arX

iv:1

908.

0794

4v2

[nu

cl-t

h] 2

5 Fe

b 20

20

Critical endpoint and universality class of neutron 3P2 superfluids in neutron stars

Takeshi Mizushima,1, ∗ Shigehiro Yasui,2, † and Muneto Nitta2, ‡

1Department of Materials Engineering Science, Osaka University, Toyonaka, Osaka 560-8531, Japan2Department of Physics & Research and Education Center for Natural Sciences,

Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan

(Dated: February 26, 2020)

We study the thermodynamics and critical behavior of neutron 3P2 superfluids in the inner cores of neutron

stars. 3P2 superfluids offer a rich phase diagram including uniaxial/biaxial nematic phases, the ferromagnetic

phase, and the cyclic phase. Using the Bogoliubov-de Gennes (BdG) equation as superfluid Fermi liquid theory,

we show that a strong (weak) magnetic field drives the first (second) order transition from the dihedral-two

biaxial nematic phase to dihedral-four biaxial nematic phase in low (high) temperatures, and their phase bound-

aries are divided by the critical endpoint (CEP). We demonstrate that the set of critical exponents at the CEP

satisfies the Rushbrooke, Griffiths, and Widom equalities, indicating a new universality class. At the CEP, the3P2 superfluid exhibits critical behavior with nontrivial critical exponents, indicating a new universality class.

Furthermore, we find that the Ginzburg-Landau (GL) equation up to the 8th-order expansion satisfies three

equalities and properly captures the physics of the CEP. This implies that the GL theory can provide a tractable

way for understanding critical phenomena which may be realized in the dense core of realistic magnetars.

I. INTRODUCTION

A neutron star is a compact star which is composed al-

most entirely of neutrons under extreme conditions such as

high density, rapid rotation, and a strong magnetic field (see

Refs. [1, 2] for recent reviews). The most recent discover-

ies include the observations of massive neutron stars whose

masses are almost twice as large as the solar mass [3, 4]

and the observation of gravitational waves from a binary neu-

tron star merger [5]. In the inner structure, neutron super-

fluidity and proton superconductivity are key ingredients for

understanding the evolution of neutron stars (see Refs. [6–

8] for recent reviews). As the superfluid and superconduct-

ing components reorganize low-lying elementary excitations,

their presence profoundly affects neutrino emissivities and

specific heats and can explain the long relaxation time ob-

served in the sudden speed-up events of neutron stars [9–11]

and the enhancement of neutrino emission at the onset of su-

perfluid transition [12–17]. Sudden changes of spin periods

observed in pulsars (pulsar glitches) may also be explained by

the existence of superfluid components with quantized vor-

tices [18, 19].

We notice that the 1S 0 channel, which is attractive at low

density, becomes repulsive in the high density regime.1 In-

stead, the neutron 3P2 superfluids can be realized at the high

density regime ρ & 1014g/cm3 (ρ is the density of neutrons),

where the 3P2 interaction stems from a strong spin-orbit force

between two nucleons [22–40].2 Hence, the neutron 3P2 su-

∗ mizushima@mp.es.osaka-u.ac.jp† yasuis@keio.jp‡ nitta(at)phys-h.keio.ac.jp1 In the literature, the 1S 0 superfluidity at low density was proposed in

Ref. [20]. However, it was pointed out in Ref. [21] that this channel turns

to be repulsive due to the strong core repulsion at higher densities.2 It is noted that the interaction in the 3P0 and 3P1 channels are repulsive

one at high density, and hence they are irrelevant to the formation of the

superfluidity [41].

perfluids are expected to be realized in the inner cores of neu-

tron stars. Furthermore, the neutron 3P2 superfluids have tol-

erance against the strong magnetic field, such as 1015 − 1018

G in magnetars, because the spin-triplet pairing is not bro-

ken through the spin-magnetic field interaction by Zeeman ef-

fects.3 It has recently been proposed that the observation of

the rapid cooling of the neutron star in Cassiopeia A may be

explained by enhanced neutrino emissivities due to the for-

mation and dissociation of neutron Cooper pairs in the 3P2

channel which is a short-ranged attraction in the total angu-

lar momentum J = 2 [16, 17, 52] (see also Refs. [53–55]).

Theoretically, neutron 3P2 superfluids provide a fertile ground

for exploring exotic superfluidity. The superfluid states with

J = 2 are classified into several phases: Nematic, cyclic, and

ferromagnetic phases [27, 28, 56–61]. The nematic phase is

further divided into the uniaxial nematic (UN) phase and the

dihedral-two and dihedral-four biaxial nematic (D2-BN and

D4-BN) phases. All these phases are accompanied by topo-

logically protected Bogoliubov quasiparticles. The nematic

phase is a prototype of class-DIII topological superconductors

and a harbor of Majorana fermions [62]. The other phases are

non-unitary states with broken time-reversal symmetry and

promising platforms to host Weyl superfluidity [62, 63]. In ad-

dition to such exotic fermions, the 3P2 order parameters also

bring about rich bosonic excitations [64–76], which might be

relevant to the cooling process by neutrino emission4, as well

as exotic topological defects, including spontaneously mag-

netized vortices [27, 57, 58, 60] and vortices with Majorana

fermions [78], solitonic excitations on a vortex [79], and half-

3 The origin of the strong magnetic fields in neutrons stars or in magnetars

has been studied in several types of mechanisms such as spin-dependent

interactions [42–45], pion domain walls [46, 47], spin polarizations in the

quark-matter in the neutron star core [48–50] and so on. However, this

problem is not settled yet. Recently, a negative result for the generation

of strong magnetic fields was reported in the study based on the nuclear

many-body calculations [51].4 The cooling process is related not only to low-energy excitations but also

to quantum vortices [77].

2

FIG. 1. (a, b) Phase diagram of 3P2 superfluids under a magnetic field computed with the superfluid Fermi liquid theory for the Fermi liquid

parameter G(n)

0= −0.7 (a) and G

(n)

0= −0.4 (b). The thick solid (thin broken) curve is the first (second) order phase boundary and “CEP”

denotes the critical endpoints. (c, d) Phase diagram in G(n)

0= −0.75 computed from the GL theory with the 8th-order expansion. The color

map in (a-d) represents the nematic order parameter r(T, B). In the GL theory [(c) and (d)], the critical endpoint is given by Tcep/Tc = 0.774597

and γnBcep/(πTc) = 0.004465.

quantized non-Abelian vortices [61], domain walls [80], and

surface topological defects (boojums) on the boundary of 3P2

superfluids [81]. Those states share common properties in

the condensed matter systems, such as D-wave superconduc-

tors [82], P-wave superfluidity in 3He liquid [83–85], chi-

ral P-wave superconductivity e.g. in Sr2RuO4 [86, 87] and

U-based ferromagnetic superconductors [88], spin-2 Bose-

Einstein condensates [89], and so on.

The neutron 3P2 superfluidity can be described by the Fermi

liquid theory which is composed of the set of self-consistent

equations based on the Luttinger-Ward thermodynamic func-

tional. Microscopically, the most fundamental equation of the

neutron 3P2 superfluidity is provided by the Bogoliubov-de

Gennes (BdG) equation where the order parameter, i.e., gap

function, should be solved self-consistently with the wave-

functions of the gapped neutrons [22–25, 30, 32–40]. The

BdG equation was successfully applied to study the topologi-

cal properties of the neutron 3P2 superfluidity [62]. The phase

digram with respect to the magnetic field and temperature

was obtained in Ref. [62], where the first and second-order

phase transitions between the D2-BN and D4-BN phases are

present and these transitions meet at a critical endpoint (CEP),

as shown in Figs. 1(a) and 1(b).5 The existence of the CEP is

cerntainly important for transport coefficients and equations of

state of neutron matter when neutron stars are cooled down.

Around the transition temperature from the normal phase

to the superfluid phase, the Ginzburg-Landau (GL) theory

can be induced by the systematic expansion of the functional

with respect to the order parameter field and the magnetic

field [27, 28, 56–61, 81, 90–92]. Unlike the ordinary cases,

the GL expansion up to the 4th order in terms of the order pa-

rameter cannot determine the unique ground state, but there

exists a continuous degeneracy among the UN, D2-BN and

D4-BN phases.6 The GL expansion up to the 6th order deter-

5 See Eqs. (30) and (34) for the definitions of G(n).6 We notice that, at the 4th order, there happens to exist an SO(5) symme-

mines the unique ground state [60] but it is stable only locally

and there exists the instability for a large value of the order

parameter. Recently, in order to solve this problem, the GL

equation up to the 8th order term in the condensates was ob-

tained [92], in which it was shown that the 8th order term

ensures the global stability with respect to the variation of the

order parameter in the ground state. As a byproduct, it was

also found that the phase diagram in the expansion up to the

8th oder possesses the CEP as shown in Fig. 1(c), in contrast

to the GL equation up to the 6th order in which no CEP ex-

ists, although the positions of the CEPs in the BdG and GL

formalism are rather different as shown in Fig. 1(d).

In this paper, we study the critical exponents at the CEP in

the BdG equation and in the GL equation. Under the scaling

hypothesis, a set of critical exponents, (α, β, γ, δ),7 at the CEP

should satisfy the the universal relations, i.e., the Rushbrooke,

Griffiths, and Widom equalities

α + 2β + γ = 2 (Rushbrooke), (1)

α + β(1 + δ) = 2 (Griffiths), (2)

−γβ+ δ = 1 (Widom). (3)

In the both cases of the BdG equation and of the GL equation,

we extract the critical behavior of neutron 3P2 superfluids by

directly computing all the critical exponents at the CEP. We

demonstrate that the CEP in the GL approach properly cap-

tures the critical phenomena in the BdG equation, and the ex-

ponents satisfy all three equalities reasonably in both the BdG

and GL equations within a numerical error. We find that the

try as an extended symmetry in the potential term, which is absent in the

original Hamiltonian. In this case, the spontaneous breaking eventually

generate a quasi-Nambu-Goldstone mode which should be irrelevant to the

excitations in the true ground state [93]. This is nothing but the origin of

the continuous degeneracy.7 See Eqs. (57)-(59) for the definitions of (α, β, γ, δ).

3

3P2 superfluid at the CEP exhibits critical behavior with non-

trivial critical exponents in such a manner that the exponents

associated with the critical behaviors of the specific heat and

magnetization exhibits α ∼ 0.6 and γ ∼ 0.5. In particular,

the exponent γ < 1 is unique and essentially different from

γ ≥ 1 in ordinary universality classes [94, 95], except for a

few models, e.g., O(n) models with n < 0 [96] and the tricriti-

cal Ising model coupled to massless Dirac fermions [97]. This

indicates the CEP in neutron 3P2 superfluids belongs to a new

universality class.

