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Comptes Rendus Physique

Effects of spin–orbit coupling and Rabi fields in Tomonaga–Luttinger liquids: current status and open questions

Xiaoyong Zhang, Carlos A. R. Sá de Melo

Published Jul 15, 2025• 1 citation

Abstract

We discuss the effects of spin–orbit coupling and Rabi fields in Tomonaga–Luttinger liquids for SU(2) and SU(3) Fermi systems. In the SU(2) case, we show that spin–orbit coupling and Rabi fields mix separated spin and charge excitations producing helical massless bosons, which we call Weyl bosons in analogy to their cousins, the Weyl fermions. We discuss the phase diagram and the velocities of bosonic modes, showing where different flavors of Weyl bosons emerge. We suggest that the dispersion and helicity of Weyl bosons can be detected through measurements of the the dynamical structure factor tensor. In the SU(3) case, we preliminarily discuss the effects of spin–orbit coupling and Rabi fields, and conjecture that the emergent collective modes have a scalar (charge), vector (spin) and tensor (quadrupolar) components, suggesting that these modes are more complex than Weyl bosons. To describe spin–orbit coupling, we use the terminology color-orbit coupling, where the three internal states are labeled as colors Red, Green and Blue. We discuss the phase diagram and velocities of boson modes in the non-interacting regime and ponder over several open questions that need to be addressed for SU(3) systems. Lastly, we make some concluding remarks and suggest potential experimental candidates, with two and three internal states, where spin–orbit or color-orbit coupling and Rabi fields could be used to investigate the emergence of unusual collective modes with scalar, vector and tensor properties.

DOI: 10.5802/crphys.254

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We discuss the effects of spin–orbit coupling and Rabi fields in Tomonaga–Luttinger liquids for SU(2) and SU(3) Fermi systems. In the SU(2) case, we show that spin–orbit coupling and Rabi fields mix separated spin and charge excitations producing helical massless bosons, which we call Weyl bosons in analogy to their cousins, the Weyl fermions. We discuss the phase diagram and the velocities of bosonic modes, showing where different flavors of Weyl bosons emerge. We suggest that the dispersion and helicity of Weyl bosons can be detected through measurements of the the dynamical structure factor tensor. In the SU(3) case, we preliminarily discuss the effects of spin–orbit coupling and Rabi fields, and conjecture that the emergent collective modes have a scalar (charge), vector (spin) and tensor (quadrupolar) components, suggesting that these modes are more complex than Weyl bosons. To describe spin–orbit coupling, we use the terminology color-orbit coupling, where the three internal states are labeled as colors Red, Green and Blue. We discuss the phase diagram and velocities of boson modes in the non-interacting regime and ponder over several open questions that need to be addressed for SU(3) systems. Lastly, we make some concluding remarks and suggest potential experimental candidates, with two and three internal states, where spin–orbit or color-orbit coupling and Rabi fields could be used to investigate the emergence of unusual collective modes with scalar, vector and tensor properties.

Motivated by the experimental observation of spin-charge separation in one-dimensional interacting Fermi gases, we investigate these systems in the presence of spin-orbit coupling and Rabi fields. We demonstrate that spin-charge-separated modes evolve into helical collective modes due to the special mixing of spin and charge induced by spin-orbit coupling and Rabi fields. We obtain the phase diagram of chemical potential versus Rabi fields for given spin-orbit coupling and interactions, and find several topological quantum phase transitions of the Lifshitz type. We show that the velocities of the collective modes are nonanalytic at the boundaries between different phases. Lastly, we analyze the charge-charge, spin-charge and spin-spin dynamical structure factors to show that the dispersions, spectral weights and helicities of the collective modes can be experimentally extracted in systems such as

^{6}{\rm Li}

,

^{40}{\rm K}

and

^{173}{\rm Yb}

.

