A monotone Lagrangian torus is a Lagrangian torus in a monotone symplectic manifold satisfying a certain balanced condition. Each monotone Lagrangian torus assigns a Laurent polynomial (called a potential function) which is an invariant of a Lagrangian up to Hamiltonian isotopy. In this talk, we will construct an infinitely many monotone Lagrangian tori in a full flag manifold and distinguish them by calculating their potential functions using the theory of cluster algebras. If time permits, I will also talk about a generalization of this result to the case of partial flag manifolds. This talk is based on joint works with Myungho Kim, Yoosik Kim, and Euiyong Park.