Reexamining Ginzburg-Landau theory for neutron P23 superfluidity in neutron stars

Abstract

The Ginzburg-Landau (GL) effective theory is a useful tool to study a superconductivity or superfluidity near the critical temperature, and usually the expansion up to the 4th order in terms of order parameters is sufficient for the description of the second-order phase transition. In this paper, we discuss the GL equation for the neutron 3 P2 superfluidity relevant for interior of neutron stars. We derive the GL expansion up to the 8th order in the condensates and find that this order is necessary for the system to have the unique ground state, unlike the ordinary cases. Starting from the LS potential, which provides the dominant attraction between two neutrons at the high density, we derive the GL equation in the path-integral formalism, where the auxiliary field method and the Nambu-Gor'kov representation are used. We present the detailed description for the trace calculation necessary in the derivation of the GL equation. As numerical results, we show the phase diagram of the neutron 3 P2 superfluidity on the plane spanned by the temperature and magnetic field, and find that the 8th order terms lead to a first-order phase transition, whose existence was predicted in the Bogoliubov-de Gennes equation but has not been found thus far within the framework of the GL expansion up to the 6th order. The first-order phase transition will affect the interior structures inside the neutron stars. I. INTRODUCTIONNeutron stars are one of the important astrophysical objects providing us laboratories of nuclear physics (see Refs. [1,2] for recent reviews). The extreme environments in the neutron stars, such as high density state, rapid rotation, strong magnetic field, strong gravitational field, and so on, lead to interesting questions about the unconventional states of nuclear systems. For example, it is considered that there can exist a various kind of matter phases inside the neutron stars: neutron rich gas and crusts at the surface, and neutron superfluidity, hyperon matter, π, K condensates, quark matter, etc., in the inside. Those states can be accessible from observations of the mass-radius relation, timeevolution of the surface temperature, neutrino emissions, and so on. The research of the equation of state is still an open question, and many efforts have been devoted to understand the massive neutron stars whose masses are almost twice as large as the solar mass [3,4]. One of the most recent observational developments were provided by the gravitational waves from the binary neutron star merger [5]. This finding opens a new era of researches of neutron *

Vector-boson condensates, spin-triplet superfluidity of paired neutral and charged fermions, and 3P2 pairing of nucleons

“…In order not to complicate consideration we further choose the simplest form of the {γ i ψ i } 6 term (γ(ψ * i ψ i ) 3 ) assuming γ > 0. Note here that in the BCS weak coupling theory one obtains γ < 0 and expansion of the Gibbs free-energy should be continued up to the 8-th order [56].…”

“…Assuming (for γ = 0) a second-order phase transition to the paired state, in absence of the external fields one may useᾱ =ᾱ 0 t for |t| < ∼ 1. As we have mentioned, being computed in BCS approximation, the γ 6 A 6 term proves to be negative [54] that implies necessity to continue the Ginzburg-Landau expansion up to γ 8 A 8 positive contribution [56]. Simplifying consideration, as in previous sections, we will employ the simplest form of the {γ A 6 } interaction with γ > 0.…”

“…where ζ(5) ≈ 1.0369, that forces to keep A 8 term in expansion of G. Reference [56] uses a more complicated structure of the A 6 term and keeps A 8 term in expansion of G. To avoid these complications in our rough numerical estimates, e.g. for m J = −1 and −2, we take…”

“…The phases of the 3P 2 nn pairing were studied in [54][55][56][57][58] within the BCS weak coupling approximation, when the ground state corresponds to the symmetric (magnetically neutral) phase. The order parameter for the 3P 2 nn pairing is the 3x3 matrix.…”

“…However the 6th order term calculated within the BCS approximation proves to be negative [54] causing a possibility of the first-order phase transitions in the system. Recently the Ginzburg-Landau free-energy functional was expanded in the order parameter up to 8-th power and coefficients of expansion were found in the BCS approximation [56]. A particular role of the Zeeman and gradient terms, which are of our key interest here, was not studied, cf.…”