Anomalous discrete chiral symmetry in the Gross–Neveu model and loop gas simulations

We investigate the discrete chiral transformation of a Majorana fermion on a torus. Depending on the boundary conditions the integration measure can change sign. Taking this anomalous behavior into account we define a chiral order parameter as a ratio of partition functions with differing boundary conditions. Then the lattice realization of the Gross–Neveu model with Wilson fermions is simulated using the recent ‘worm’ technique on the loop gas or all-order hopping representation of the fermions. An algorithm is formulated that includes the Gross–Neveu interaction for N fermion species. The critical line m_c(g) is constructed for a range of couplings at N=6 and for N=2, the Thirring model, as examples.