Conservation of energy–momentum tensor in fermionic superfluid phase: Effects of U(1) broken symmetry

• We have constructed a new vertex for the external metric field, such that the stress tensor Ward identity is satisfied.
• The conservation of momentum and energy is maintained at each local points rather than for the whole system.
• The collective modes of broken U(1) symmetry have made nontrivial contribution to the vertex.
• Our results are important for computing viscosity from stress tensor correlations.

Respecting the conservation laws of momentum and energy in a many body theory is very important for understanding the transport phenomena. The previous conserving approximation requires that the self-energy of a single particle could be written as a functional derivative of a full dressed Green's function. This condition can not be satisfied in the G0GG0G t-matrix or pair fluctuation theory which emphasizes the fermion pairing with a stronger than the Bardeen–Cooper–Schrieffer (BCS) attraction. In the previous work [1], we have shown that when the temperature is above the superfluid transition temperature TcTc, the G0GG0G t-matrix theory can be put into a form that satisfies the stress tensor Ward identity (WI) or local form of conservation laws by introducing a new type of vertex correction. In this paper, we will extend the above conservation approximation to the superfluid phase in the BCS mean field level. To establish the stress tensor WI, we have to include the fluctuation of the order parameter or the contribution from the Goldstone mode. The result will be useful for understanding the transport properties such as the behavior of the viscosity of Fermionic gases in the superfluid phases.