Universal collective modes in 2 dimensional chiral superfluids

# Universal collective modes in 2 dimensional chiral superfluids

WeiHan Hsiao Kadanoff Center for Theoretical Physics, University of Chicago, Chicago, Illinois 60637, USA
August 2018
###### Abstract

Order parameter collective modes provide the information of ground state symmetry and could manifest themselves in forms of resonance in the spectrum of other physical responses. In this work, we adopt semi-classical kinetic equations to investigate the order parameter collective modes of a class of 2 dimensional superfluids. Extending the known results for -wave superfluids, we show for any chiral ground states, there exists at least a pair of modes with mass in the weak-coupling limit. We further investigate effects that may modify these universal massive modes. We show that they receive corrections from fermionic vacuum polarization and gap anisotropy. Moreover, we determine the most substantial angular modes which contribute to the mass renormalization. These results provide potential diagnostics for distinguishing 2 dimensional chiral ground states of different angular momenta with order parameter collective modes.

preprint: EFI-xx

## I Introduction

In the studies of interacting quantum many-body systems, collective modes allow us to explore the correlated motions of underlying microscopic degrees of freedom. Should a superfluid phase exist, the order parameter component enriches the nature of collective excitations. To name a few instances, the number of gapless modes is related to the symmetry breaking pattern of the ground state because of Goldstone theorem. Massive sub-gap modes could provide thermodynamic information at the scale below the pairing breaking threshold. Moreover, via their coupling with particle-hole channel, order parameter collective modes may also result in resonant signature in transport properties Higashitani and Nagai (2000).

More intriguingly, it was first emphasized by Nambu Nambu (1985) that the bosonic (order parameter) collective modes in the Bardeen-Cooper-Schrieffer (BCS) theory, Nambu-Jona-Lasinio (NJL) model, and 3 dimensional superfluid He-B obey a relation at the level of RPA calculation. Suppose the mass of fundamental fermion in the theory is given by . Collective modes can be classified into pairs with masses according the total angular momentum of the Cooper-pairs, and these masses satisfy

 m21+m22=4m2F. (1)

To be concrete, in the BCS theory, the fermion is the Bogoliubov fermion of mass , while the 2 bosonic modes are the Goldstone boson of and the Higgs boson of . In NJL model, the fermion mass is generated by spontaneously breaking chiral symmetry. Two bosons are the pion and -boson of masses and respectively. He is a relatively richer system due to its pairing nature. The Cooper pairs in He have orbital angular momentum and spin , resulting in complex degrees of freedom and 18 bosonic collective modes. The B-phase ground state is characterized by total angular momentum . Bosonic collective modes are labeled by different and their parity under charge conjugation The relation assumes the form .

Such a compelling relation was later promoted to a sum rule and adapted to examine the elementary particle physics Volovik and Zubkov (2013). The mass of a possible Nambu partner of the celebrated 125 GeV Higgs boson was computed using an extended NJL model, which yielded the result of 325 GeV. Following this proposal, it was indicated that such calculation may suffer the effect of strong coupling nature and overlook the correction from the underlying fermionic ground state Sauls and Mizushima (2017). These bosonic masses would receive self-energy correction from particle-hole channels.

Natural questions following these developments include the generalizations in 2 dimensions and other pairing channels such as and -wave superconductors because of the following developments. Order parameter collective modes and their corrections in 2 dimensional superconductors could be interesting beyond the studies of thin film He Levitin et al. (2013, 2014) since spin-polarized low dimensional -wave superfluids could be realized in optical lattices. In addition, they may provide us with more clues for the nature of the quantum Hall state. In terms of spin-polarized non-relativistic composite fermions coupled to fluctuating gauge fields, the Pfaffian, PH-Pfaffian, and anti-Pfaffian states are chiral ground states of angular momenta and respectively Son (2018).

In particular, we ask the following 2 questions.

1. Does any Nambu sum rule exists for 2 dimensional superconductor models?

2. If any Nambu sum rule exists, what effect could possibly modify or invalidate it?

The first question has been partially addressed for chiral -wave superconductors/superfluids. Beside the well-known subgap collective modes in 3 dimensional superfluid He Wölfle (1977), their counterparts in 2 dimensions also possess order parameter collective modes Brusov and Popov (1981, 1982); Tewordt (1999). It is known that the 2 dimensional analog of B-phase hosts 4 modes of mass with angular momenta , and the analog of A-phase, whose fermionic spectrum is fully gapped in 2 dimensions, hosts six modes of mass . Clearly, in 2 dimensions both isotropic (B) and axial (A) phases have massive sub-gap modes obeying Nambu’s relation.

