Tomonaga-Luttinger liquid theory for one-dimensional attractive Fermi gases

The gist: This work develops a universal Tomonaga-Luttinger liquid (TLL) theory to describe the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in one-dimensional attractive Fermi gases for both weak and strong binding regimes, revealing a two-component TLL structure with spin and charge coupling or charge separation depending on the binding strength

Theoretical Framework

The study employs the Tomonaga-Luttinger liquid (TLL) theory as a universal framework to describe the low-energy physics of one-dimensional (1D) quantum many-body systems. The model considered is the one-dimensional Yang-Gaudin model, which describes an integrable δ-function interacting Fermi gas with a Hamiltonian given by H = −ħ/2m X i 2/2 + 2c X N i,j=1 δ(x i - x j) − hM− µN,(1) The low-energy properties of these gases are effectively described by Luttinger liquid theory, where fermionic excitations are bosonized into bosonic collective modes.

Weak Binding Regime

In the weak binding regime, the effective Hamiltonian is derived for a Fermi gas with weak interaction subject to an external magnetic field. The magnetic field induces a mismatch of the Fermi surfaces for the two spin components, leading to δkF ≡ kF ↑ − kF ↓ ≠ 0 and δvF ≡ vF ↑ − vF ↓ ≠ 0 in general. The most general effective Hamiltonian includes terms describing interactions between particles on the same branch (g4 process), interactions between different branches (g1 and g2 processes), and a sine-Gordon term.

The effective Hamiltonian is rewritten in terms of spin and charge degrees of freedom as H = Hc + Hs + Hcs + HSG. The term Hcs describes the coupling between spin and charge degrees of freedom, characterized by coefficients ∆cs1, ∆cs2 proportional to δvF. This spin-charge coupling hybridizes the excitation spectra that prevents factorization into individual spin or charge quasiparticles.

Renormalization Group Analysis

For weak interaction (g1⊥ ≪ 1), the sine-Gordon term (HSG) is treated as a perturbation in the Hamiltonian. The renormalization group (RG) equations for the sine-Gordon term are derived by considering a two-point correlation function of the field operator ϕ1(r). The resulting RG equations are dK1(l)dl = −y 2/1⊥K2 1 sin 2(ζ) squared J0(ϵ), dy1⊥(l) = 2 − 2 sin 2(ζ)K1 + cos 2(ζ)K2 y1⊥, where y1⊥ ≡ g1⊥/(2πu0), and ε ≡ 2δkF α0.

The behavior of the sine-Gordon term is analyzed based on the cutoff parameter α0, which is typically taken as the mean distance between fermions. If δkF < 1/(2α0), the renormalization equation resembles the spin-balanced case, where y1⊥ flows to 0 for repulsive interaction but to infinity for attractive ones. If δkF > 1/(2α0), the sine-Gordon term becomes irrelevant, and the system is described by a two-component Luttinger liquid with H = H1 + H2.

Strong Binding Regime

In the strong coupling regime, where ∆σ ≫ (cn), the deeply bound pairs remain stable at low energies and effectively suppress pair-breaking excitations. This suppression naturally gives rise to a decoupling of spin and charge degrees of freedom, allowing the effective Hamiltonian to separate into two distinct components of Luttinger liquid: H = HU + HB.

The Hamiltonian for the unpaired fermions (HU) takes the form of a standard charge-sector Luttinger liquid. For the bound pairs (HB), the low energy effective Hamiltonian should contain a sine-Gordon term in the form ∼ g1⊥ R cos h 2/2√2ϕ(B)s(x)i dx. However, for strong attractive interaction, ϕ(B)s(x) is pinned to zero in order to minimize the energy, and thus the sine-Gordon term can be neglected.

Pair Correlation Function

The pair correlation function in momentum space at k ≃ δkF reads P(k) ∼ k − δkF ν (15), where ν = 1 − η. The exponent η is determined by the parameters K1(2), ζ and β, which are defined in Eq. (10).

