Universal breathing mode scaling in harmonically trapped Fermi gases

TL;DR

The paper addresses universal scaling of the monopole (breathing) mode in harmonically trapped Fermi gases by decomposing the problem into fixed hyperangular channels and exploiting a contact-weighted Laguerre identity. A gamma-ratio simplification of hyperradial overlaps yields analytic, per-channel results: a level-resolved breathing shift with universal large-$q$ scaling $rac{δω}{2ω}∝Q^{-1}$, a first-order quantum anomaly with exactly two intermediate states producing a finite leakage tail $rac{W_{\rm leak}}{(q+s+1)^{2}}$, and a closed-form finite-temperature average with a low-$T$ plateau and a high-$T$ tail $\propto 1/T$, all parameter-free after calibrating the Tan contact $λ_s$ at $q=0$. A mixed anomaly–quartic correction and robustness to weak anisotropy are established, with exact $R^2$ tridiagonality ensuring a closed $SO(2,1)$ structure and saturation of sum rules. The framework connects to Tan relations, OPE, and hydrodynamic sum rules, and remains applicable under finite-range and quasi-2D geometry by absorbing corrections into channel parameters; a single calibration per channel enables parameter-free predictions for the entire spectrum and temperature dependence. Overall, the work provides actionable analytic predictions for breathing-mode spectra and temperature dependence in low-dimensional Fermi gases, guiding precision measurements and offering insight into universal symmetry-breaking phenomena in quantum gases.

Abstract

We derive universal, experiment ready analytic laws for the breathing (monopole) mode of harmonically trapped Fermi gases. Within a fixed hyperangular channel $s>0$, contact-weighted products of associated Laguerre polynomials reduce to an elementary gamma ratio, yielding: (i) a level resolved fractional breathing mode shift with scaling $δω/(2ω)\propto Q^{-1}$, where $Q\equiv 2q+s+1$, with $q$ the radial quantum number; (ii) a first order quantum anomaly correction involving exactly two intermediate states, producing a $Q^{-2}$ falloff of the leaked monopole strength with an explicit prefactor; and (iii) a closed form finite temperature average exhibiting a low-$T$ plateau and a $1/T$ high-$T$ tail. We also obtain a mixed anomaly\nobreakdash-quartic correction for weak anharmonicity. All expressions become parameter free after a single per-channel calibration of the Tan contact $λ_s$ at $q=0$.

Figures (2)

• Figure 1: (a) Level-resolved shift obeys the $Q^{-1}$ law with $Q\equiv 2q+s+1$. At $q=0$ (leftmost point on each curve), the shift matches the standard sum‑rule/hydrodynamic expression once $\lambda_s$ is calibrated; for $q>0$, our level‑resolved prediction falls below that single‑number estimate with the universal $Q^{-1}$ decay. (b) Finite-$T$ average with low-$T$ plateau and high-$T$$1/T$ tail.
• Figure 2: (c) Anomaly-induced leaked weight vs $q$ with slope $-2$ guide. (d) Channel $s{=}2$ normalization: $\mathcal{Y}_q=1/(q+3)^2$.