The organization of this paper is as follows. In Sec. II,

we present the superfluid Fermi liquid theory, where the self-

consistent equations for the gap functions and Fermi liquid

corrections are described in detail. This theory is based on

the quasiclassical approximation which is relevant to 3P2 su-

perfluids of neutrons. Based on the theory, we show that the

phase diagram of 3P2 superfluids under strong magnetic fields

has the CEP and compute the critical exponents, indicating a

new universality class. Furthermore, in Sec. III, we present

the GL theory up to the 8th-order expansion to examine the

critical phenomena at the CEP, showing that the critical expo-

nents in the GL theory coincide with those in the BdG theory

within a certain accuracy. Sec. IV is devoted to a summary

and discussion.

II. SUPERFLUID FERMI LIQUID THEORY

A. General formalism

Here we start with the Hamiltonian for neutrons interacting

through the potentialVc,d

a,b,

H =∫

drψ†a(r)ξab(−i∇)ψb(r)

+1

2

∫

dr1

∫

dr2Vc,d

a,b(r12)ψ†a(r1)ψ

†b(r2)ψc(r2)ψd(r1), (4)

where r12≡ r1 − r2 denotes the relative coordinate and ψa and

ψ†a (a =↑, ↓ for spins) denote the fermionic field operators.

The single-particle energy for a neutron under a magnetic field

B is given by

ξ(k) = ξ0(k) − 1

2γnσ · B, (5)

with ξ0(k) = k2/(2m) − µ for the neutron mass m and the

chemical potential µ. Here γn = 1.2 × 10−13 MeV/T is the

gyromagnetic ratio for a neutron,8 and σ = (σ1, σ2, σ3) de-

notes the Pauli matrices in the spin space. In Eq. (4),Vc,d

a,b(r12)

contains microscopic informations on neutron-neutron inter-

action potentials. The repeated Roman and Greek indices im-

ply the sum over the spin degrees of freedom and the three-

dimensional spatial component (x, y, z), respectively. In this

paper, we set ~ = kB = 1.

8 Notice the unit conversion 1 T = 104 G for the strength of a magnetic field.

Let us define the Nambu-Gor’kov (NG) Green’s function

in terms of a grand ensemble average of the fermion-field

operators in the Nambu space, Ψ ≡ (ψ↑, ψ↓, ψ†↑, ψ

†↓)

tr, as

G(x1, x2) = −〈TτΨ(x1)Ψ†(x2)〉, where xi ≡ (ri, τi) with the

three dimensional space position ri and the imaginary time

τi for the neutron i = 1, 2. atr denotes the transpose of the

matrix a. In this paper, we consider translationally invari-

ant neutron matter and transform the space-time position x

to momentum p and Matsubara frequency εn = (2n + 1)πT

(n = 0,±1,±2, . . . ): x → (p, εn). The self-consistent formal-

ism is derived from the Luttinger-Ward thermodynamic func-

tional which is given in terms of the full NG Green’s function

G and the self-energy Σ as

Ω[G,Σ] = − 1

2Sp

ΣG + ln(

−G−10 + Vext + Σ

)

+ Φ[G], (6)

where

Sp · · · ≡ T∑

n

∫

d3 p

(2π)3Tr · · · , (7)

with the trace (Tr) taken over the spin space and the

NG (particle-hole) space. The inverse propagator for free

fermions is given by G−10

(p, εn) =[

iεn − ξ0(p)τ3

]

δ(x−x′), and

Vext is an external field including a magnetic Zeeman term in

Eq. (5). Here we use τ = (τ1, τ2, τ3) to denote the matrices

in the NG space. The Green’s function and the self-energy

are related to the functionalΦ[G] by the stationary conditions

with respect to the Green’s function, δΩ/δGtr = 0, and the

self-energy, δΩ/δΣtr = 0. The former is recast into the defini-

tion of the self-energy in terms of the functional derivative

Σ[G] = 2δΦ[G]

δGtr. (8)

The Dyson’s equation for the full Green’s function is obtained

from the latter stationary condition as

G−1 = G−10 − Vext − Σ[G]. (9)

The above set of equations from Eq. (6) to Eq. (9) provide

a starting point for deriving the quasiclassical Fermi liquid

theory for 3P2 superfluids.

B. Quasiclassical approximation

In general, the quasiclassical approximation provides a

powerful tool for describing phenomena when the charac-

teristic lengths are much greater than the Fermi wavelength,

λF ∼ 2π/pF (pF the Fermi momentum), and characteristic fre-

quencies are much smaller than the Fermi energy, ω ≪ εF/~

(εF the Fermi energy) [98, 99]. The typical scales in the super-

fluid state of 3He and superconducting states are the coherence

length, ξc ≡ ~vF/2πkBTc and the excitation gap ∆0 ∼ kBTc.

The quasiclassical theory uses the fact that all relevant param-

eters, such as temperature T and external potentials V , are

very small relative to the atomic scales which are given by

Fermi temperature TF, Fermi energy εF and Fermi momen-

tum pF. This difference in scales allows one to perform an

4

asymptotic expansion of full many-body propagators in small

parameters T/TF and |V |/εF, and it leads eventually to inte-

grate out all quantities that vary on the atomic scales.

A key feature of the quasiclassical approximation is that G

is sharply peaked at the Fermi surface, and depends weakly on

energies far away from it. We use this assumption to split the

propagator into low and high energy parts, G = Glow + Ghigh,

where Glow(p, εn) = G(p, εn) for |ε| < εc and Glow(p, εn) = 0

for |ε| > εc . The cutoff energy εc is taken to be εc ≪ εF.

As shown in Fig. 2, we introduce the renormalized vertices

(filled circles) that sum an infinite set of diagrams composed

of the high energy part of the propagator and the bare vertices

(open circles). The low energy part of the propagator obeys

the Dyson equation,

Glow(p, εn) = Glow0 (p, εn) +Glow

0 (p, εn)Σ(p, εn)Glow(p, εn),(10)

where Glow0

denotes the low energy part for the free propaga-

tor. This equation will play an important role in the following

discussion.

For the convenience of the analysis, we define the quasi-

classical propagator for the low energy part, g(pF, εn), as an

integral over a shell, |vF · (p − pF)|, in momentum space near

the Fermi surface:

g(pF, εn) =1

a

∫ εc

−εc

dξpτ3Glow(p, ǫn), (11)

where ξp = vF ·(p− pF) and vF = v(pF) with v(p) = ∂ξ(p)/∂p is

the Fermi velocity. The propagator is normalized by dividing

by the weight of the quasiparticle pole in the spectral function,

a. The quasiclassical propagator matrix is parameterized as

g =

(

g0 + g · σ iσy f0 + iσ · fσy

iσy f0 + iσyσ · f g0 + g · σtr

)

, (12)

where f0 and f represent the spin-singlet and spin-triplet com-

ponents of anomalous propagators, and g0 and g represent the

spin-singlet and spin-triplet components of normal propaga-

tors. The Matsubara propagators maintain the following sets

of symmetry relations in the NG space,

(

g(pF, εn))†= τ3g(pF,−εn)τ3, (13)

(

g(pF, εn))tr= τ2g(−pF,−εn)τ2. (14)

To convert the Dyson equation (9) to a transport-like equa-

tion, we first perform the “left-right subtraction trick” for qua-

siclassical propagators, i.e.,

G−1τ3 ⊗ τ3G − τ3GG−1τ3 = 0. (15)

The inverse Green’s function for free fermions, G−10

, is re-

placed with a−1(ε − ξ(p)τ3) if we include renormalization of

the normal propagator by the zeroth order self-energy in the

small parameter, i.e., Tc/TF ≪ 1 or εc/εF ≪ 1. We remember

that a is the weight of the quasiparticle pole in the spectral

function. The kinetic equation is then reduced to

[ετ3 − aVextτ3 − aΣτ3, τ3G] = 0. (16)

An important property of the self-energies is their weak

dependence on momentum. We suppose that their charac-

teristic momentum scale is set by the Fermi momentum, pF.

FIG. 2. Leading order contributions to quasiclassical self-energies.

Filled vertices, which couple to low-energy propagators (solid lines),

show the particle-hole and particle-particle vertices, Γph and Γpp,

that sum all orders of the bare interaction (open circle) and high-

energy intermediate states, Ghigh (dashed lines). The particle-hole

and particle-particle vertices determine the leading-order quasiparti-

cle self-energy and pair potential, respectively.

For the quasiclassical renormalized perturbation, we can in-

troduce vext andσMF, which are related to an external potential

Vext and self-energy Σ taken at the Fermi level by

vext(pF) = aVext(p)τ3, σMF(pF) = aΣ(p)τ3, (17)

respectively, with p ≈ pF. The factors, a and τ3, are included

in vext and σMF for convenience. After the ξp-integration,

Eq. (16) reduces to

[iετ3 − vext − σMF, g] = 0, (18)

which is the equation to determine the quasiclassical propa-

gator g. Notice that Eq. (18) holds for homogeneous systems.

The mean-field self-energiesσMF are composed of the Fermi-

liquid corrections (diagonal parts) and the pair potentials (off-

diagonal parts) as

σMF =

(

Σ0 + Σ · σ ∆

∆ Σ0 + Σ · σtr

)

. (19)

The spin-triplet pair potentials are parametrized by

∆(pF) = iσµσ2dµ(pF), (20)

and

∆(pF) = iσ2σµd∗µ(pF), (21)

in terms of the three-dimensional vector dµ(pF) (µ = 1, 2, 3)

which is called the d-vector for the spin-triplet superfluidity.

In the above equation, the sum is taken over µ. The explicit

form of the d-vector will be expressed by the Green’s function

in Eq. (37) in the next subsection. The solution of Eq. (18) is

not uniquely determined per se, because a + bg satisfies the

same equation as g (a and b are arbitrary constants). To deter-

mine uniquely a solution for g, Eq. (18) must be supplemented

by the normalization condition on the quasiclassical propaga-

tor: g2 = −π2I with the unit matrix I in the NG space.