In the strong binding regime, the ground-state pair correlation in coordinate space is given by P(x, τ) ∼ e − iδkF x (2πa) squared e − η squared Fa((x,τ),u2) × e − 2 cos(ζ) sin(β)Fb((x,τ),u1) × e − η1 Fa((x,τ),u1). The pair correlation function in momentum space at k ≃ δkF reads P(k) ∼ k - δkF ν (15), where ν = 1 − η.

The analysis shows a good consistency between the various approaches, as the method described in the present paper allows to reach regimes of sizable polarization usually out of reach for the bosonization approach. The spin and charge dynamical structure factors can be calculated analytically using the strong-coupling effective Hamiltonian developed herein, with both quantities capturing characteristic signatures of the FFLO pairing state.

Conclusion and Outlook

The paper presents an analytical study of the TLL theory for attractive Fermi gases in both weak and strong binding regimes and arbitrary polarizations. The analysis reveals a description of the FFLO-like state in terms of two-component TLL liquids, exhibiting a spin-charge coupled regime in the weakly bound limit and charge-charge separation in the strongly bound limit. The results advance experimental observations of the LutherEmery liquid and charge-charge separation in 1D attractive Fermi gases. The spin and charge dynamical structure factors can be measured via Feshbach resonance and Bragg spectroscopy, extending beyond the small-polarization regime of the recent experiment.

Appendix Details

The appendix provides additional details on analytical results from the main text, including the bosonization of the Hamiltonian in the weak coupling regime. Section A details this bosonization, while Section B derives the RG flow equations for the sine-Gordon term. Section C discusses bound pairs’ DSF in detail. The determination of parameters Kc(s) and uc(s) is achieved by mapping the attractive Fermi gas onto a corresponding repulsive Hubbard model in the limit of n ≪ 1.

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The gist: This work develops a universal Tomonaga-Luttinger liquid (TLL) theory to describe the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state in one-dimensional attractive Fermi gases for both weak and strong binding regimes, revealing a two-component TLL structure with spin and charge coupling or charge separation depending on the binding strength

Theoretical Framework

The study employs the Tomonaga-Luttinger liquid (TLL) theory as a universal framework to describe the low-energy physics of one-dimensional (1D) quantum many-body systems. The model considered is the one-dimensional Yang-Gaudin model, which describes an integrable δ-function interacting Fermi gas with a Hamiltonian given by H = −ħ/2m X i 2/2 + 2c X N i,j=1 δ(x i - x j) − hM− µN,(1). The low-energy properties of these gases are effectively described by Luttinger liquid theory, where fermionic excitations are bosonized into bosonic collective modes.

The effective Hamiltonian is rewritten in terms of spin and charge degrees of freedom as H = Hc + Hs + Hcs + HSG.

1. Improved Simulation of 1D Quantum Many-Body Dynamics: The AI can now simulate the low-energy effective Hamiltonian derived from bosonization for a two-component Tomonaga-Luttinger liquid across both weak and strong binding regimes, allowing for the study of spin-charge coupling in the weak binding regime, and another featuring charge-charge separation in the strong binding regime.

2. Predictive Modeling of FFLO Pairing States: The system can now compute the dynamical correlation functions for the FFLO pairing state using the effective Hamiltonian, enabling predictions for experimental observables related to spin-charge coupling and charge-charge separation, which complements previous experimental efforts.

3. Accurate Characterization of Correlation Exponents: The AI can calculate the correlation exponent η using parameters derived from the effective Hamiltonian given in Eq.(9), providing a method to determine the critical behavior of the system, which is crucial for comparing theoretical predictions with experimental data from methods like Bragg spectroscopy.

4. Simulation of Dynamical Structure Factors (DSFs): The system can analytically calculate spin and charge dynamical structure factors using the strong-coupling effective Hamiltonian, capturing distinct signatures of the FFLO pairing state, which allows for the prediction of excitation spectra measurable via Bragg spectroscopy on systems like 6Li atoms.

5. Mapping to Repulsive Models for Parameter Extraction: The AI can utilize a method that maps the attractive Fermi gas onto a corresponding repulsive Hubbard model in the low-filling limit to extract necessary parameters like Luttinger parameters and velocities, overcoming difficulties in obtaining them directly.