Part of the second questions has also been studied for SrRuO Sauls et al. (2015). It was shown that strong coupling effect and gap anisotropy induced by band mixing are able to modify the magnitude of the masses and break the spectrum degeneracy.

In this work, we aim to address these 2 questions for a larger class of models. More specifically, we study -wave superconductor, B-phase -wave superconductor and chiral superconductors of angular momentum of . We find that in the weak-coupling limit, there is a least a pair of bosonis modes of universal mass for all . Hence, the Nambu relation holds at this level. We investigate corrections to these degenerate modes owing to fermionic vacuum and gap anisotropy in a phenomenological approach and determine the angular momentum channels substantial for mass renormalization and degeneracy breaking.

These results indicate the following: (1) Though the oddness or the evenness of the ground state angular momentum can be indicated by the number of modes in the mass spectrum, different ground states cannot be distinguished in the weak-coupling limit. (2) Given a certain chiral ground state of angular momentum , the spectrum can be modified, but it only receives significant corrections from few specific angular channels.

This paper is organized as follows. In Sec. II, we review the semi-classical equation approach for the computation of collective excitations. The equations derived are used in Sec. III to compute collective excitations for various ground states. In Sec. IV, we calculate the Fermi liquid ground state effect upon the bare bosonic spectra. Finally, Sec. V studies the effect of anisotropic perturbation of the superconducting ground state with a phenomenological polar mode decomposition. A summary and several open directions are composed at the end to close the main text. The full solutions to the kinetic equation (II) without assuming and are present along with the method in appendix.

## Ii Formalism

Bosonic collective modes in superfluids or superconductors 111In this work, we turn off the gauge field and therefore do not strictly distinguish these 2 terminologies. can be computed with various approaches. In this work, we adopt the time-dependent mean field approximation to include the Fermi liquid corrections. This approach can be formulated in terms of generalized Landau-Boltzmann kinetic equations Vollhardt and Wolfle (2013), or the linearized non-equilibrium Eilenberger equation Serene and Rainer (1983). Though we will not provide a derivation starting from defining Green’s functions, which we refer the readers to Ref.Serene and Rainer (1983), we will give a complete elaboration of the machinery.

In the semi-classical limit, physical quasi-particle distribution is related to the Keldysh Green’s function . In its argument are the Fourier transformed variables of the fast coordinates, where as are ones of the center of mass coordinates. In clean limit, the linear response of a non-relativistic fermion without spin-orbital coupling is given by the following kinetic equation

 ε+τ3δˆg−δˆgτ3ε−− vF^p⋅qδˆg−[ˆσ0,δˆg] = δˆσˆg0(ε−)−ˆg0(ε+)δˆσ, (2)

where denotes . The operators and are the molecular mean field at equilibrium and the linear perturbation respectively. Similarly, represents the Keldysh propagator at equilibrium and is related to retarded and advanced propagators via , which yields

 ˆg0=−2πi(τ3ε−ˆΔ)√ε2−|Δ|2Θ(ε2−|Δ|2)sgn(ε)tanhε2T. (3)

The low energy fluctuation of quasi-particles and the deduced physical quantities are given by the -integrated

 ∫∞−∞dε2πiˆg(ε,^p;ω,q). (4)

For example, the particle-current is given by

 δj=N(0)∫dθ2π∫dε2πi12vptr[τ3δˆg(ε;ω,q)]. (5)

In particular, the molecular field, or self-energy, is self-consistently determined by the convolution of inter-particle potentials and .

To further elaborate, we note that has a general structure

 δˆg=(δg+δg⋅σ(δf+δf⋅σ)iσ2iσ2(δf′+δf′⋅σ)δg′+δg′⋅σt), (6)

and accordingly so does ,

 δˆσ=(δε+δε⋅σ(d+d⋅σ)iσ2iσ2(d′+d′⋅σ)δε′+δε′⋅σt), (7)

where the primed variables are

 δg′(^p;ω,q)=δg(−^p;ω,q) (8a) δε′(^p;ω,q)=δε(−^p;ω,q) (8b) δf′(^p;ω,q)=δf∗(^p;−ω,−q) (8c) d′(^p;ω,q)=d∗(^p;−ω,−q). (8d)

Physical observables are usually expressed in terms of the symmetric and anti-symmetric combination of , and their primed partners. In this work, we define and combinations of a function as

 f(±)=f±f′. (9)

The eigenvalues represent the parity under charge conjugation. As we will see, the charge density and energy stress tensor correspond to the scalar and quadrupole modes of respectively, whereas the current density is proportional to the vector mode of . Similarly, and stand for the amplitude and phase fluctuations of the pairing fields.

