| Summary: | We calculate the sound velocity and the damping rate of the collective excitations of a 2D fermionic superfluid in a non-perturbative manner. Specifically, we focus on the Anderson−Bogoliubov excitations in the BEC-BCS crossover regime, as these modes have a sound-like dispersion at low momenta. The calculation is performed within the path-integral formalism and the Gaussian pair fluctuation approximation. From the action functional, we obtain the propagator of the collective excitations and determine their dispersion relation by locating the poles of this propagator. We find that there is only one kind of collective excitation, which is stable at <inline-formula> <math display="inline"> <semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula> and has a sound velocity of <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>v</mi> <mi>F</mi> </msub> <mo>/</mo> <msqrt> <mn>2</mn> </msqrt> </mrow> </semantics> </math> </inline-formula> for all binding energies, i.e., throughout the BEC-BCS crossover. As the temperature is raised, the sound velocity decreases and the damping rate shows a non-monotonous behavior: after an initial increase, close to the critical temperature <inline-formula> <math display="inline"> <semantics> <msub> <mi>T</mi> <mi>C</mi> </msub> </semantics> </math> </inline-formula> the damping rate decreases again. In general, higher binding energies provide higher damping rates. Finally, we calculate the response functions and propose that they can be used as another way to determine the sound velocity.
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