Unitary Fermi gases and universal scales

For two equally populated states, define total density as the sum of both components. The Fermi wavenumber of either component can then be written using three pi squared times total density, or equivalently six pi squared times one component's density. Mixing these conventions changes the inferred energy and temperature scales. The Fermi energy is a density reference scale even for an interacting gas; it is not automatically equal to the actual chemical potential.

Opposite internal states permit low-energy s-wave scattering. To reach the zero-range unitary regime, require inverse scattering length small compared with kF and effective range small compared with inverse kF. Merely tuning to a pole does not remove a large effective range near a narrow resonance. The gas must also be described by the intended dimensionality, spin balance, and near-equilibrium state. Trap averages require a spatial model rather than direct substitution of a single global density.

On the weak negative-a side, attractive fermions can form extended Cooper pairs through a many-body instability even though there is no universal free-space shallow dimer. At positive a, tightly bound dimers can form and eventually behave as composite bosons when their size is small relative to spacing. The balanced zero-temperature crossover connects these limits through strong pairing. A pairing signal and a superfluid phase are related but distinct: a finite-temperature system can have pairing correlations without long-range phase coherence.

At resonance, the two-body cross section is limited at finite relative momentum instead of diverging with a squared. Thermodynamic properties depend on universal functions of reduced temperature and any additional declared dimensionless parameters such as polarization. These functions must be measured or calculated using controlled many-body methods. Inserting an infinite scattering length into a weak-coupling GP interaction energy is not a valid way to predict a unitary Fermi gas.

In a homogeneous balanced zero-temperature unitary gas, dimensional analysis makes the ground-state energy proportional to the noninteracting Fermi energy per particle with one dimensionless coefficient. Differentiating that energy at fixed particle number gives pressure, and differentiating at fixed volume gives chemical potential. The same coefficient appears in all three relations. Its numerical value is a many-body result, not determined by dimensional analysis alone.

Finite temperature replaces that one ground-state coefficient with thermodynamic functions and introduces a superfluid transition at a reduced temperature determined by the interacting system. Precision pressure and compressibility measurements provide stronger evidence for thermodynamics than an arbitrary fit of a cloud width. Finite effective range, spin imbalance, nonequilibrium expansion, and losses can all complicate comparison. Pauli suppression can make a resonant two-component Fermi gas long lived, but it does not imply that every inelastic channel is absent.