At one-loop, the correction to the self-energy is determined by the two-body vertex. Evaluating internal momentum integral over the Fermi surface, we have, in the Landau channel,

 δε(^p;ω,q )=δεext(^p;ω,q) +∫dθ′2πAs(θ,θ′)∫dε′4πiδg(ε′,^p′;ω,q), (10a) δε(^p;ω,q )=δεext(^p;ω,q) +∫dθ′2πAa(θ,θ′)∫dε′4πiδg(ε′,^p′;ω,q). (10b)

where is the spin-independent (exchange) forward scattering amplitude which can be rewritten in terms of Landau parameters via

 A(^p,^p′)=F(^p,^p′)−∫dθ′′2πF(^p,^p′′)A(^p′′,^p′). (11)

Similarly in Cooper channel, the off-diagonal components are related by the linearized gap equations.

 d(^p;ω,q)=∫dθ′2πVe(θ,θ′)∫dε′4πiδf(ε′,^p′;ω,q) (12a) d(^p;ω,q)=∫dθ′2πVo(θ,θ′)∫dε′4πiδf(ε′,^p′;ω,q) (12b)

where () is the pairing potentials in even (odd) angular momentum channel.

Analogous to 3 dimensions, where scattering amplitudes and pairing potentials are expanded in terms of spherical harmonics, in 2 dimensions we have

 A=∞∑ℓ=−∞Aℓe−iℓ(θ−θ′), Aℓ=A−ℓ, (13a) Ve=∑ℓ∈{even}Vℓ[e−iℓ(θ−θ′)+h.c.] (13b) Vo=∑ℓ∈{odd}Vℓ[e−iℓ(θ−θ′)+h.c.]. (13c)

from which we can derive , where is the conventional dimensionless Landau parameter of channel . For other functions evaluated on Fermi surface , the angular decomposition is defined as

 f=∞∑ℓ=−∞e−iℓθfℓ. (14)

We can then provide a recipe for the computation. We first invert (II) to obtain as a function of . Taking the the convolution as in (10a), (10b), (12a), and (12b) establishes integral equations for . Projecting equations (12a), and (12b) to different angular modes gives us the dynamical equations of and , which are sourced by and . The bare bosonic collective modes are given by the normal modes of the homogeneous part of the equations. To include the Fermi liquid corrections, we project (10a), and (10b) to their th angular modes as well and solve and in terms of , and . Plugging the results back into the equations for and yields inhomogeneous equations sourced solely by external fields. The renormalized mass spectrum is solved as the poles of the solution kernels.

In the rest of this section, we use the above formulation to derive the integral equation for 2 dimensional spin-singlet and spin-triplet superfluids and compute the collective modes and Fermi liquid corrections in the sections following.

### ii.1 spin-singlet pairing

In a spin-singlet pairing channel, the equilibrium self-energy is characterized by a complex gap field .

 ˆσ0=ˆΔ=(0Δiσ2Δ∗iσ20). (15)

The fluctuation of the spin-singlet order parameter can be parametrized by a complex number . It transforms as a scalar under spin rotation SO(3) and can have internal structures, i.e., tensor indices under orbital rotation SO(2) depending on pairing symmetries. In the absence of magnetic field, spin-triplet fluctuations decouple from . Hence we consider them separately in the present work. Plugging (15) into (II), inverting it using the variables defined in (6) and (7), and taking the convolution as in (12a) and (12b) give us, in the long-wavelength limit, the off-diagonal components of the molecular fields

 d(^p;ω)=∫dθ′2πVe(θ,θ′) [(γ+14¯λ[ω2−2|Δ|2])d −¯λ2Δ2d′−ω4¯λΔδε(+)], (16a) d′(^p;ω)=∫dθ′2π Ve(θ,θ′)[(γ+14¯λ[ω2−2|Δ|2])d′ −¯λ2(Δ∗)2d+ω¯λ4Δ∗δε(+)]. (16b)

The definition of is given in Appendix (67). The function , often called the Tsunedo function, whose complete form is given in appendix A. In limit,

 ¯λ=λ(^p;ω)|Δ|2=∫∞|Δ|dε√ε2−|Δ|2tanhε2Tε2−ω2/4. (17)

We see there could be some residual angular dependence through the anisotropy in even in the long wavelength limit. This point will be crucial as we examine the effect of gap anisotropy. Suppose only a single pairing channel is significant, i.e., that . Taking on both sides of (16) and (16) eliminates s with their right-hand sides. Thus, we obtain the dynamical equations of motion

 (18a) ⟨eiLθ¯λ([ω2−2|Δ|2]d′−2(Δ∗)2d+ωΔ∗δε(+))⟩=0, (18b)

where we use the angle bracket to denote the angular average .

