitations and are thus governed by the Tomonaga-Luttinger liquid (TLL) theory, in which collective excitations decouple into charge and spin modes instead of particle-like excitations. Over the course of this work we have realized analog quantum simulations of the TLL model using a 6-Li atomic system. By taking advantage of a magnetic Feshbach resonance we readily tune the interactions of the sample so that we can transition from a non-interacting gas to either a strongly repulsive or a strongly attractive system, and probe the dynamics of the system within these regimes. By using low-momentum Bragg spectroscopy we were able to measure and characterize the spin-charge separation of a TLL. Our work also presents the first observation of spin and charge density waves that are separately excited from the ground state through a Bragg perturbation. The observed dynamical response function of the spin and charge excitations remarkably captures both the universal behavior of the Tomonaga-Luttinger liquid, as well as the non-linear effects related to the band curvature effect in the charge sector and the back-scattering interaction effect in the spin sector. Up to now considering higher-order corrections to the linear dispersion in either the spin or charge degrees of freedom had remained out of reach for both theory and experiment. In addition to this, we also realized a spin-incoherent Luttinger liquid and characterized its temperature crossover between the coherent and incoherent regimes. On the attractive interactions regime we observe an inversion of spin-charge separation, and we also have seen preliminary signatures of confinement-induced 1D bound states.
In the last decade it has become evident that information-theoretic quantities, which quantify a quantum entanglement encoded in a quantum state, are extremely useful to analyze a state of mane-body systems.Celebrated examples are the entanglement entropy and the Shannon entropy, which extract universal information of the underlying field theory of the system.Furthermore, a significant concept, the entanglement spectrum, has been proposed recently.It contains more complete information than the entanglement entropy.They have achieved a considerable success in characterizing exotic quantum phases that are beyond conventional descriptions.Inspired by these developments, we propose a new state, which we name a Rényified state, as follows: for a given quantum state |Ψ⟩ = ∑ i ψ i |i⟩ and the basis states |i⟩, the Rényified state is defined by raising the wave-function coefficients ψ i to the power of n (and normalized).The Shannon entropy can be obtained from the Rényified state.Moreover, the latter contains more information than the former.We study the Rényified state starting from a Tomonaga-Luttinger liquids (TLL), which is an important universality class of one-dimensional quantum systems.A TLL is characterized by a TLL parameter K, and described by the free boson field theory with a central charge c = 1 in the conformal field theory (CFT) context.We will show, using numerical calculations, that the Rényified state behaves as a TLL described by a modified TLL parameter K = K/n for small n.For larger n, the Rényified state behaves in a different way: the longitudinal correlation has long-range order while the transverse one remains algebraic.Based on the numerical results, we apply a replica field theory formulation to an analysis of the Rényified state.It suggests that the Rényified state is also a TLL described by a modified TLL parameter K = K/n, when n < 4K.The field theoretical analysis indicates that the TLL description breaks down at n = 4K, which is related to a phase transition in the Rényi-Shannon entropy.Beyond the transition, n > 4K, it suggests that the Rényified state is no longer a TLL since the longitudinal correlation has long-range order while the transverse one remains i algebraic.This exceptional behavior is unlikely to be realized in the ground state of a Hamiltonian with only short-range interactions.This indicates that the Rényified state beyond the transition belongs to a new class of exotic quantum phase.We explain an origin of the exotic behavior by constructing a particular conformal invariant boundary state of a two-component massless free boson.The relationship between the Rényified state and the Rényi-Shannon entropy is also elucidated in boundary CFT formalism.