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The field theory of nonrelativistic fermions interacting via contactinteractions can be used to calculate the properties of few-body systems ofcold atoms confined in harmonic traps. The state-operator correspondence ofNon-Relativistic Conformal Field Theory (NRCFT) shows that the energyeigenvalues (in oscillator units) of N harmonically trapped fermions can becalculated from the scaling dimensions of N-fermion operators in the NRCFT.They are also in one-to-one correspondence with zero-energy, scale-invariantsolutions to the N-body problem in free space. We show that these two mappingsof the trapped fermion problem to free space problems are related by anautomorphism of the SL(2,R) algebra of the conformal symmetry of fermions atthe unitary limit. This automorphism exchanges the internal Hamiltonian of thegas with the trapping potential and hence provides a novel method for derivingvirial theorems for trapped Fermi gases at the unitary limit. We also show thatthe state-operator correspondence can be applied directly in three spatialdimensions by calculating the scaling dimensions of two- and three-fermionoperators and finding agreement with known exact results for energy levels oftwo and three trapped fermions at the unitary limit.

On Non-Relativistic Conformal Field Theory and Trapped Atoms: Virial Theorems and the State-Operator Correspondence in Three Dimensions