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Journal of High Energy Physics

The unitary Fermi gas at large charge and large N

Simeon Hellerman, Daniil Krichevskiy, Domenico Orlando, Vito Pellizzani, Susanne Reffert, Ian Swanson

Published May 29, 2024• 4 citations
Publisher

Abstract

A bstract We study the unitary Fermi gas in a harmonic trapping potential starting from a microscopic theory in the limit of large charge and large number of fermion flavors N . In this regime, we present an algorithmic procedure for extracting data from perturbation theory, order-by-order, without the need for other assumptions. We perform a gradient expansion in the interior of the particle cloud, sufficiently far from the cloud edge where the particle density drops rapidly to zero. In this latter region we present the first microscopic computation characterizing the contribution of the edge terms. The microscopic theory reproduces the predictions of the superfluid eft , including the action, the form of the gap equation, and the energy of the system in a harmonic trap (which maps, via the non-relativistic state-operator correspondence, to the scaling dimension of the lowest operator of charge Q ). We additionally give the Wilsonian coefficients at leading order in N up to nnlo in the large-charge expansion.

DOI
10.1007/jhep05(2024)323

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A bstract We study the unitary Fermi gas in a harmonic trapping potential starting from a microscopic theory in the limit of large charge and large number of fermion flavors N . In this regime, we present an algorithmic procedure for extracting data from perturbation theory, order-by-order, without the need for other assumptions. We perform a gradient expansion in the interior of the particle cloud, sufficiently far from the cloud edge where the particle density drops rapidly to zero. In this latter region we present the first microscopic computation characterizing the contribution of the edge terms. The microscopic theory reproduces the predictions of the superfluid eft , including the action, the form of the gap equation, and the energy of the system in a harmonic trap (which maps, via the non-relativistic state-operator correspondence, to the scaling dimension of the lowest operator of charge Q ). We additionally give the Wilsonian coefficients at leading order in N up to nnlo in the large-charge expansion.
We study the unitary Fermi gas in a harmonic trapping potential starting from a microscopic theory in the limit of large charge and large number of fermion flavors N. In this regime, we present an algorithmic procedure for extracting data from perturbation theory, order-by-order, without the need for other assumptions. We perform a gradient expansion in the interior of the particle cloud, sufficiently far from the cloud edge where the particle density drops rapidly to zero. In this latter region we present the first microscopic computation characterizing the contribution of the edge terms. The microscopic theory reproduces the predictions of the superfluid EFT, including the action, the form of the gap equation, and the energy of the system in a harmonic trap (which maps, via the non-relativistic state-operator correspondence, to the scaling dimension of the lowest operator of charge Q). We additionally give the Wilsonian coefficients at leading order in N up to NNLO in the large-charge expansion.
We extend and clarify the large-charge expansion of the conformal dimension ${\mathrm{\ensuremath{\Delta}}}_{Q}$ of the lowest operator of charge QQQ in nonrelativistic conformal field theories using the state-operator correspondence. The latter requires coupling the theory to an external harmonic trap that confines the particles to a spherical cloud, at the edge of which the effective theory breaks down and leads to divergences. Only recently has this issue been overcome by constructing appropriate counterterms at the edge of the cloud [S. Hellerman and I. Swanson, arXiv:2010.07967]. In this paper, we extend these results by systematically analyzing the degree of divergence of operators in the effective action and show that there always exist appropriate edge counterterms that make the final contributions to ${\mathrm{\ensuremath{\Delta}}}_{Q}$ finite. On the other side of the correspondence, this also provides new corrections to the Thomas-Fermi approximation of the unitary Fermi gas, and we comment on their relevance for ultracold atom physics.
Cold-atom experiments that measure Fermi-gas properties near unitarity confine fermionic atoms to a region of space using trapping potentials of various shapes. The presence of a trapping potential introduces a new characteristic physical scale in the superfluid effective-field theory (EFT) description of the unitary Fermi gas which, among other things, describes the acoustic branch of excitations in the far infrared well below the scale of the superfluid gap. In this EFT there is a clear hierarchy of scales, and corrections to the homogeneous system due to the trapping potential may be organized into three regions with distinct power counting that relies on both the EFT derivative expansion and the Wentzel--Kramers--Brillouin approximation, which is an expansion in gradients of the trapping potential. The energy spectrum of the superfluid system is obtained in each of the regions by explicit computation of the phonon-field fluctuations and by the modifications to the dynamic structure factor due to the corresponding density fluctuations. This work presents a systematic and quantitative method for treating the presence of a trapping potential, which is essential for interpreting experimental realizations of the unitary Fermi gas. It provides clear predictions for how the trapping potential modifies the dispersion relation's curvature, a key factor in characterizing the relaxation mechanisms of the superfluid. The most significant deviations from linear dispersion due to the trapping potential are found in the far-infrared region of the superfluid EFT.
We investigate the odd-even staggering in the harmonically-trapped unitary Fermi gas at large particle-number charge QQQ. Using both a large-NNN BdG description and a complementary large-charge EFT method, we show that for odd particle number the extra fermion forms an edge-localized quasiparticle near the Thomas-Fermi surface rather than a bulk excitation. In the edge limit, the microscopic BdG problem reduces to a universal coupled Airy system whose lowest positive eigenvalue fixes the leading odd-even splitting energy, χ ξ1/6(24Q)1/9 ℏω+⋯χ\,ξ^{1/6}(24Q)^{1/9}\,\hbarω+ \cdotsχξ1/6(24Q)1/9ℏω+⋯ where ξξξ is the Bertsch parameter, and χχχ is a universal edge coefficient. The associated EFT describes a fermionic mode confined to the boundary and coupled to the superfluid Goldstone field, reproducing the same QQQ scaling while introducing a dependence on two low-energy constants. Finally, we numerically compute the spectrum and confirm the predicted scaling and localization properties.
We investigate the inhomogeneous unitary Fermi gas and use the long-wavelength properties to predict the energies of small clusters of unitary fermions trapped in harmonic potentials. The large pairing gap and scale invariance place severe restrictions on the form of the density functional. We determine the relevant universal constants needed to constrain the functional from calculations of the bulk in oscillating external potentials. Comparing with exact quantum Monte Carlo calculations, we find that the same functional correctly predicts the lack of shell closures for small clusters of fermions trapped in harmonic wells as well as their absolute energies. A rapid convergence to the bulk limit in three dimensions, where the surface-to-volume ratio is quite large, is demonstrated. The resulting functional can be tested experimentally, and is a key ingredient in predicting possible polarized superfluid phases and the properties of the unitary Fermi gas in optical lattices.
Abstract We study operators in Schrödinger invariant field theories (non-relativistic conformal field theories or NRCFTs) with large charge (particle number) and spin. Via the state-operator correspondence for NRCFTs, such operators correspond to states of a superfluid in a harmonic trap with phonons or vortices. Using the effective field theory of the Goldstone mode, we compute the dimensions of operators to leading order in the angular momentumLand chargeQ. We find a diverse set of scaling behaviors for NRCFTs in bothd= 2 andd= 3 spatial dimensions. These results apply to theories with a superfluid phase, such as unitary fermions or critical anyon systems.