Abstract
We investigate the finite-temperature and -density chiral Gross-Neveu model with an axial U$_A$(1) symmetry in $1+1$ dimensions on the lattice. In the limit where the number of flavors $N_\mathrm{f}$ tends to infinity the continuum model has been solved analytically and shows two phases: a symmetric high-temperature phase with a vanishing condensate and a low-temperature phase in which the complex condensate forms a chiral spiral which breaks translation invariance. In the lattice simulations we employ chiral SLAC fermions with exact axial symmetry. Similarly to $N_\mathrm{f}\to\infty$, we find for $8$ flavors, where quantum and thermal fluctuations are suppressed, two distinct regimes in the $(T,\mu)$ phase diagram, characterized by qualitatively different behavior of the two-point functions of the condensate fields. More surprisingly, at $N_\mathrm{f}=2$, where fluctuations are no longer suppressed, the model still behaves similarly to the $N_\mathrm{f}\to\infty$ model and we conclude that the chiral spiral leaves its footprints even on systems with a small number of flavors. For example, at low temperature the two-point functions are still dominated by chiral spirals with pitches proportional to the inverse chemical potential, although in contrast to large-$N_\mathrm{f}$ their amplitudes decrease with distance. We argue that these results should not be interpreted as the spontaneous breaking of a continuous symmetry, which is forbidden in two dimensions. Finally, using Dyson-Schwinger equations we calculate the decay of the U$_A$(1)-invariant fermion four-point function in search for a BKT phase at zero temperature.
Highlights
A surprising amount of physical phenomena in particleand condensed-matter physics are well described by fourFermi theories
The effective four-Fermi theory describing the dynamics of nucleons and mesons goes back to Nambu and Jona-Lasinio (NJL) [1] and is built upon interacting Dirac fermions with chiral symmetry, paralleling the construction of Cooper pairs from electrons in the BCS theory of superconductivity
In the present work we studied the (1 þ 1)-dimensional chiral Gross-Neveu model with chiral SLAC fermions and exact axial UAð1Þ symmetry on the lattice
Summary
A surprising amount of physical phenomena in particleand condensed-matter physics are well described by fourFermi theories. Interacting Fermi theories at finite temperature and density were mainly investigated in the limit of a large number of fermion flavors Nf. For Nf → ∞ the saddle-point approximation becomes exact and one can solve the corresponding gap equation analytically on the set of homogeneous condensates. They have been constructed in [16] for the GN model with discrete and in [17,18] for the chiral GN model with continuous chiral symmetry These remarkable analytic results for Nf → ∞ prove the existence of inhomogeneous phases, which are regions in parameter space where the chiral condensate acquires a spatial dependence, indicating the spontaneous breakdown of chiral symmetry alone but in a combination with spacetime symmetries (see [19] for a review). Towards the end we exploit Dyson-Schwinger equations to study the UAð1Þ-invariant fermion four-point function in the infrared
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