In this thesis we present contributions in the field of the applications of quantum field theories techniques to condensed matter models. In chapter 3 we investigate on the non covariant fermionic determinant and its connection to Luttinger liquids. We address the problem of the regularization of the theory. In chapter 4 we treat spin flipping interactions in the non local Thirring model and we obtain an effective bosonic actions that describe separated spin and charge degrees of freedom. In chapter 4 we apply the self consistent harmonic approximation to previously derived bosonic action and we obtain potential depending equations for the spectrum gap. In chapter 5 we include spin-orbit couplings and compute correlations functions. We show that the spin orbit interactions modify the exponents and the phase diagram of the system and makes new susceptibilities diverge for low temperature. Finally in chapter 6 we summarize the main results and the conclusions.

I give a brief introduction to Luttinger liquids. Luttinger liquids are paramagnetic one-dimensional metals without Landau quasi-particle excitations. The elementary excitations are collective charge and spin modes, leading to charge-spin separation. Correlation functions exhibit power-law behavior. All physical properties can be calculated, e.g. by bosonization, and depend on three parameters only: the renormalized coupling constant Kρ, and the charge and spin velocities. I also discuss the stability of Luttinger liquids with respect to temperature, interchain coupling, lattice effects and phonons, and list important open problems.

In this thesis we present contributions in the ﬁeld of the applications of quantum ﬁeld theories techniques to condensed matter models. In chapter 3 we investigate on the non covariant fermionic determinant and its connection to Luttinger liquids. We address the problem of the regularization of the theory. In chapter 4 we treat spin ﬂipping interactions in the non local Thirring model and we obtain an eﬀective bosonic actions that describe separated spin and charge degrees of freedom. In chapter 4 we apply the self consistent harmonic approximation to previously derived bosonic action and we obtain potential depending equations for the spectrum gap. In chapter 5 we include spin-orbit couplings and compute correlations functions. We show that the spin orbit interactions modify the exponents and the phase diagram of the system and makes new susceptibilities diverge for low temperature. Finally in chapter 6 we summarize the main results and the conclusions.

We theoretically investigate a three-dimensional Fermi gas with Rashba spin–orbit coupling in the presence of both out-of-plane and in-plane Zeeman fields. We show that, driven by a sufficiently large Zeeman field, either out-of-plane or in-plane, the superfluid phase of this system exhibits a number of interesting features, including inhomogeneous Fulde–Ferrell pairing, gapped or gapless topological order, and exotic quasi-particle excitations known as Weyl fermions that have linear energy dispersions in momentum space (i.e., massless Dirac fermions). The topological superfluid phase can have either four or two topologically protected Weyl nodes. We present the phase diagrams at both zero and finite temperatures and discuss the possibility of their observation in an atomic Fermi gas with synthetic spin–orbit coupling. In this context, topological superfluid phase with an imperfect Rashba spin–orbit coupling is also studied.

Motivated by the recent experimental success in realizing synthetic spin-orbit coupling in ultracold atomic systems, we consider N -component atoms coupled to a non-Abelian SU(N) gauge field. More specifically, we focus on the case, referred to here as “SU(3) spin-orbit-coupling,” where the internal states of three-component atoms are coupled to their momenta via a matrix structure that involves the Gell-Mann matrices (in contrast to the Pauli matrices in conventional SU(2) spin-orbitcoupled systems). It is shown that the SU(3) spin-orbit-coupling gives rise to qualitatively different phenomena and in particular we find that even a homogeneous SU(3) field on a simple square lattice enables a topologically non-trivial state to exist, while such SU(2) systems always have trivial topology. In deriving this result, we first establish an equivalence between the Hofstadter model with a 1/N Abelian flux per plaquette and a homogeneous SU(N) non-Abelian model. The former is known to have a topological spectrum for N > 2, which is thus inherited by the latter. It is explicitly verified by an exact calculation for N = 3, where we develop and use a new algebraic method to calculate topological indices in the SU(3) case. Finally, we consider a strip geometry and establish the existence of three gapless edge states – the hallmark feature of such an SU(3) topological insulator.