Determination of Landau Fermi-liquid parameters of strongly interacting fermions by means of a nonlinear scaling transformation

A nonlinear transformation approach is formulated for the correlated fermions’ thermodynamics through a medium-scaling effective action. An auxiliary implicit variable-effective chemical potential is introduced to characterize the non-Gaussian fluctuations physics. By incorporating the nonlocal correlation effects, the achieved grand partition function is made of coupled highly nonlinear parametric equations. Analytically, the low temperature expansions for the strongly interacting unitary Fermi gas are performed for the adiabatic compressibility-sound speed and specific heat with the Sommerfeld lemma. The expressions for the Landau Fermi-Liquid parameters $F_0^s$ and $F_1^s$ of the strongly interacting fermion system are obtained. As a universal constant, the effective fermion mass ratio is $m^*/m={10/9}$ at unitarity.

💡 Research Summary

The paper presents a novel theoretical framework for extracting Landau Fermi‑liquid parameters of a strongly interacting fermionic system, specifically the unitary Fermi gas, by means of a nonlinear scaling transformation. The authors begin by constructing a medium‑scaled effective action that incorporates non‑Gaussian fluctuations through an auxiliary implicit variable, the “effective chemical potential” (\tilde\mu). This variable absorbs the complex many‑body correlations and is linked to the physical chemical potential (\mu) by a set of highly nonlinear parametric equations. Consequently, the grand partition function (Z) is expressed not as a high‑dimensional functional integral but as a coupled system of nonlinear algebraic relations, which still retains the essential non‑local correlation effects that are usually lost in mean‑field or perturbative treatments.

With the formalism in place, the authors perform a low‑temperature expansion ( (T\ll T_F) ) using the Sommerfeld lemma. They derive explicit series for the adiabatic compressibility (\kappa(T)), the sound speed (c(T)), and the specific heat at constant volume (C_V(T)) up to the leading temperature‑dependent terms ((\kappa\sim \kappa_0 + \alpha T^2), (c\sim c_0 + \beta T^2), (C_V\sim \gamma T + \delta T^3)). By matching these thermodynamic quantities to the standard Landau Fermi‑liquid expressions, they obtain closed‑form formulas for the symmetric Landau parameters (F_0^s) and (F_1^s). In particular, the relation (c^2 = \frac{n}{m}\frac{\partial\mu}{\partial n}\left(1+\frac{F_1^s}{3}\right)) yields (F_1^s), while the compressibility provides (F_0^s).

A striking result is the universal effective‑mass ratio at unitarity:
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