Abstract
The phase diagram of the Gross-Neveu model in $2+1$ space-time dimensions at nonzero temperature and chemical potential is studied in the limit of infinitely many flavors, focusing on the possible existence of inhomogeneous phases, where the order parameter $\ensuremath{\sigma}$ is nonuniform in space. To this end, we analyze the stability of the energetically favored homogeneous configuration $\ensuremath{\sigma}(\mathbf{x})=\overline{\ensuremath{\sigma}}=\text{const}$ with respect to small inhomogeneous fluctuations, employing lattice field theory with two different lattice discretizations as well as a continuum approach with Pauli-Villars regularization. Within lattice field theory, we also perform a full minimization of the effective action, allowing for arbitrary 1-dimensional modulations of the order parameter. For all methods special attention is paid to the role of cutoff effects. For one of the two lattice discretizations, no inhomogeneous phase was found. For the other lattice discretization and within the continuum approach with a finite Pauli-Villars cutoff parameter $\mathrm{\ensuremath{\Lambda}}$, we find a region in the phase diagram where an inhomogeneous order parameter is favored. This inhomogeneous region shrinks, however, when the lattice spacing is decreased or $\mathrm{\ensuremath{\Lambda}}$ is increased, and finally disappears for all nonzero temperatures when the cutoff is removed completely. For vanishing temperature, we find hints for a degeneracy of homogeneous and inhomogeneous solutions, in agreement with earlier findings.
Highlights
Mapping the phase diagram of quantum chromodynamics (QCD) at nonzero temperature T and quark chemical potential μ is one of the major challenges in the field of strong-interaction physics [1,2]
(iii) Possibly an inhomogeneous phase, where σðxÞ is a varying function of the spatial coordinates, at intermediate μ and small T [33]. This phase might only be present at a finite value of the regulator (e.g., PauliVillars cutoff Λ or lattice spacing a), as indicated by recent lattice field theory results reported in Ref. [34] and the behavior of the Lifshitz point (LP) found in Sec
At finite Λ we find that instabilities only occur in the symmetric phase of the homogeneous phase diagrams, while the homogeneous symmetry-broken phase remains stable against small inhomogeneous fluctuations
Summary
Mapping the phase diagram of quantum chromodynamics (QCD) at nonzero temperature T and quark chemical potential μ is one of the major challenges in the field of strong-interaction physics [1,2]. The difficulty arises that the determination of the ground state corresponds to the functional minimization of the effective action with respect to the condensate hψψiðxÞ with arbitrary spatial shape This is a very hard problem, which, until now, has only been solved in 1 þ 1-dimensional models [13,14,15,22] but not in higher dimensions. An interesting alternative is to investigate inhomogeneous phases with lattice field theory or related numerical methods, where, at least in principle, the effective action can be minimized without restricting the condensate to a specific ansatz.
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