5

C. Mean-field self-energies and self-consistent equations

The interaction between two neutrons is modified by the

“polarization effects”. In the vicinity of the Fermi surface,

a perturbation that couples to the quasiparticle states gener-

ates a polarization of the fermionic vacuum. Such polariza-

tion leads to a correction to the self-energy with respect to the

energy of a fermionic quasiparticle. The leading-order cor-

rection is given by mean-field interaction energy associated

with a particle-hole excitation. As we mentioned above, the

two-body interaction between fermionic quasiparticles is rep-

resented by a four-point renormalized vertex (Fig. 2),

Γph

ab;cd(p, p′) = Γs(p, p′)δacδbd + Γ

a(p, p′)σac · σbd, (22)

which is composed of the amplitudes for spin-independent

scattering (Γ(s)) and spin-dependent exchange scattering (Γ(a)),

where p ≡ (p, ε). It sums the bare two-body interactions to

all orders involving all possible intermediate states of high-

energy fermions. We use the notation (p,−p) and (p′,−p′) to

stand for the in-coming and out-going momenta, respectively,

for the fermion 1 and 2. As the quasiclassical approximation

takes account of quasiparticles confined to a low-energy shell

near the Fermi surface, the vertex function can be evaluated

with p = pF and ε→ 0. The resulting vertex function reduces

to

A(s,a)(pF, p′F) = 2NFΓ(s,a)(p ≈ pF, ε = 0; p′ ≈ p′F, ε

′ = 0).

(23)

Then, the scalar (Σ0) and vector (Σ) components in the di-

agonal parts in the mean-field self-energy σMF, Eq. (19), are

determined as

Σ0(pF) = T∑

n

⟨

A(s)(pF, p′F)g0(p′F, εn)⟩′, (24)

Σ(pF) = T∑

n

⟨

A(a)(pF, p′F)g(p′F, εn)⟩′, (25)

respectively, where∑

n denotes the Matsubara sum with the

cutoff energy εc, and

〈· · · 〉′ = 1

4π

∫

dθ′p sin θ′p

∫

dφ′p · · · , (26)

is the average over the Fermi surface with p′ ≈|p′

F| (cosφ′p sin θ′p, sinφ′p sin θ′p, cos θ′p

)

with the polar angles θ′pand φ′p for p′.

Let f (p1, p2) be the particle-hole interaction between two

nucleons which is generalized in spin (σ) and isospin (τ)

spaces as

f (p1, p2) =NF

F(p1, p2) + F′(p1, p2)τ1 · τ2

+G(p1, p2)σ1 · σ2 +G′(p1, p2)[σ1 · σ2][τ1 · τ2]

,

(27)

where p1 (p2) are the three-dimensional momentum for the

in-coming (out-going) particle, σ1 (σ2) and τ1 (τ2) stand for

SU(2) spin and isospin interactions for the particle 1 (2). The

first two terms with F(p1, p2) and F′(p1, p2) represent sym-

metric (spin-independent) quasiparticle scattering processes,

while the latter two terms with G(p1, p2) and G′(p1, p2) rep-

resent the antisymmetric (spin-dependent) quasiparticle scat-

tering processes. The single-particle momenta are taken at the

Fermi surface, |pi| ≈ |pF| for i = 1, 2. We introduce the fac-

tor NF = mpF/(2π2) for the density-of-state of the fermion on

the Fermi surface, so that the parameters F, F′, G, and G′ are

dimensionless quantities. Near the Fermi surface, F, F′, G,

and G′ (= G) are approximately regarded as functions only of

the angle between p1 and p2, and thus they can be expanded

in terms of the Legendre polynomials Pℓ(x) (ℓ = 0, 1, 2, · · · ):

G(pF · p′F) =∑

ℓ

GℓPℓ(pF · p′F), (28)

for the function G(pF · p′F) depending on pF · p′F. Here Gℓ is

the coefficient in the channel ℓ. For neutron matter, one has

τ1 · τ2 = 1 and the interaction potentials can be reduced

to a more compact form by defining F(n) = F + F′ and

G(n) = G +G′. The superscript (n) stands for the neutron mat-

ter. The self-energies describe the Fermi liquid corrections

due to symmetric (A(s)) and antisymmetric (A(a)) quasiparti-

cle scattering processes. The symmetric and antisymmetric

quasiparticle scattering amplitudes are parametrized with the

Landau’s Fermi-liquid parameters F(n)

ℓand G

(n)

ℓas

A(s)(pF, p′F) =∑

ℓ

F(n)

ℓ

1 + F(n)

ℓ/(2ℓ + 1)

Pℓ(pF · p′F). (29)

for the spin-symmetric case, and

A(a)(pF, p′F) =∑

ℓ

G(n)

ℓ

1 +G(n)

ℓ/(2ℓ + 1)

Pℓ(pF · p′F), (30)

for the spin-asymmetric case. We notice that, among several

coefficients, F(n)

ℓ=1and G

(n)

ℓ=0give the Fermi liquid corrections

to mass and spin susceptibility of a free neutron.

By taking into account the high-energy vertex corrections,

the Zeeman energy in Eq. (5), −(1/2)γnσ·B, in the NG space

is recast into

vext = −1

1 +G(n)

0

(

12γnσ·B 0

0 12γnσ

tr ·B

)

. (31)

with the factor 1/(1 + G(n)

0). Here we introduce the magneti-

zation density

M = MN +γnNF

1 +G(n)

0

T∑

n

〈g(pF, εn)〉 , (32)

as a sum of the magnetization in the normal state and the cor-

rection by the superfluid state. The first term is explicitly

given by MN = χNB, where χN = (1/2)γ2nNF/(1 + G

(n)

0) is

the spin susceptibility renormalized by the Fermi-liquid cor-

rection (G(n)

0) in the normal state. The nonvanishing magne-

tization density M is fed back to the effective magnetic field

Beff through the Fermi-liquid correction (G(n)

0),

vext +

(

Σ · σ 0

0 Σ · σtr

)

≡ − 1

1 +G(n)

0

(

12γnσ·Beff 0

0 12γnσ

tr ·Beff

)

,

6

(33)

by referring Eqs. (31) and (32). Thus, the effective magnetic

field including the corrections of spin-polarization density is

given by

Beff =

1 +G(n)

0

(

1 − M

MN

)

B. (34)

This gives rise to a nonlinear effect of the Zeeman magnetic

field.

In the same manner, the polarization effects exist for the

four-fermion vertex which is denoted by Γpp

ab;cd(p, ε; p′, ε′)

with the three dimensional momentum p (p′), the energy ε

(ε′), and spin indices a, c =↑, ↓ (b, d =↑, ↓) for the in-coming

(out-going) particles. This vertex is irreducible in the particle-

particle channel that sums bare two-body interactions to all or-

ders involving all possible intermediate states of high-energy

fermions (Fig. 2). As fermion pairs with binding energy

|∆| ≪ εc are confined to a low-energy band near the Fermi sur-

face |ε| ≤ εc ≪ εF, the particle-particle vertex varies slowly

on p in the neighborhood of the Fermi surface. Thus, the ver-

tex reduces to functions only of the relative momenta,

Vab;cd(pF, p′F) ≡ 2NFΓpp

ab;cd(p ≈ pF, ε→ 0; p′ ≈ p′F, ε

′ → 0),

(35)

with the Fermi momenta pF and p′F. The particle-particle ver-

tex function is decomposed into the spin-singlet (e: even par-

ity) and spin-triplet (o: odd parity) functions for the particle-

particle channels:

Vab;cd = (iσy)abV (e)(iσy)cd + (iσµσy)abV (o)µν (iσyσν)cd. (36)

Using the effective interaction potential, one obtains the gap

equation

dµ(pF) = −T∑

n

⟨

V (o)µν (pF, p′F) fν(p′F, εn)

⟩′, (37)

with µ = 1, 2, 3, which determines the equilibrium d-vector,

d = (d1, d2, d3). Here we have taken only the negative part

of the 3P2 channel (odd parity) as the effective pairing inter-

action for dense neutrons, and have discarded the even-parity

channel, i.e., V (e) = 0. The interaction in the 3P2 channel

is supposed to be the short-range one so that the momentum

dependence can be safely neglected.

For the representation of the interaction potential, let us in-

troduce the spherical tensors, t(m)

µim=−J,··· ,+J with µ, i = 1, 2, 3,

that form bases for representations of the rotational symmetry

SO(3). Here m = −J, · · · ,+J are the eigenvalues of Jz. For

the 3P2 channel, i.e., J = 2, the interaction potential can be ex-

pressed as the separable form of the symmetric and traceless

tensors as

V (o)µν (pF, p′F) = −v

J∑

m=−J

3∑

i, j=1

[t(m)

µipF,i][t

(m)

ν jp′F, j]

∗, (38)

where v > 0 is the coupling constant of the zero-range attrac-

tive 3P2 interaction. Here pF,i (p′F, j

) is the ith ( jth) component

of the three-dimensional momentum pF (p′F) for the in-coming

(out-going) states of the scattering neutron. It is important that

the momentum dependence appears because the P-wave inter-

action potential is adopted. Eq. (38) is recast into

V(o)

ab;cd(pF, p′F) = −v

3∑

µ,ν=1

Tµν,ab(pF)T ∗µν,dc(p′F), (39)

with the traceless and symmetric tensor

Tµν,ab(pF) =

(

i

(

1

2√

2

(

σµpF,ν + σνpF,µ

)

− 1

3√

2δµνσ· pF

)

σy

)

ab

,

(40)

which obeys Tµν(pF) = Tνµ(pF) and tr(T (pF)) ≡ ∑

µ Tµµ(pF) =

0 [100].

The order parameter of spin-triplet superfluids, d(pF), is pa-

rameterized as

dµ(pF) =

3∑

i=1

Aµi pi, (41)

with the rank-2 tensor Aµi, where the index µ (i) denotes the

spin (orbital) degrees of freedom of the Cooper pair and we

have introduced p = ( p1, p2, p3) with p ≡ pF/pF. In general,

a rank-2 tensor can be expanded as a sum of the terms of the

total angular momentum J = 0, 1, and 2 as

Aµi = A(0)δµi +A(1)

µi+A(2)

µi, (42)

where the scalar function A(0) ≡ tr(A)/3, the antisymmetric

matrixA(1)

µi≡ (Aµi − Aiµ)/2, and the symmetric traceless ma-

trix A(2)

µi≡ (Aµi + Aiµ)/2 − 1

3δµitr(A) are the eigenstates of

J = 0, 1, and 2, respectively. Thus, the number of indepen-

dent components in Aµi is then given as 3⊗3 = 1⊕3⊕5, where

the numbers in the right-hand side represent the multiplicities

of eigenstates of the total angular momentum J = 0, 1, 2,

respectively. In the followings, we consider the neutron 3P2

superfluidity. Thus, we neglect the J = 0 and 1 components

(A(0) = A(1)

µi= 0), and express the tensor of the 3P2 order

parameter by Aµi = A(2)

µi. Aµi is then determined by solving

the gap equation

Aµi =1

2

[

Fµi + Fiµ

]

− 1

3δµitr [F ] , (43)

with

Fµi ≡ vT∑

n

⟨

fµ(p′F, εn) p′i⟩′, (44)

where the average calculation for the momentum has been

adopted by Eq. (26).

