Inhomogeneous chiral phases away from the chiral limit

Abstract

The effect of explicit chiral-symmetry breaking on inhomogeneous chiral phases is studied within a Nambu-Jona-Lasinio model with nonzero current quark mass. Generalizing an earlier result obtained in the chiral limit, we show within a Ginzburg-Landau analysis that the critical endpoint of the first-order chiral phase boundary between two homogeneous phases gets replaced by a "pseudo-Lifshitz point" when the possibility of inhomogeneous order parameters is considered. Performing a stability analysis we also show that the unstable mode along the phase boundary is in the scalar but not in the pseudoscalar channel, suggesting that modulations which contain pseudoscalar condensates, like a generalized dual chiral density wave, are disfavored against purely scalar ones. Numerically we find that the inhomogeneous phase shrinks as one moves away from the chiral limit, but survives even at significantly large values of the current quark mass.

Inhomogeneous chiral condensates in three-flavor quark matter

“…In particular it was found for the two-flavor NJL model in mean-field approximation that the inhomogeneous phase completely covers the first-order phase boundary between the homogeneous phases [11] and that the LP exactly coincides with the CP [10]. 1 Recently we have generalized this result to the case away from the chiral limit [14]. A similar picture also emerged from a truncated Dyson-Schwinger approach to QCD, although the coincidence of CP and LP has not yet been shown rigorously [15].…”

“…In the context of chiral symmetry breaking the order parameter is usually identified with the chiral condensate, and the expansion is performed about the chirally restored solution where the condensate vanishes. Of course, this only works in the chiral limit, while in the presence of nonzero bare quark masses there is no chirally restored solution and therefore the expansion has to be performed around nonvanishing condensate values [14].…”

“…In this case a two-flavor restored solution σ = π = 0 exists, which we take as the expansion point of the GL analysis. In the strange-quark sector we proceed similarly as in [14] for the case of light quarks away from the chiral limit and expand around a T and µ-dependent but spatially constant condensate σ (0) s , corresponding to a stationary point of Ω at σ = π = 0. To this end we write…”

“…Here, the orders in ∆ and gradients are treated equally, O(∇ n ) = O(∆ n ), as we already did when we wrote down Eq. (14). From Eq.…”

“…The GL analysis is valid in regimes where both the amplitudes and the gradients of the order parameters are small, i.e., in the vicinity of second-order phase boundaries between homogeneous phases or in the inhomogeneous phase close to the LP. It is possible however to determine the entire second-order boundary between a homogeneous and an inhomogeneous phase as well if we drop the assumption of small gradients while keeping the assumption of small amplitudes [9,14]. To this end we start from Eq.…”