Abstract
The phase diagram of the two-dimensional Nambu--Jona-Lasinio model with isospin is explored in the large Nc limit with semiclassical methods. We consider finite temperature and include chemical potentials for all conserved charges. In the chiral limit, a full analytical solution is presented, expressed in terms of known results for the single-flavor Gross-Neveu and Nambu--Jona-Lasinio models. A novel crystalline structure appears and is shown explicitly to be thermodynamically more stable than the homogeneous phase at zero temperature. If we include a bare fermion mass, the problem reduces again to solved problems in one-flavor models provided that either the fermionic or the isospin chemical potentials vanish. In the general case, a stability analysis is used to construct the perturbative phase boundary between homogeneous and inhomogeneous phases. This is sufficient to get a good overview of the complete phase diagram. Missing non-perturbative phase boundaries requiring a full numerical Hartree-Fock calculation will be presented in future work.
Highlights
The two most widely studied versions of the GrossNeveu (GN) model in 1 þ 1 dimensions are the original one [1] with Lagrangian LGN 1⁄4 ψði∂ −m0Þψ þ g2 2 ðψψÞ2ð1Þ and the chiral GN or 1 þ 1 dimensional Nambu–JonaLasinio (NJL) model [2], LNJL g2 21⁄2ðψψÞ2 þ ðψiγ5ψÞ2: ð2ÞIn the massless limit (m0 1⁄4 0), LGN has a discrete Zð2ÞL × Zð2ÞR chiral symmetry, promoted to a continuous Uð1ÞL × Uð1ÞR chiral symmetry in LNJL
The Dirac fermions come in Nc “colors”, and such models are typically solved in the large Nc limit [3] with semiclassical methods
We have studied the phase diagram of the isoNJL model in 1 þ 1 dimensions
Summary
The two most widely studied versions of the GrossNeveu (GN) model in 1 þ 1 dimensions are the original one [1] with Lagrangian. The Moscow group [14,15] has presented several variational calculations of the phase diagram including isospin and chiral imbalance, both with homogeneous and inhomogeneous mean fields. They emphasize the phenomenon of charged pion condensation and a certain duality. The phase diagram of the massive isoNJL model has been addressed within several variational calculations in recent years [16,17,18]. We believe that the toolbox developed in the past for solving models (1) and (2) should contain everything necessary for determining the complete phase diagram of the isoNJL model as well, using a combination of numerical and analytical methods. V, we summarize our findings and point out areas where further numerical work is needed
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