In calculating Eqs. (43) and (44), we utilize the fact that

the cutoff energy εc and the coupling constant v are related to

measurable quantity, i.e., the bulk transition temperature Tc,

through linearized gap equation

1

v=

5

9πT

∑

|εn |<εc

1

|εn|≈ 5

9ln

1.13εc

T. (45)

7

Eliminating εc and v from the above gap equation, Eq. (44)

reduces to

(

lnT

Tc

)

Fµi = 3T∑

n

⟨

fµ(p′F, εn) p′i⟩′ −

∑

n

πT

|εn|. (46)

This is free from the ultraviolet divergence. Thus, the regular-

ization of the gap equation leads to rapidly convergent series

defined in terms of Tc.

D. Thermodynamic potential

The thermodynamic potential in Eq. (6), which is the Φ-

functional, generates the diagonal components and the off-

diagonal components in the self-energy (8). To derive the ther-

modynamic potential within the quasiclassical approximation,

we subtract the normal-state contributions from the Luttinger-

Ward functional as ∆Ω ≡ Ω[G,Σ] − Ω[GN,ΣN], where GN

and ΣN are the Green’s function and self-energy in the normal

state, respectively. In this approximation, the Luttinger-Ward

thermodynamic potential is then given by [101, 102]

∆Ω[g] =1

2

∫ 1

0

dλ Sp′ σMF (gλ − g) + ∆Φ[g], (47)

where ∆Φ[g] is the Φ-functional confined to the low-energy

region of the phase space. In the diagrammatic representa-

tion, ∆Φ is formally constructed by a number of low-energy

propagators (Glow), and the higher-energy propagator (Ghigh)

is renormalized into vertices as in Fig. 2. In Eq. (47) we set

Sp′· · · ≡ NFT∑

n

〈· · · 〉. (48)

The quasiclassical auxiliary function gλ is given by replacing

σMF → λσMF in Eq. (18). Here we determine the quasiclassi-

cal Φ-functional so as to consistently generate the self-energy

through the functional derivative, σMF = 2δ∆Φ[g]/δgT. It is

found that the Φ-functional is constructed as

∆Φ[g] =1

4Sp′ σMFg , (49)

which generates the self-consistent equations (24), (25), and

(37).

E. Critical exponents and new universality class

The superfluid states subject to the total angular momen-

tum J = 2 are classified into several phases: Uniaxial/biaxial

nematic (UN/BN) phases, the ferromagnetic phase, and the

cyclic phase [62, 103, 104]. The nematic phases preserve the

time-reversal symmetry (TRS) and occupy the almost region

of the phase diagram under a uniform magnetic field, while

the latter two are nonunitary states with spontaneously broken

time-reversal-symmetry. The ground state at the weak cou-

pling limit is the uniaxial/biaxial nematic phase in which the

rank-2 tensor Aµi is represented by

Aµi(T, B) = ∆(T, B)

1 0 0

0 r(T, B) 0

0 0 −1 − r(T, B)

µi

, (50)

where ∆ = ∆(T, B) ≥ 0 is the amplitude and r = r(T, B) ∈[−1,−0.5] is the internal parameter that characterizes the bi-

axiality of the nematic state. The state with r = −1/2 is called

the uniaxial nematic (UN) phase where Aµi is invariant un-

der D∞ including SO(2). The state with r = −1 is called

the D4-biaxial nematic (D4-BN) phase where Aµi is invari-

ant under dihedral-four (D4) symmetry with C4 and C2 axes.

The intermediate r is called the D2-biaxial nematic (D2-BN)

phase where Aµi is invariant under dihedral-two (D2) symme-

try with three C2 axes.9 The order parameters ∆(T, B) and

r(T, B) are determined by self-consistently solving the quasi-

classical equation (18), the spin polarization in Eq. (25), and

the gap equation (43). Notice that we consider a spatially uni-

form magnetic field along the z-axis, without loss of general-

ity: B = (0, 0, B).

Let us see the phase diagram on the T -B plane. Figures 1(a)

and 1(b) show the phase diagram of 3P2 superfluids under a

magnetic field for the Landau parameter G(n)

0= −0.7 and−0.4,

respectively. The UN phase (r = −1/2) is thermodynamically

stable in zero fields, while the magnetic field drives the tran-

sition from the D2-BN phase (−1 < r < −1/2) to the D4-BN

phase (r = −1). The behavior of r on the T -B plane is shown

in detail in Fig. 3(a). In this figure, we find that r continuously

reduces to −1 with increasing T in the lower B region, and

hence the D2-BN state undergoes the second-order phase tran-

sition to the D4-BN. Under higher B fields, however, the order

parameter r shows the finite jump at a finite B, leading to the

first-order phase transition from the D2-BN phase to the D4-

BN phase, as indicated by white blobs in the figure. The first

and second order phase boundaries meet at the critical end-

point (CEP) at (Tcep/Tc, γnBcep/(πTc)) ≈ (0.48950, 0.079063)

for G(n)

0= −0.7 and at (0.28568, 0.184375) for G

(n)

0= −0.4.

We remark that the first-order transition and the CEP are

attributed to the screening effect of the external magnetic field

due to the spin-polarized molecular field. The magnetic Zee-

man field gives rise to the Pauli paramagnetic depairing of the

uniaxial nematic state that suppresses the component of the d-

vector along the B field, i.e., |Azz|/∆ = 1 + r < 1/2 in Eq. (50)

for B , 0. The suppression of the magnetization in the D2-BN

state, |M| < MN, is fed back to the effective magnetic field in

Eq. (34), giving rise to the screening of the external magnetic

field, |Beff | < B, for G(n)

0< 0. In contrast, the D4-BN state

always satisfies the configuration d ⊥ H, which is most fa-

vored under B and free from the paramagnetic depairing, i.e.,

M = MN and Beff = B. As G(n)

0approaches the Pomeranchuk

instability at G(n)

0= −1, therefore, the D4-BN phase can be

stabilized in lower fields and the position of the CEP shifts to

9 See e.g. Appendix B in Refs. [81, 92] for more information on the defini-

tions of the UN, D2-BN, and D4-BN phases.

8

-1.0

-0.6

-0.7

-0.8

-0.90.088

0.086

0.0780.0800.0820.084

0.520.50

0.440.420.400.38

0.480.46

(a)

1.0

1.2

1.4

1.6

0.8

0.6

0.0880.086

0.0780.0800.0820.084

0.520.50

0.440.420.400.38

0.480.46

(b)

0

-0.02

-0.04

-0.12

-0.10

-0.08

-0.06

0.0880.086

0.0780.0800.0820.084

0.520.50

0.440.420.400.38

0.480.46

(c)

FIG. 3. (a) Temperature dependence of the order parameter r(B,T ). (b) the heat capacity CS(T, B). (c) the magnetization M(T, B) − MN(B) ≡−∂δΩ/∂B around the CEP. The open (closed) circles correspond to the first (second) order phase transition between D2-BN and D4-BN states.

Here we set G(n)

0= −0.7.

the region of lower fields and higher temperatures. We note

that the position of the CEP reads (Tcep/Tc, γnBcep/(πTc)) ≈(0.48950, 0.079063), (0.28568, 0.184375), (0.2225, 0.24875),

and (0.15, 0.3111) for G(n)

0= −0.7, −0.4, −0.2, and 0, re-

spectively. The CEP shifts toward low temperatures with

G(n)

0→ −η with a small positive number η (0 < η ≪ 1),

and vanishes at a positive value of G(n)

0.

The consequence of the CEP is captured by thermodynamic

quantities. First, in Fig. 3(b), we plot the heat capacity in the

superfluid state per volume, C(T, B), which is obtained from

the Luttinger-Ward thermodynamic potential, ∆Ω[g], as

CV (T, B) ≡ CN(T ) − T∂2∆Ω

∂T 2, (51)

where the heat capacity of the normal gas of neutrons is

given by CN(T ) = (2π2/3)NFk2B

T . The heat capacity con-

tains critical information on the thermal evolution of neutron

stars [105]. The heat capacity shows the jump at the lower T .

Another quantity which captures the consequence of the CEP

is the magnetization M. This is defined as the first derivative

of ∆Ω,

M(T, B) = MN(B) − ∂∆Ω∂B

, (52)

which coincides with Eq. (32). It is seen from Fig. 3(c) that the

T -dependence of M has the jump in the higher B region, indi-

cating the first-order phase transition from the D2-BN phase

to the D4-BN phase. The jump in M decreases as the magnetic

field approaches the CEP. The discontinuity of M implies the

divergence of the spin susceptibility,

χ(T, B) =∂M

∂B= χN −

∂2∆Ω

∂B2. (53)

To extract the critical behaviors of the 3P2 superfluids, we

compute the critical exponents around the CEP at (Tcep, Bcep).

We note that the contributions of the normal gas of neutrons

to Eqs. (51) and (52), CN(T ) and MN(B), are negligible in

the vicinity of the CEP, and the critical behaviors of the heat

capacity CV , the magnetization M, and the spin susceptibility

(b)

(d)

1

0

-1

-2-5 -4 -3 -2

-1

-3

-5 -4 -3 -2

-2

-4

-1

-3

-2

-4

(a)

1

2

3

4

-5 -4 -3 -2

(c)

-5 -4 -3 -2

FIG. 4. The scaling behavior of CV(T, Bcep), M(T, Bcep), M(Tcep, B),

and χ(T, Bcep) around the CEP (Tcep/Tc, γnBcep/(πTc)) ≈(0.48950, 0.079063) for G

(n)

0= −0.7 (circles) and

(0.28568, 0.184375) for G(n)

0= −0.4 (triangles). Here we set

CV (T ) ≡ CV (T, Bcep), M(T ) ≡ M(T, Bcep), M(B) ≡ M(Tcep, B),

χ(T ) ≡ χ(T, Bcep), Ccep

V≡ CV (Tcep, Bcep), Mcep ≡ M(Tcep, Bcep), and

χcep ≡ χ(Tcep, Bcep).

