A low-energy effective Hamiltonian for Landau quasiparticles: II. Application to the contact Fermi gas

A low-energy effective Hamiltonian for Landau quasiparticles provides a systematic framework for studying strongly-correlated Fermi systems, and its application to an atomic Fermi gas with contact interactions allows for the derivation of renormalized parameters and corrections to static properties and dynamical transport.

The gist: This article applies a new renormalization scheme to construct a quantized theory of Fermi liquids by applying it to a low-temperature atomic Fermi gas where short-range interactions are parametrized by the s-wave scattering length, benchmarking the theory against known perturbative results for static properties and demonstrating preexponential corrections to zero sound velocity in the collisionless regime.

Effective Description and Renormalization

The paper begins by establishing a low-energy effective Hamiltonian for a two-component gas of ultracold fermionic atoms with contact interactions, where bare atom-atom interactions are replaced by an effective contact potential characterized by the scattering length 'a'. This potential is regularized perturbatively using the Schrieffer-Wolff decomposition, leading to an expansion of quasiparticle parameters up to subleading order in kFa. The resulting effective Hamiltonian is shown to be sextic in the quasiparticle creation/annihilation operators when truncated to second order in Vˆ.

Key steps include:

1. Using a cubic lattice regularization for the contact potential and expanding the Lippmann-Schwinger relation perturbatively for a → 0 at fixed l.

2. Constructing an effective Hamiltonian Hˆ′ using the quasiparticle expansion, which is sextic in ˆγ, and linearizing about the quasiparticle Fermi sea FS⟩ to obtain a form resembling Eq. (I.49).

3. Evaluating effective parameters like the eigenenergy ϵp and effective mass m∗ through perturbative calculations up to second order in g, recovering results such as the Lee-Huang-Yang equation of state to second order in kFa [36].

Static Properties and Interaction Functions

The formalism is used to benchmark known static properties. The Gor’kov Melik-Barkhudarov (GMB) correction to the critical temperature Tc follows from the second-order corrections to the renormalized pair interaction Gσσ′. The paper verifies Bethe-Salpeter relations derived in Section II on forward and frontal collision amplitudes by comparing perturbative expressions of Aσσ′ with those of fσσ′ and gσσ'.

The collision amplitudes are explicitly expressed on the Fermi surface (pF) in terms of angular functions IΛ and JΛ, which characterize the crossed Λ, θ dependence. These functions converge to known limits:

(II.26)

I(θ) ≡ limΛ¯→0IΛ(θ) = 1 − s squared ln1 + s1 − s, with s = sin θ2.

(II.27)

J(θ) ≡ limΛ¯→0JΛ(θ) = 1/2 (1 + c2 / (2s ln1 + s1 − s)), where c = cos θ2.

Zero Sound in the Collisionless Regime

The paper investigates the dynamical properties, focusing on the collisionless regime where ω0τ → +∞. The dispersion equation for collective modes is derived by projecting the transport equation onto Legendre polynomials, leading to a determinant condition (Eq. (II.62)).

The log-perturbative expansion of the zero-sound velocity c0 in powers of a = kFa yields crucial results:

1. For the density mode, c+0 = 1 + 2e4 e − π/a, for a > 0 [II.71]. Second-order corrections shift this velocity to higher values by a factor exp(6) ≈ 403.

2. For the polarization mode, c−0 = 1 + 2e−4 e − π/a, for a < 0 [II.72]. The second-order correction reduces this velocity by a factor exp(−2) ≈ 0.14, shifting it closer to the continuum edge.

Numerical Solution and Crossover

A numerical method is developed to solve the transport equation exactly throughout the crossover from collisionless to hydrodynamic regimes, based on an orthogonal decomposition in an angular basis (Legendre polynomials). This method involves:

1. Decomposing the quasiparticle distribution ν into components νl±(y) using Legendre polynomials Pl(cos θ) and energy-dependent polynomials Qn(y).

2. Solving the projected transport equation via a backward recurrence on l, with a truncation parameter nmax to handle the infinite matrix Hl.

3. Analyzing the response functions Im[χρ] and Im[χp] across different regimes, showing how second-order corrections modify the spectral density features, such as shifting zero-sound resonances away from the continuum edge in repulsive cases or smoothing sharp behavior in attractive cases.

As a fastidious researcher, I have analyzed this manuscript for its potential impact on AI system development by leveraging its sophisticated theoretical framework for many-body physics.

The core contribution of this paper is the development of a unified, renormalized effective Hamiltonian for Fermi liquids and its application to transport phenomena (zero sound) in ultracold atomic Fermi gases, specifically addressing the crossover between collisionless and hydrodynamic regimes.

Here are the specific improvements that can be made to AI systems by integrating this physics:

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Improving AI Systems via This Research: Specific Enhancements

The primary improvement is moving beyond classical machine learning models (which often struggle with complex many-body correlations and non-equilibrium dynamics) toward developing specialized, physically informed simulation and predictive engines. The improved AI system can perform the following tasks:

1. [][]

Predictive Modeling of Strongly Correlated Matter Dynamics:

By integrating the derived effective Hamiltonian (Eqs. II.7 to II.17) into a machine learning framework (e.g., Physics-Informed Neural Networks or Graph Neural Networks), the system can accurately model the time evolution of quasiparticle distributions in strongly correlated regimes where traditional kinetic equations fail.

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Accurate Zero Sound Velocity Prediction:

The system can predict the zero sound velocity, including its preexponential corrections (GMB correction), by learning from the functional forms derived in Section VIII (Eqs. II.71 and II.72). This is crucial for designing experiments or simulations aiming to observe specific collective modes in cold atom setups.

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Non-Equilibrium Transport Simulation:

The numerical method developed in Section IX (Orthogonal Decomposition, Eq. II.84 onwards) allows the AI system to solve the full transport equation across the collisionless-to-hydrodynamic crossover without relying on simple grid discretization. This enables accurate simulation of density and polarization response functions under realistic, time-dependent driving potentials (Eqs. II.107–II.109).

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Characterization of Interaction Regimes:

The AI can be trained to classify the interaction parameter space (defined by the scattering length 'a') using the phase diagram (Fig. 1) and response function behaviors (Figs. 6, 7, 8, 9, 10). This allows for rapid identification of whether a system is operating in a normal Fermi liquid phase, BEC-BCS crossover, or superfluid regime based on observable spectral signatures.

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Damping and Lifetime Estimation:

By utilizing the results from Section VIII (Eqs. II.83 and II.81), the AI can estimate the collisional damping rates of collective excitations (zero sound) as a function of temperature and interaction strength, offering a tool for characterizing experimental resolution limits or system stability in quantum simulators.

This article follows up on arxiv:2511.15938, in which we developed an Hamiltonian renormalization scheme to construct a quantized theory of Fermi liquids. Here, we apply this formalism to a low-temperature atomic Fermi gas where the short-range interactions are fully parametrized by the s-wave scattering length a. We benchmark our renormalized theory by recovering known perturbative results on the static properties of the Fermi gas, such as the Lee-Huang-Yang expansion of the equation of state, the Galitskii expansion of the momentum distribution, and the Gor'kov-Melik Barkhudarov preexponential correction to the critical temperature. We then turn to the transport dynamics and demonstrate the presence of a preexponential correction to the speed of zero sound when including corrections of second order in a. Finally, we develop an efficient numerical method to solve the transport equation exactly, and we apply it to study the crossover from the collisionless to the hydrodynamic regime in the density and polarisation response functions.