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

TL;DR

This paper explores the application of Non-Relativistic Conformal Field Theory to trapped fermionic atoms, deriving virial theorems and establishing a state-operator correspondence that links energy levels to operator scaling dimensions in three dimensions.

Contribution

It introduces a novel automorphism of the conformal algebra that relates trapped and free space problems, enabling new derivations of virial theorems and confirming the state-operator correspondence in three dimensions.

Findings

Derived virial theorems for trapped Fermi gases at unitarity.

Calculated scaling dimensions matching known energy levels for few-fermion systems.

Demonstrated the applicability of the state-operator correspondence in three dimensions.

Abstract

The field theory of nonrelativistic fermions interacting via contact interactions can be used to calculate the properties of few-body systems of cold atoms confined in harmonic traps. The state-operator correspondence of Non-Relativistic Conformal Field Theory (NRCFT) shows that the energy eigenvalues (in oscillator units) of N harmonically trapped fermions can be calculated from the scaling dimensions of N-fermion operators in the NRCFT. They are also in one-to-one correspondence with zero-energy, scale-invariant solutions to the N-body problem in free space. We show that these two mappings of the trapped fermion problem to free space problems are related by an automorphism of the SL(2,R) algebra of the conformal symmetry of fermions at the unitary limit. This automorphism exchanges the internal Hamiltonian of the gas with the trapping potential and hence provides a novel method for deriving virial theorems for trapped Fermi gases at the unitary limit. We also show that the state-operator correspondence can be applied directly in three spatial dimensions by calculating the scaling dimensions of two- and three-fermion operators and finding agreement with known exact results for energy levels of two and three trapped fermions at the unitary limit.