χ, are governed by the superfluid contributions,

CV (T, B) ≈ −T∂2∆Ω

∂T 2, (54)

M(T, B) ≈ −∂∆Ω∂B

, (55)

9

χ(T, B) ≈ −∂2∆Ω

∂B2. (56)

Then, we consider the set of the critical exponents (α, β, γ, δ)

from the scaling behavior, which are parametrized by

CV (T, Bcep) −CV (Tcep, Bcep) ∝ |T − Tcep|−α, (57)

M(T, Bcep) − M(Tcep, Bcep) ∝ |T − Tcep|β, (58)

M(Tcep, B) − M(Tcep, Bcep) ∝ |B − Bcep|1/δ, (59)

χ(T, Bcep) − χ(Tcep, Bcep) ∝ |T − Tcep|−γ, (60)

for T < Tcep and B < Bcep. Under the scaling hypothesis, the

set of the critical exponents, (α, β, γ, δ), satisfies three equal-

ities in Eqs. (1)-(3), i.e., Rushbrooke, Griffiths, and Widom

equalities, that should hold at the CEP for any systems irre-

spective to the different interactions and dimensions. These

three relations relate the critical exponents of magnetic sys-

tems, which have the endpoint of a first-order phase transition

in non-zero temperatures.

Figure 4 shows the scaling behavior of the specific heat

CV(T, B), the magnetization M(T, B) and the spin suscep-

tibility χ(T, B) around the CEP, (Tcep, Bcep), which are di-

rectly computed with self-consistent solutions of the super-

fluid Fermi liquid theory. From these data, we read the values

of the critical exponents as

α = 0.68, β = 0.41, γ = 0.57, δ = 2.3, (61)

for G(n)

0= −0.7. We find that the values of Eq. (61) satisfy the

three equalities Eqs. (1)-(3) within the error range of 10% at

most:

α + 2β + γ = 2.07, (62)

α + β(1 + δ) = 2.03, (63)

−γβ+ δ = 0.91, (64)

indicating that the superfluid Fermi liquid theory properly

captures the critical behavior of the CEP in neutron 3P2

superfluids. For G(n)

0= −0.4, we read (α, β, γ, δ) =

(0.60, 0.45, 0.59, 2.3) from the data in Fig. 4, which satisfy

the above equalities within the error range of 5% at most, such

that α+2β+γ = 2.09, α+β(1+ δ) = 2.09, and − γβ+ δ = 0.99.

In Table I, we summarize the values of the critical exponents,

α, β, γ, δ, for the Landau parameters G(n)

0= −0.7 and −0.4.

It turns out that the resulting exponents are insensitive to the

screening effect of the external magnetic field due to the spin-

polarized molecular field.

We propose that the set of the critical exponents, (α, β, γ, δ),

in Eq. (61) belongs to a new type of university class. First

of all, one may notice the large value of α (α ∼ 0.6). It is

known that the value of α is usually much smaller than one in

the phase transitions accompanied with a continuous symme-

try breaking at least in known models thus far. However, the

value of α can be larger in the cases accompanied with dis-

crete symmetry breaking, such as the Potts model in two di-

mensions [94, 95]. In our case, the CEP appears in the phase

transition with a discrete symmetry breaking, i.e., from D2 to

D4. Thus, it may be natural to have the large value of α in

the neutron 3P2 superfluid. Phenomenologically, the large α

indicates that the heat capacity is much enhanced at the CEP

(cf. Eq. (57)), which may affect the cooling process in the evo-

lution of neutron stars. Naively to say, the large heat capacity

will lead to a slow cooling in the evolution of neutron stars.

Another feature of the critical exponents in Eq. (61) is that

the value of γ is smaller than one (γ ∼ 0.5). In the literature,

there are only a limited number of examples which indicate

γ < 1. One example is the O(n) model with n < 0 [96].

The O(n) model induces the Ising model at n = 1 and the

self-avoiding polymer/walk model at n = 0. If the value of

γ is expressed in the asymptotic series up to the second-order

terms in the vicinity of four dimensions, it is found that γ can

be smaller than one if n is extrapolated to the negative region

(n < 0). Another example for γ < 1 is the tricritical Ising

model coupled to massless Dirac fermions [97]. In conclu-

sion, the large α and the small γ are the unique feature of the

critical exponents in the neutron 3P2 superfluid, implying a

new universality class.

III. GINZBURG-LANDAU THEORY FOR THE CRITICAL

ENDPOINT

A. Ginzburg-Landau free energy

Let us turn to a discussion based on the Ginzburg-Landau

(GL) theory [27, 28, 56–61, 81, 90–92]. In the weak coupling

limit for the neutron-neutron interaction, we obtain the GL

free energy density

∆Ω[A] = Ω(0)

8[A] + Ω

(≤4)

2[A] + Ω

(≤2)

4[A] + O(BmAn)m+n≥7,

(65)

as an expansion series in terms of the condensate Aµi and

the magnetic field B [90, 92]. We have adopted the quasi-

classical approximation for the momentum integrals for the

neutron loops. Notice that the free energy part for the non-

interacting neutron is not included, because they are irrelevant

to the the condensate. Each term in Eq. (65) is explained as

follows. Ω(0)

8[A] includes Aµi up to the 8th order with no mag-

netic field, Ω(≤4)

2[A] includes Aµi up to the 2nd order with the

magnetic field up to |B|4, and Ω(≤2)

4[A] includes Aµi up to the

4th order with the magnetic field up to |B|2. Their explicit

forms are

Ω(0)

8[A] = K(0)

∑

i, j,µ=1,2,3

(

∇ jA∗iµ∇ jAµi + ∇iA

∗iµ∇ jAµ j + ∇iA

∗jµ∇ jAµi

)

10

+α(0)(trA∗A)

+β(0)(

(

tr A∗A)2 − (

tr A∗2A2))

+γ(0)(

−3(

tr A∗A)(

tr A2)(tr A∗2)

+ 4(

tr A∗A)3+ 6

(

tr A∗A)(

tr A∗2A2) + 12(

tr A∗A)(

tr A∗AA∗A)

−6(

tr A∗2)(

tr A∗A3) − 6(

tr A2)(tr A∗3A) − 12

(

tr A∗3A3) + 12(

tr A∗2A2A∗A)

+ 8(

tr A∗AA∗AA∗A)

)

+δ(0)(

(

tr A∗2)2(

tr A2)2+ 2

(

tr A∗2)2(

tr A4) − 8(

tr A∗2)(

tr A∗AA∗A)(

tr A2) − 8(

tr A∗2)(

tr A∗A)2(

tr A2)

−32(

tr A∗2)(

tr A∗A)(

tr A∗A3) − 32(

tr A∗2)(

tr A∗AA∗A3) − 16(

tr A∗2)(

tr A∗A2A∗A2)

+2(

tr A∗4)(

tr A2)2+ 4

(

tr A∗4)(

tr A4) − 32(

tr A∗3A)(

tr A∗A)(

tr A2)

−64(

tr A∗3A)(

tr A∗A3) − 32(

tr A∗3AA∗A)(

tr A2) − 64(

tr A∗3A2A∗A2) − 64(

tr A∗3A3)(tr A∗A)

−64(

tr A∗2AA∗2A3) − 64(

tr A∗2AA∗A2)(tr A∗A)

+ 16(

tr A∗2A2)2+ 32

(

tr A∗2A2)(tr A∗A)2

+32(

tr A∗2A2)(tr A∗AA∗A)

+ 64(

tr A∗2A2A∗2A2) − 16(

tr A∗2AA∗2A)(

tr A2) + 8(

tr A∗A)4

+48(

tr A∗A)2(

tr A∗AA∗A)

+ 192(

tr A∗A)(

tr A∗AA∗2A2) + 64(

tr A∗A)(

tr A∗AA∗AA∗A)

−128(

tr A∗AA∗3A3) + 64(

tr A∗AA∗2AA∗A2) + 24(

tr A∗AA∗A)2+ 128

(

tr A∗AA∗AA∗2A2)

+48(

tr A∗AA∗AA∗AA∗A)

)

, (66)

Ω(≤4)

2[A] = β(2)BtA∗AB + β(4)|B|2BtA∗AB, (67)

Ω(≤2)

4[A] = γ(2)

(

−2 |B|2(tr A2)(tr A∗2) − 4 |B|2(tr A∗A

)2+ 4 |B|2(tr A∗AA∗A

)

+ 8 |B|2(tr A∗2A2)

+BtA2B(

tr A∗2) − 8 BtA∗AB

(

tr A∗A)

+ BtA∗2 B(

tr A2) + 2 BtAA∗2AB

+2 BtA∗A2A∗B − 8 BtA∗AA∗AB − 8 BtA∗2A2B)

, (68)

with the derivative ∇i for the spatial direction i = 1, 2, 3 and

the GL coefficients defined by

K(0) =7 ζ(3)NFp4

F

240m2(πTc)2, α(0) =

NF p2F

3log

T

Tc

,

β(0) =7 ζ(3)NFp4

F

60 (πTc)2, β(2) =

7 ζ(3)NFp2Fγ2

n

48(1 +G(n)

0)2(πTc)2

,

β(4) = −31 ζ(5)NFp2

Fγ4

n

768(1 +G(n)

0)4(πTc)4

, γ(0) = −31 ζ(5)NFp6

F

13440 (πTc)4,

γ(2) =31 ζ(5)NFp4

Fγ2

n

3840(1+G(n)

0)2(πTc)4

, δ(0) =127 ζ(7)NFp8

F

387072 (πTc)6. (69)

ζ(n) is the zeta function. In the above expression, µn has been

replaced to µ∗n = (γn/2)σ/(1+G(n)

0), i.e., the magnetic momen-

tum of a neutron modified by the Landau parameter G(n)

0. We

notice that the Landau parameter stems from the Hartree-Fock

approximation which are not taken into account explicitly in

the present procedure for the fermion-loop expansion. The

choice of the value of G(n)

0does not affect the values of the

critical exponents, because the magnetic field is scaled uni-

formly. This is different from the analysis in the BdG equation

in Sec. II, where the spin-polarization leads to the non-linear

effect for the Zeeman magnetic field (cf. Eq. (34)).

We notice that the β(4) and γ(2) terms were derived beyond

the leading-order term for the magnetic field [90], and the

δ(0) term was calculated to recover the global stability of the

ground state which was absent at the 6th order [92]. We em-

phasize that, as discussed in detail in Ref. [92], the δ(0) term

(the 8th order term) produces the first-order transition in the

GL equation, which was absent in the analysis up to the 6th

order term, and hence it leads to the existence of the CEP in

the GL equation. In the derivation of the GL equation, we

have supposed that the temperature T is close to the critical

temperature at zero magnetic field Tc, and hence the applica-

ble region of the GL equation is limited in |1 − T/Tc| ≪ 1.

Notice that the critical temperature is the unique energy scale

in the GL theory in the above.

