Fermion bag approach for the massive Thirring model at finite density

We consider the 2 + 1-dimensional massive Thirring model with one flavor at finite density. Two numerical methods, the fermion bag approach and complex Langevin dynamics, are used to calculate the chiral condensate and fermion density of this model. The numerical results obtained by the fermion bag approach are compared with those obtained by complex Langevin dynamics. They are also compared with those obtained under a phase-quenched approximation. We show that in some range of fermion coupling strength and chemical potential, the sign problem in the fermion bag approach is mild, while it becomes severe for the complex Langevin dynamics.