The low-energy magnetic excitations of a three-band Hubbard model with a strong spin-orbit coupling for 5d transition metal oxide Sr$_2$IrO$_4$

5$d$ transition metal oxides in a layered perovskite structure such as Sr$_2$IrO$_4$ have attracted much attention because of their unique properties caused by a strong relativistic spin-orbit coupling of 5$d$ transition element. Recent experiments on Sr$_2$IrO$_4$ have revealed that the low-energy magnetic excitations can be described by an ``isospin''-1/2 Heisenberg model with an effective exchange interaction as large as $\sim$ 60-100 meV. Motivated by these experiments, we study theoretically the ground state magnetic structure and the low-energy magnetic excitations for Sr$_2$IrO$_4$ using a three-band Hubbard model with the spin-orbit coupling. Our results demonstrate that the low-energy magnetic excitations are well described by an effective antiferromagnetic Heisenberg model composed of a local Kramers doublet. The estimated value of the effective exchange interaction is as large as 79 meV, which is in good quantitative agreement with the experiments.