B. Critical endpoint of 3P2 superfluid phase diagrams

Let us consider the phase diagram drawn by the variational

calculation for the GL free energy. For the magnetic field as

B = (0, 0, B), we consider to minimize the GL free energy

with respect to ∆ and r in Eq. (50). We show the phase di-

agram on the plane spanned by the temperature (T/Tc) and

the magnetic field (γnB/(πTc)) in Fig. 1(d). The CEP is

(Tcep/Tc, γnBcep/(πTc)) = (0.774597, 0.004465). The phase

boundary at T < Tcep and B < Bcep is the first order transition,

as indicated by the cyan lines in the figure. It should be noted

that the existence of the CEP is due to the 8th order term as

it was discovered in the previous work [92]. Thus, the GL

theory shares common properties with the BdG theory about

the existence of the CEP, though the positions of the CEP are

different.

We consider the thermodynamical quantities, i.e., the heat

capacity, the magnetization, and the spin susceptibility, which

have been introduced in Eqs. (54), (55), and (56), and inves-

11

(a) (b)

(c) (d)

-3.0-4.0-5.0

-3.2

-3.4

-3.8

-3.6

-4.0

-4.2-3.0-4.0 -2.0-3.5 -2.5-4.5 -2.5-3.5 -1.5

0

0.2

1.0

0.6

0.4

0.8

1.6

1.4

1.2

-0.6

-0.8

-1.2

-1.0

-1.4

-1.6

-0.4

-1.8-3.0-4.0-5.0 -3.5 -2.5-4.5

-3.2

-3.4

-3.8

-3.6

-4.0

-4.2

-3.0

-2.0-3.0 -1.0-1.5-2.5 -0.5

FIG. 5. The scaling behavior of CV (T, B), M(T, B) and χ(T, B)

around the CEP (Tcep/Tc, γnBcep/(πTc)) = (0.774597, 0.004465).

Here we use the same abbreviations as those in Fig. 4.

tigate their scaling behaviors at the CEP. Around the critical

endpoint, we introduce the critical exponents α, β, γ and δ

for the heat capacity, the magnetization, and the spin suscep-

tibility as defined in Eqs. (57), (58), (59), and (60). We plot

CV (T, B), M(T, B) and χ(T, B) in Fig. 5. From those plots, we

find that the values of the critical exponents read as:

α = 0.60, β = 0.49, γ = 0.52, δ = 1.95. (70)

When we substitute the values in Eq. (70) to the left hand sides

of the identities, Eqs. (1), (2), and (3), we obtain

α + 2β + γ = 2.10, (71)

α + β(1 + δ) = 2.04, (72)

−γβ+ δ = 1.11. (73)

Those values agree with the values in the right-hand-sides in

Eqs. (1), (2), and (3) by the exact values within the 10% nu-

merical error.

In Table I, we summarize the values of the critical expo-

nents from the BdG equation with different values of G(n)

0and

the ones from the GL equation. Interestingly, we observe that

they are close to each other, even though the positions of the

CEP on the T -B plane are different (cf. Fig. 1). The coinci-

dence between the two suggests that the GL equation up to the

8th order term (δ0 term) captures the essence of the CEP of the

neutron 3P2 superfluid. Thus, the GL equation also supports a

new universality class discussed in Sec. II E.

TABLE I. Critical exponents (α, β, γ, δ) computed by the superfluid

Fermi liquid theory with G(n)

0= −0.7 and −0.4 in the BdG theory and

the GL theory. The Fermi liquid correction with G(n) < 0 leads to the

screening effect of a magnetic field due to spin-polarized molecular

field. Notice that the values of the critical exponents in the GL theory

are independent of G(n)

0.

G(n)

0α β γ δ

BdG -0.7 0.68 0.41 0.57 2.3

-0.4 0.60 0.45 0.59 2.3

GL 0.60 0.49 0.52 1.95

IV. SUMMARY AND DISCUSSION

We have discussed the critical exponents at the CEP in the

phase diagram of the neutron 3P2 superfluidity, which can ex-

ist inside of neutron stars. Adopting the BdG equation with

the spin-polarization effect, we have obtained the critical ex-

ponents and have confirmed that they satisfy the universal re-

lations, i.e. the Rushbrooke, Griffiths, and Widom equalities,

which hold for the spin systems. We have argued that the set

of the critical exponents with large α and small γ belongs to a

new universality class. One of the interesting features of the

obtained critical exponents is the large α and small γ, in which

the former indicates the slow cooling in the evolution of the

neutron stars. We have checked that the spin-polarization ef-

fect induces the unique values of the critical exponents within

the 10% numerical errors. We also have investigated the crit-

ical exponents in the GL equation up to the 8th order, and

have confirmed that they satisfy the same universality rela-

tions, again within 10% errors. In spite of the different loca-

tions of the CEP in the phase diagram for the BdG equation

and for the GL equation, we have found that the values of the

critical exponents from the GL equation are properly regarded

to be the same to the ones from the BdG equation, although

there are still some discrepancies between the two due to the

limited number of terms in the GL equation. Thus, we reach

the conclusion that the GL equation up to the 8th order cap-

tures correctly the behaviors at the CEP in the neutron 3P2

superfluids.

For more advanced study in future, we leave comments

on the Fermi liquid parameter G(n)

0in dense neutron matter.

Backman et al. [106] computed the Fermi liquid parameters

including the spin channel and isospin channel of quasipar-

ticle scattering processes and find that G(n)

0in Eq. (29) may

be negligible. This result stems from the short-range property

of the ρ-meson exchange interaction. It will be important to

more carefully study the short-range behavior of the nucleon-

nucleon interaction, which will be attributed by the quark-

exchange contributions at the nucleon core, the core polariza-

tion in the nuclear medium, and so on. As for another ques-

tion, we may consider how the evolution of the neutron stars

are influenced by the enhancement of the heat capacity, the

magnetization, and the spin susceptibility at the CEP. Those

information will be useful to research the internal structures

of the neutron stars through the astrophysical observations.

Finally we would like to raise two issues. First, it is im-

12

portant to identify a key factor of strong deviation of critical

exponents at the CEP from those of the mean-field theory. We

would like to mention that a multiple-superfluid phase dia-

gram with a CEP was theoretically predicted in the superfluid3He under a magnetic field [107]. In addition, a Pauli-limited

superconductor under a magnetic field or ultracold atomic

gases with population imbalance also show the phase diagram

with a CEP [108–111], but the ordered state is characterized

by a single order parameter. A comprehensive study on uni-

versality class at CEP in such single-component and multi-

component superfluids may give clues for understanding the

origin of nontrivial critical behaviors. The second issue is the

impact of order parameter fluctuations on the critical expo-

nents. The nematic phases in 3P2 superfluids are characterized

by multiple component of the order parameter represented by

the traceless symmetric tensor, leading to rich bosonic exci-

tation spectra [64–76]. How do bosonic fluctuations alter the

critical behaviors at the CEP? This remains as a future issue.

ACKNOWLEDGMENTS

The authors would like to thank Michikazu Kobayashi for

useful discussion. This work was supported by the Grant-in-

Aids for Scientific Research from MEXT of Japan [Grant No.

JP15H05855 (KAKENHI on Innovative Areas “Topological

Materials Science”)] and the Ministry of Education, Culture,

Sports, Science (MEXT)-Supported Program for the Strate-

gic Research Foundation at Private Universities “Topological

Science” (Grant No. S1511006). This work is also supported

in part by JSPS Grant-in-Aid for Scientific Research [KAK-

ENHI Grant No. JP16K05448 (T. M.), No. 16H03984 (M.

N.), No. 18H01217 (M. N.), and No. 17K05435 (S. Y.)].

[1] V. Graber, N. Andersson, and M. Hogg, Neutron Stars

in the Laboratory, Int. J. Mod. Phys. D26, 1730015 (2017),

arXiv:1610.06882 [astro-ph.HE].

[2] G. Baym, T. Hatsuda, T. Kojo, P. D. Powell, Y. Song,

and T. Takatsuka, From hadrons to quarks in neu-

tron stars: a review, Rept. Prog. Phys. 81, 056902 (2018),

arXiv:1707.04966 [astro-ph.HE].

[3] P. Demorest, T. Pennucci, S. Ransom, M. Roberts,

and J. Hessels, Shapiro Delay Measurement of A Two

Solar Mass Neutron Star, Nature 467, 1081 (2010),

arXiv:1010.5788 [astro-ph.HE].

[4] J. Antoniadis, P. C. C. Freire, N. Wex, T. M. Tau-

ris, R. S. Lynch, M. H. van Kerkwijk, M. Kramer,

C. Bassa, V. S. Dhillon, T. Driebe, J. W. T. Hessels, V. M.

Kaspi, V. I. Kondratiev, N. Langer, T. R. Marsh, M. A.

McLaughlin, T. T. Pennucci, S. M. Ransom, I. H. Stairs,

J. van Leeuwen, J. P. W. Verbiest, and D. G. Whe-

lan, A Massive Pulsar in a Compact Relativistic Binary,

Science 340 (2013), 10.1126/science.1233232.

[5] B. Abbott et al. (Virgo, LIGO Scientific), GW170817:

Observation of Gravitational Waves from a Binary Neu-

tron Star Inspiral, Phys. Rev. Lett. 119, 161101 (2017),

arXiv:1710.05832 [gr-qc].

[6] N. Chamel, Superfluidity and Superconductivity in Neutron

Stars, Journal of Astrophysics and Astronomy 38, 43 (2017).

[7] B. Haskell and A. Sedrakian, Superfluid-

ity and Superconductivity in Neutron Stars,

Astrophys. Space Sci. Libr. 457, 401 (2018),

arXiv:1709.10340 [astro-ph.HE].

[8] A. Sedrakian and J. W. Clark, Superfluidity in nuclear systems

and neutron stars, (2018), arXiv:1802.00017 [nucl-th].

[9] G. Baym, C. Pethick, D. Pines, and M. Ruderman, Spin

Up in Neutron Stars : The Future of the Vela Pulsar,

Nature 224, 872 (1969).

[10] D. Pines, J. Shaham, and M. Ruderman, Corequakes and the

Vela Pulsar, Nature Phys. Sci. 237, 83 (1972).

[11] T. Takatsuka and R. Tamagaki, Corequake Model of Pul-

sar Glitches for Neutron Stars With Pion Condensed Core,

Prog. Theor. Phys. 79, 274 (1988).

[12] D. G. Yakovlev, A. D. Kaminker, O. Y. Gnedin, and

P. Haensel, Neutrino emission from neutron stars,

Phys. Rept. 354, 1 (2001), arXiv:astro-ph/0012122 [astro-ph].

[13] A. Y. Potekhin, J. A. Pons, and D. Page, Neutron stars

- cooling and transport, Space Sci. Rev. 191, 239 (2015),

arXiv:1507.06186 [astro-ph.HE].

[14] D. G. Yakovlev, K. P. Levenfish, and Yu. A.

