Precise Phase Structure in Four-fermion Interaction Model on Torus

We investigate finite-size effects on chiral symmetry breaking in a four-fermion interaction model at a finite temperature and a chemical potential. Applying the imaginary time formalism, the thermal quantum field theory is constructed on an $S^1$ in the imaginary time direction. In this paper, the finite-size effect is introduced by a compact $S^1$ spatial direction with a $\mathrm{U}(1)$-valued boundary condition. Thus, we study the model on a $\mathbb{R}^{D-2} \times S^1 \times S^1$ torus. Phase diagrams are obtained by evaluating the local minima of the effective potential in the leading order of the $1/N$ expansion. From the grand potential, we calculate the particle number density and the pressure, then we illustrate the correspondence with the phase structure. We obtain a stable size for which the sign of the pressure flips from negative to positive as the size decreases. Furthermore, the finite chemical potential expands the parameter range that the stable size exists.