Abstract
We explore the thermodynamics of the 1+1-dimensional Gross-Neveu (GN) model at finite number of fermion flavors $N_f$, finite temperature and finite chemical potential using lattice field theory. In the limit $N_f \rightarrow \infty$ 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 $N_f \rightarrow \infty$ 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 $N_f \rightarrow \infty$, 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.
Highlights
The Gross-Neveu (GN) model describes Dirac fermions with Nf flavors interacting via quartic interactions in 1 þ 1 dimensions
We explore the thermodynamics of the 1 þ 1-dimensional Gross-Neveu (GN) model at a finite number of fermion flavors Nf, finite temperature, and finite chemical potential using lattice field theory
We could not answer the question, whether in GN models with a finite number of flavors translation invariance is spontaneously broken at low T and large μ, or whether the system is in a BerezinskiiKosterlitz-Thouless–like phase, we clearly spotted a low temperature and high density region, where the model exhibits oscillating spatial correlators
Summary
The Gross-Neveu (GN) model describes Dirac fermions with Nf flavors interacting via quartic interactions in 1 þ 1 dimensions. A full lattice simulation and investigation of the phase diagram of any of these models at a finite number of fermion flavors, where quantum fluctuations are taken into account, is still missing. In this work we shall use naive fermions and SLAC fermions to study the multiflavor GN model These fermion discretizations are all chiral, and no fine-tuning is required to end up with a chirally symmetric continuum limit. II we summarize some known features of the GN model that are relevant for its thermodynamical properties These include properties of the fermion determinant in the continuum and on the lattice, homogeneous and inhomogeneous phases in the Nf → ∞ limit, and some comments concerning the spontaneous symmetry breaking (SSB) of translation invariance. In the Appendix we discuss why the lattice GN model with naive fermions may have an incorrect continuum limit, and how the interaction term can be modified to end up with an (almost) naive fermion discretization with a correct continuum limit
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