Shibanov, Cooling neutron stars and superfluid-

ity in their interiors, Phys. Usp. 42, 737 (1999),

arXiv:astro-ph/9906456 [astro-ph].

[15] C. O. Heinke and W. C. G. Ho, Direct Ob-

servation of the Cooling of the Cassiopeia A

Neutron Star, Astrophys. J. 719, L167 (2010),

arXiv:1007.4719 [astro-ph.HE].

[16] P. S. Shternin, D. G. Yakovlev, C. O. Heinke, W. C. G. Ho,

and D. J. Patnaude, Cooling neutron star in the Cassiopeia

A supernova remnant: evidence for superfluidity in the core,

Mon. Notices Royal Astron. Soc. 412, L108 (2011).

[17] D. Page, M. Prakash, J. M. Lattimer, and A. W.

Steiner, Rapid Cooling of the Neutron Star in Cas-

siopeia A Triggered by Neutron Superfluidity in

Dense Matter, Phys. Rev. Lett. 106, 081101 (2011),

arXiv:1011.6142 [astro-ph.HE].

[18] P. E. Reichley and G. S. Downs, Second Decrease in the Pe-

riod of the Vela Pulsar, Nature Phys. Sci. 234, 48 (1971).

[19] P. W. Anderson and N. Itoh, Pulsar glitches and restlessness

as a hard superfluidity phenomenon, Nature 256, 25 (1975).

[20] A. B. Migdal, Superfluidity and the Moments of Inertia of

Nuclei, Zh. Eksp. Teor. Fiz. 37, 249 (1960), [Sov. Phys.

JETP10,no.1,176(1960)].

[21] R. A. Wolf, Some Effects of the Strong Interac-

tions on the Properties of Neutron-Star Matter,

Astrophys. J. 145, 834 (1966).

[22] F. Tabakin, Single Separable Potential with Attraction and Re-

pulsion, Phys. Rev. 174, 1208 (1968).

[23] M. Hoffberg, A. E. Glassgold, R. W. Richardson, and M. Ru-

derman, Anisotropic Superfluidity in Neutron Star Matter,

Phys. Rev. Lett. 24, 775 (1970).

[24] R. Tamagaki, Superfluid State in Neutron Star

Matter. I: Generalized Bogoliubov Transforma-

tion and Existence of 3P2 Gap at High Density*),

Progress of Theoretical Physics 44, 905 (1970).

[25] T. Takatsuka and R. Tamagaki, Superfluid State in Neutron

13

Star Matter. IIProperties of Anisotropic Energy Gap of 3P2

Pairing, Progress of Theoretical Physics 46, 114 (1971).

[26] T. Takatsuka, Superfluid State in Neutron Star Mat-

ter. III: Tensor Coupling Effect in 3P2 Energy Gap,

Progress of Theoretical Physics 47, 1062 (1972).

[27] T. Fujita and T. Tsuneto, The Ginzburg-Landau Equa-

tion for 3P2 PairingSuperfluidity in Neutron Stars,

Progress of Theoretical Physics 48, 766 (1972).

[28] R. W. Richardson, Ginzburg-landau theory of anisotropic su-

perfluid neutron-star matter, Phys. Rev. D5, 1883 (1972).

[29] L. Amundsen and E. Ostgaard, Superfluidity of neutron matter

(II). triplet pairing, Nucl. Phys. A442, 163 (1985).

[30] T. Takatsuka and R. Tamagaki, Superfluidity in neu-

tron star matter and symmetric nuclear matter,

Prog. Theor. Phys. Suppl. 112, 27 (1993).

[31] J. A. Sauls, Superfluidity in the Interiors of Neutron Stars,

in Timing Neutron Stars, edited by H. Ogelman and E. P. J.

van den Heuvel (Springer Netherlands, Dordrecht, 1989) pp.

457–490.

[32] M. Baldo, J. Cugnon, A. Lejeune, and U. Lombardo,

Proton and neutron superfluidity in neutron star matter,

Nucl. Phys. A536, 349 (1992).

[33] O. Elgaroy, L. Engvik, M. Hjorth-Jensen, and

E. Osnes, Triplet pairing of neutrons in beta sta-

ble neutron star matter, Nucl. Phys. A607, 425 (1996),

arXiv:nucl-th/9604032 [nucl-th].

[34] V. A. Khodel, V. V. Khodel, and J. W.

Clark, Universalities of triplet pairing in neu-

tron matter, Phys. Rev. Lett. 81, 3828 (1998),

arXiv:nucl-th/9807034 [nucl-th].

[35] M. Baldo, O. Elgaroey, L. Engvik, M. Hjorth-

Jensen, and H. J. Schulze, Triplet P-3(2) to F-3(2)

pairing in neutron matter with modern nucleon-

nucleon potentials, Phys. Rev. C58, 1921 (1998),

arXiv:nucl-th/9806097 [nucl-th].

[36] V. V. Khodel, V. A. Khodel, and J. W. Clark, Triplet

pairing in neutron matter, Nucl. Phys. A679, 827 (2001),

arXiv:nucl-th/0001006 [nucl-th].

[37] M. V. Zverev, J. W. Clark, and V. A. Khodel, (3)p(2)

- (3)F(2) pairing in dense neutron matter: The

Spectrum of solutions, Nucl. Phys. A720, 20 (2003),

arXiv:nucl-th/0301028 [nucl-th].

[38] S. Maurizio, J. W. Holt, and P. Finelli, Nuclear pair-

ing from microscopic forces: singlet channels and

higher-partial waves, Phys. Rev. C90, 044003 (2014),

arXiv:1408.6281 [nucl-th].

[39] S. K. Bogner, R. J. Furnstahl, and A. Schwenk,

From low-momentum interactions to nuclear

structure, Prog. Part. Nucl. Phys. 65, 94 (2010),

arXiv:0912.3688 [nucl-th].

[40] S. Srinivas and S. Ramanan, Triplet Pairing in

pure neutron matter, Phys. Rev. C94, 064303 (2016),

arXiv:1606.09053 [nucl-th].

[41] D. J. Dean and M. Hjorth-Jensen, Pairing in

nuclear systems: From neutron stars to fi-

nite nuclei, Rev. Mod. Phys. 75, 607 (2003),

arXiv:nucl-th/0210033 [nucl-th].

[42] D. H. Brownell and J. Callaway, Ferromagnetic

transition in superdense matter and neutron stars,

Il Nuovo Cimento B (1965-1970) 60, 169 (1969).

[43] M. Rice, The hard-sphere Fermi gas and ferromagnetism in

neutron stars, Physics Letters A 29, 637 (1969).

[44] S. D. Silverstein, Criteria for Ferromagnetism

in Dense Neutron Fermi Liquids-Neutron Stars,

Phys. Rev. Lett. 23, 139 (1969).

[45] P. Haensel and S. Bonazzola, Ferromagnetism of dense matter

and magnetic properties of neutron stars, Astron. Astrophys.

314, 1017 (1996), arXiv:astro-ph/9605149 [astro-ph].

[46] M. Eto, K. Hashimoto, and T. Hatsuda, Ferromag-

netic neutron stars: axial anomaly, dense neutron

matter, and pionic wall, Phys. Rev. D88, 081701 (2013),

arXiv:1209.4814 [hep-ph].

[47] K. Hashimoto, Possibility of ferromagnetic neutron matter,

Phys. Rev. D91, 085013 (2015), arXiv:1412.6960 [hep-ph].

[48] T. Tatsumi, Ferromagnetism of quark

liquid, Phys. Lett. B489, 280 (2000),

arXiv:hep-ph/9910470 [hep-ph].

[49] E. Nakano, T. Maruyama, and T. Tatsumi,

Spin polarization and color superconductivity

in quark matter, Phys. Rev. D68, 105001 (2003),

arXiv:hep-ph/0304223 [hep-ph].

[50] K. Ohnishi, M. Oka, and S. Yasui, Possible fer-

romagnetism in the large N(c) and N(f) limit

of quark matter, Phys. Rev. D76, 097501 (2007),

arXiv:hep-ph/0609060 [hep-ph].

[51] G. H. Bordbar and M. Bigdeli, Spin polarized asym-

metric nuclear matter and neutron star matter within

the lowest order constrained variational method,

Phys. Rev. C77, 015805 (2008), arXiv:0809.3498 [nucl-th].

[52] C. O. Heinke and W. C. G. Ho, Direct Observa-

tion of the Cooling of the Cassiopeia A Neutron Star,

The Astrophysical Journal Letters 719, L167 (2010).

[53] D. Blaschke, H. Grigorian, D. N. Voskresensky, and F. We-

ber, On the Cooling of the Neutron Star in Cassiopeia A,

Phys. Rev. C85, 022802 (2012), arXiv:1108.4125 [nucl-th].

[54] D. Blaschke, H. Grigorian, and D. N. Voskresensky, Nu-

clear medium cooling scenario in the light of new Cas

A cooling data and the 2M⊙ pulsar mass measurements,

Phys. Rev. C88, 065805 (2013), arXiv:1308.4093 [nucl-th].

[55] H. Grigorian, D. N. Voskresensky, and D. Blaschke,

Influence of the stiffness of the equation of state and

in-medium effects on the cooling of compact stars, Pro-

ceedings, Compact Stars in the QCD Phase Diagram

IV (CSQCD IV): Prerow, Germany, September 26-30,

2014, (2016), 10.1140/epja/i2016-16067-4, [Eur. Phys.

J.A52,no.3,67(2016)], arXiv:1603.02634 [astro-ph.HE].

[56] J. A. Sauls and J. Serene, 3P2 pairing near the transition tem-

perature in neutron-star matter, Phys. Rev. D17, 1524 (1978).

[57] P. Muzikar, J. A. Sauls, and J. W. Serene, 3P2 pairing

in neutron star matter: magnetic field effects and vortices,

Phys. Rev. D21, 1494 (1980).

[58] J. A. Sauls, D. L. Stein, and J. W. Serene, magnetic vortices in

a rotating 3P2 neutron superfluid, Phys. Rev. D25, 967 (1982).

[59] V. Z. Vulovic and J. A. Sauls, influence of strong-coupling cor-

rections on the equilibrium phase for 3P2 superfluid neutron

star matter, Phys. Rev. D29, 2705 (1984).

[60] K. Masuda and M. Nitta, Magnetic Properties of Quan-

tized Vortices in Neutron 3P2 Superfluids in Neutron Stars,

Phys. Rev. C93, 035804 (2016), arXiv:1512.01946 [nucl-th].

[61] K. Masuda and M. Nitta, Half-quantized Non-Abelian

Vortices in Neutron 3P2 Superfluids inside Magnetars,

arXiv:1602.07050 [nucl-th].

