Inhomogeneous phases in the Gross-Neveu model in 1+1 dimensions at finite number of flavors

We explore the thermodynamics of the 1 + 1-dimensional Gross-Neveu (GN) model at a finite number of fermion flavors N-f, finite temperature, and finite chemical potential using lattice field theory. In the limit N-f -> infinity the model has been solved analytically in the continuum. In this limit three phases exist: a massive phase, in which a homogeneous chiral condensate breaks chiral symmetry spontaneously; a massless symmetric phase with vanishing condensate; and most interestingly an inhomogeneous phase with a condensate, which oscillates in the spatial direction. In the present work we use chiral lattice fermions (naive fermions and SLAC fermions) to simulate the GN model with 2, 8, and 16 flavors. The results obtained with both discretizations are in agreement. Similarly as for N-f -> infinity we find three distinct regimes in the phase diagram, characterized by a qualitatively different behavior of the two-point function of the condensate field. For N-f = 8 we map out the phase diagram in detail and obtain an inhomogeneous region smaller as in the limit N-f -> infinity, where quantum fluctuations are suppressed. We also comment on the existence or absence of Goldstone bosons related to the breaking of translation invariance in 1 + 1 dimensions.