Collective modes of an imbalanced trapped Fermi gas in two dimensions at finite temperatures

\u3cp\u3ePhase diagrams and collective excitations are investigated for a trapped imbalanced Fermi gas in two dimensions with s-wave pairing at finite temperatures. The treatment of collective modes is performed within the hydrodynamic approach using the Euler and continuity equations. The equations of state for different phases are simulated by a polytropic law with parameters obtained using a fit to the microscopic equilibrium distributions of the pressure and density. These are determined within the whole range of the BCS-BEC crossover using a path-integral description and taking into account fluctuations about the saddle point. We focus on the case of imbalanced gases, when the number of spin-up and spin-down fermions that form the pair is not equal. The superfluid-to-normal transition in the trapped Fermi gas is governed by the Berezinskii-Kosterlitz-Thouless mechanism. The eigenfrequencies of collective modes behave differently from those in the zero-temperature case. They can be used to study a realistic equation of state of a trapped Fermi gas at finite temperatures.\u3c/p\u3