[62] T. Mizushima, K. Masuda, and M. Nitta, 3P2 su-

perfluids are topological, Phys. Rev. B95, 140503 (2017),

arXiv:1607.07266 [cond-mat.supr-con].

[63] T. Mizushima and M. Nitta, Topology and sym-

metry of surface Majorana arcs in cyclic su-

perconductors, Phys. Rev. B97, 024506 (2018),

14

arXiv:1710.07403 [cond-mat.supr-con].

[64] P. F. Bedaque, G. Rupak, and M. J. Savage, Gold-

stone bosons in the 3P(Z) superfluid phase of neutron mat-

ter and neutrino emission, Phys. Rev. C68, 065802 (2003),

arXiv:nucl-th/0305032 [nucl-th].

[65] L. B. Leinson, New eigen-mode of spin oscillations

in the triplet superfluid condensate in neutron stars,

Phys. Lett. B702, 422 (2011), arXiv:1107.4025 [nucl-th].

[66] L. B. Leinson, Collective modes of the order pa-

rameter in a triplet superfluid neutron liquid,

Phys. Rev. C85, 065502 (2012), arXiv:1206.3648 [nucl-th].

[67] L. B. Leinson, Neutrino emissivity of anisotropic

neutron superfluids, Phys. Rev. C87, 025501 (2013),

arXiv:1301.5439 [nucl-th].

[68] P. F. Bedaque and A. N. Nicholson, Low lying modes

of triplet-condensed neutron matter and their effective

theory, Phys. Rev. C87, 055807 (2013), [Erratum: Phys.

Rev.C89,no.2,029902(2014)], arXiv:1212.1122 [nucl-th].

[69] P. Bedaque and S. Sen, Neutrino emissivity from

Goldstone boson decay in magnetized neutron matter,

Phys. Rev. C89, 035808 (2014), arXiv:1312.6632 [nucl-th].

[70] P. F. Bedaque and S. Reddy, Goldstone modes in

the neutron star core, Phys. Lett. B735, 340 (2014),

arXiv:1307.8183 [nucl-th].

[71] P. F. Bedaque, A. N. Nicholson, and S. Sen, Massive

and massless modes of the triplet phase of neutron matter,

Phys. Rev. C92, 035809 (2015), arXiv:1408.5145 [nucl-th].

[72] L. B. Leinson, Neutrino emission from triplet pairing of

neutrons in neutron stars, Phys. Rev. C81, 025501 (2010),

arXiv:0912.2164 [astro-ph.SR].

[73] L. B. Leinson, Neutrino emission from spin waves in

neutron spin-triplet superfluid, Phys. Lett. B689, 60 (2010),

arXiv:1001.2617 [astro-ph.SR].

[74] L. B. Leinson, Superfluid phases of triplet pair-

ing and neutrino emission from neutron stars,

Phys. Rev. C82, 065503 (2010), [Erratum: Phys.

Rev.C84,049901(2011)], arXiv:1012.5387 [hep-ph].

[75] L. B. Leinson, Zero sound in triplet-correlated super-

fluid neutron matter, Phys. Rev. C83, 055803 (2011),

arXiv:1007.2803 [hep-ph].

[76] L. B. Leinson, Neutrino emissivity of 3P2-3F2 super-

fluid cores in neutron stars, Phys. Rev. C84, 045501 (2011),

arXiv:1110.2145 [nucl-th].

[77] K. M. Shahabasyan and M. K. Shahabasyan, Vortex struc-

ture of neutron stars with triplet neutron superfluidity,

Astrophysics 54, 429 (2011), [Astrofiz.54,483(2011)].

[78] Y. Masaki, T. Mizushima, and M. Nitta, Microscopic

description of axisymmetric vortices in 3P2 superfluids,

Phys. Rev. Research 2, 013193 (2020), 1908.06215.

[79] C. Chatterjee, M. Haberichter, and M. Nitta, Collective excita-

tions of a quantized vortex in 3P2 superfluids in neutron stars,

Phys. Rev. C96, 055807 (2017), arXiv:1612.05588 [nucl-th].

[80] S. Yasui and M. Nitta, Domain walls in neutron 3P2 superflu-

ids in neutron stars, (2019), arXiv:1907.12843 [nucl-th].

[81] S. Yasui, C. Chatterjee, and M. Nitta, Symmetry and topology

of the boundary of neutron 3P2 superfluids in neutron stars:

boojums as surface topological defects, (2019), phys. Rev. C

(to be published), arXiv:1905.13666 [nucl-th].

[82] N. D. Mermin, d-wave pairing near the transition tempera-

ture, Phys. Rev. A9, 868 (1974).

[83] D. Vollhardt and P. Wolfle,

The Superfluid Phases of Helium 3, Dover Books on Physics

Series (Dover Publications, New York, 2013).

[84] G. E. Volovik, The Universe in a Helium Droplet (Clarendon,

Oxford, 2003).

[85] T. Mizushima, Y. Tsutsumi, T. Kawakami, M. Sato,

M. Ichioka, and K. Machida, Symmetry-Protected Topo-

logical Superfluids and Superconductors —From the

Basics to 3He—, J. Phys. Soc. Jap. 85, 022001 (2016),

arXiv:1508.00787 [cond-mat.supr-con].

[86] A. P. Mackenzie and Y. Maeno, The superconductiv-

ity of Sr2RuO4 and the physics of spin-triplet pairing,

Rev. Mod. Phys. 75, 657 (2003).

[87] Y. Maeno, S. Kittaka, T. Nomura, S. Yonezawa, and K. Ishida,

Evaluation of Spin-Triplet Superconductivity in Sr2RuO4,

J. Phys. Soc. Jpn. 81, 011009 (2012).

[88] D. Aoki, K. Ishida, and J. Flouquet, Review of U-based

Ferromagnetic Superconductors: Comparison between UGe2,

URhGe, and UCoGe, J. Phys. Soc. Jpn. 88, 022001 (2019).

[89] Y. Kawaguchi and M. Ueda, Spinor Bose-

Einstein condensates, Phys. Rep. 520, 253 (2012),

arXiv:1001.2072 [cond-mat.quant-gas].

[90] S. Yasui, C. Chatterjee, and M. Nitta, Phase structure of neu-

tron 3P2 superfluids in strong magnetic fields in neutron stars,

Phys. Rev. C99, 035213 (2019), arXiv:1810.04901 [nucl-th].

[91] S. Yasui, C. Chatterjee, and M. Nitta, Effects of strong mag-

netic fields on neutron 3P2 superfluidity in spin-orbit interac-

tions, in 8th International Conference on Quarks and Nuclear

Physics (QNP2018) Tsukuba, Japan, November 13-17, 2018

(2019) arXiv:1902.00674 [nucl-th].

[92] S. Yasui, C. Chatterjee, M. Kobayashi, and M. Nitta, Revis-

iting Ginzburg-Landau theory for neutron 3P2 superfluidity in

neutron stars, (2019), arXiv:1904.11399 [nucl-th].

[93] S. Uchino, M. Kobayashi, M. Nitta, and M. Ueda,

Quasi-Nambu-Goldstone Modes in Bose-Einstein

Condensates, Phys. Rev. Lett. 105, 230406 (2010),

arXiv:1010.2864 [cond-mat.quant-gas].

[94] P. M. Chaikin and T. C. Lubensky, “Principles of Condensed

Matter Physics” (Cambridge Univ. Press, Cambridge, 1995).

[95] J. Zinn-Justin, Quantum Field Theory and Critical Phenomena,

International series of monographs on physics (Clarendon

Press, 2002).

[96] R. Guida and J. Zinn-Justin, 3D Ising model: the scaling equa-

tion of state, Nuclear Physics B 489, 626 (1997).

[97] S. Yin, S.-K. Jian, and H. Yao, Chiral Tricritical

Point: A New Universality Class in Dirac Systems,

Phys. Rev. Lett. 120, 215702 (2018).

[98] J. Serene and D. Rainer, The quasiclassical approach to su-

perfluid 3He, Phys. Rep. 101, 221 (1983).

[99] J. A. Sauls, Fermi-Liquid Theory for Unconventional Super-

conductors, in Strongly Correlated Electronic Materials - The

Los Alamos Symposium 1993, edited by K. Bedell, Z. Wang,

D. Meltzer, A. Balatsky, and E. Abrahams (Addison-Wesley,

New York, 1994).

[100] R. W. Richardson, Ginzburg-Landau Theory of Anisotropic

Superfluid Neutron-Star Matter, Phys. Rev. D 5, 1883 (1972).

[101] A. B. Vorontsov and J. A. Sauls, Thermodynamic

properties of thin films of superfluid 3He − A,

Phys. Rev. B 68, 064508 (2003).

[102] T. Mizushima, Superfluid 3He in a restricted

geometry with a perpendicular magnetic field,

Phys. Rev. B 86, 094518 (2012).

[103] N. D. Mermin, d-wave pairing near the transition tempera-

ture, Phys. Rev. A 9, 868 (1974).

[104] J. A. Sauls and J. W. Serene, 3P2 pairing near

the transition temperature in neutron-star matter,

Phys. Rev. D 17, 1524 (1978).

[105] D. G. Yakovlev, A. D. Kaminker, O. Y. Gnedin, and

15

P. Haensel, Neutrino emission from neutron stars,

Phys. Rep. 354, 1 (2001).

[106] S.-O. Backman, G. E. Brown, and J. A. Niskanen, The

nucleon-nucleon interaction and the nuclear many-body prob-

lem, Phys. Rep. 124, 1 (1985).

[107] M. Ashida and K. Nagai, A-B Transition of Super-

fluid 3He under Magnetic Field at Low Pressures,

Prog. Theor. Phys. 74, 949 (1985).

[108] K. Machida, T. Mizushima, and M. Ichioka, Generic Phase

Diagram of Fermion Superfluids with Population Imbalance,

Phys. Rev. Lett. 97, 120407 (2006).

[109] T. Mizushima, M. Takahashi, and K. Machida, Fulde-

Ferrell-Larkin-Ovchinnikov States in Two-Band Superconduc-

tors, J. Phys. Soc. Jpn. 83, 023703 (2014).

[110] L. Radzihovsky and D. E. Sheehy, Imbalanced Feshbach-

resonant Fermi gases, Rep. Prog. Phys. 73, 076501 (2010).

[111] J. J. Kinnunen, J. E. Baarsma, J.-P. Martikainen, and

P. Torma, The Fulde–Ferrell–Larkin–Ovchinnikov state for ul-

tracold fermions in lattice and harmonic potentials: a review,

Rep. Prog. Phys. 81, 046401 (2018).