Crystalline chiral condensates off the tricritical point in a generalized Ginzburg-Landau approach

Abstract

We present an extensive study on inhomogeneous chiral condensates in QCD at finite density in the chiral limit using a generalized Ginzburg-Landau (GL) approach. Performing analyses on higher harmonics of one-dimensionally (1D) modulated condensates, we numerically confirm the previous claim that the solitonic chiral condensate characterized by Jacobi's elliptic function is the most favorable structure in 1D modulations. We then investigate the possibility of realization of several multidimensional modulations within the same framework. We also study the phase structure far away from the tricritical point by extending the GL functional expanded up to the eighth order in the order parameter and its spatial derivative. On the same basis, we explore a new regime in the extended GL parameter space and find that the Lifshitz point is the point where five critical lines meet at once. In particular, the existence of an intriguing triple point is demonstrated, and its trajectory consists of one of those critical lines.

Inhomogeneous chiral condensates

“…In Ref. [83] Abuki, Ishibashi and Suzuki have extended Nickel's analysis to include terms up to 8th order in the expansion. Assuming that the 8th-order terms are positive, they have then studied the influence of the sign of the 6th-order coefficient γ 6 on the phase diagram.…”

“…The GL analysis of the phase transition between the homogeneous chirally broken to the inhomogeneous phase is more difficult but has been done as well [44,83]. We will come back to this in Sec.…”

Ginzburg–Landau Phase Diagram under Magnetic Field

“…It turns out that these two ingredients bring competing effects on inhomogeneous phases. In particular, the condensate accompanied by the complex phase, the chiral spiral, is found to be favored by the magnetic field [6], and accordingly the phase diagram is to be drastically changed once the effect of magnetic field prevails.The gGL action density at the first nontrivial order in the external magnetic field B and the current quark mass m q can be derived in the same way as described in [8,9]. Up to the sixth order in the scalar and pseudoscalar condensates, σ and π a (a = 1, 2, 3), as well as in ∇ ≡ ∂ x , the result is [2] …”

“…The gGL action density at the first nontrivial order in the external magnetic field B and the current quark mass m q can be derived in the same way as described in [8,9]. Up to the sixth order in the scalar and pseudoscalar condensates, σ and π a (a = 1, 2, 3), as well as in ∇ ≡ ∂ x , the result is [2] …”

Multi-Dimensional Structure of Crystalline Chiral Condensates in Quark Matter

“…Thus, one possible way to investigate iCPs with multi-D modulations is to start with some ansätze for multi-D modulations and compare their free energies with those with 1D modulations. In the vicinity of the Lifshitz point (LP), a GL analysis with multi-D RKC ansätze [10] indicates that the 1D RKC condensate is the most thermodynamically favored, followed in order by the 2D RKC, 3D RKC, and 1D DCDW condensates. It is interesting to note here that the 2D DCDW condensate is the most favored for multi-D DCDW condensates.…”