On Non-Relativistic Conformal Field Theory and Trapped Atoms: Virial Theorems and 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.

23 pages, 3 .ps figures

References in corpus (10)

• Nonrelativistic conformal field theories
• Ultracold heteronuclear molecules in a 3D optical lattice
• Molecules of Fermionic Atoms in an Optical Lattice
• Epsilon expansion for a Fermi gas at infinite scattering length
• Analytical solutions for the dynamics of two trapped interacting ultracold atoms
• Preparation of a quantum state with one molecule at each site of an optical lattice
• Fermi gas near unitarity around four and two spatial dimensions
• Effective Theory for Trapped Few-Fermion Systems
• Density-functional theory for fermions in the unitary regime
• Density functional theory for fermions close to the unitary regime

Cited by in corpus (9)

• Gravity duals for non-relativistic CFTs
• On the spectrum of nonrelativistic AdS/CFT
• Induced p-wave superfluidity in two dimensions: Brane world in cold atoms and nonrelativistic defect CFTs
• Remark on quantum mechanics with conformal Galilean symmetry
• Remark on quantum mechanics with N=2 Schrodinger supersymmetry
• Ground-state energy of the unitary Fermi gas from the epsilon expansion
• Effective interactions and operators in no-core shell model
• Three-body contact for fermions. I. General relations
• Contact interactions, self-adjoint extensions, and low-energy scattering