### ii.2 spin-triplet pairing

In a spin-triplet pairing channel, the ground state self-energy is characterized by the vector-valued gap function

 ˆΔ=(0Δ⋅iσσ2Δ∗⋅iσ2σ0). (19)

The fluctuation is encoded in the dynamics of the vector, which transforms as a vector under SO(3), and could contain internal structure depending on pairing symmetry as well. For 2 dimensional -wave superfluids or superconductors, it can be expanded as , where . Similarly inverting the kinetic equations, the dynamical equations for in limit are

 d=∫dθ′2πVo(θ,θ′)[(γ+14¯λ(ω2−2|Δ|2))d +¯λ2[(Δ⋅Δ)d′−2(Δ⋅d′)Δ] −ω¯λ4(Δδε(+)−iΔ×δε(+))]. (20a) d′=∫dθ′2πVo(θ,θ′)[(γ+14¯λ(ω2−2|Δ|2))d′ +¯λ2[(Δ∗⋅Δ∗)d−2(Δ∗⋅d)Δ∗] +ω¯λ4(Δ∗δε(+)+iΔ∗×δε(+)). (20b)

Again we multiply (20), and (20) by and integrate over . The regularized integral s again cancel out the right-hand sides and we obtain

 ⟨eiLθ¯λ([ω2−2|Δ|2]d+2(Δ⋅Δ)d′−4(Δ⋅d′)Δ)⟩ = ω⟨eiLθ¯λ(Δδε(+)−iΔ×δε(+))⟩. (21a) ⟨eiLθ¯λ([ω2−2|Δ|2]d′+2(Δ∗⋅Δ∗)d−4(Δ∗⋅d)Δ∗)⟩ = −ω⟨eiLθ¯λ(Δ∗δε(+)+iΔ∗×δε(+))⟩. (21b)

In the following section, we will solve (18a), (18b), (21), and (21) for ground states of different pairing channels and symmetries.

## Iii Collective Modes

In this section we utilize the equations derived in the last section to compute the bare bosonic spectra for various superconducting ground states. More specifically, we will focus on unitary gaps with isotropic gap amplitudes. The dynamical equations are considerably simplified as the Tsunedo function drops out of all angular averages. The masses of order parameter collective modes appear as normal modes of the homogeneous part in  (18a), (18b), (21), and (21). The particle-hole self-energy and are treated as external sources at the zeroth order, and they will be renormalized in the next section as we conclude Fermi liquid effects.

### iii.1 s-wave pairing

For -wave pairing, it is possible to choose a gauge such that , in which limit the amplitude mode and phase mode decouple. The bosonic field has no internal structure and is simply a complex scalar, implying 2 order parameter collective modes obeying the following equations

 (ω2−4Δ2)d(+)=0 (22a) ω2d(−)=2ωΔδε(+)0. (22b)

The normal modes have and corresponding to the simplest example of Higgs and Goldstone bosons respectively. Note that if we compute (22b) to the leading non-vanishing order in , we would have obtain , entailing the Goldstone boson moves at the speed . Another observation is that the Higgs mode receives to external force and consequently it would not be renormalized by particle-hole self-energy. On the other hand, the Goldstone boson is sourced by the density mode , which would trigger Higgs mechanism in the presence of Coulomb interaction.

### iii.2 p-wave pairing

The -wave pairing states have more degrees of freedom, and thus more collective modes, owing to triplet-pairing and orbital structure. In 2 dimensions, the fluctuation of -wave superconductors can be represented by the complex tensor , which contains complex degrees of freedom, leading to 12 collective modes in total. The number of the massless modes , as we will see shortly, can be determined by ground state symmetry breaking pattern. The rest is number of sub-gap collective modes.

#### B-phase

We first consider the 2-dimensional analog of He B-phase, where the gap function assumes the form.

 Δ=ΔpF(^xpx+^ypy), Δ∈R. (23)

In this phase, the global symmetry breaks following the pattern SO(3)SO(2)U(1) SO(2), which immediately indicates the existence of 4 Goldstone modes. Besides, the residual symmetry is SO(2) rotation and we expect the fluctuations can be characterized by total angular momentum . Owing to this fact, it is also convenient to choose another basis for , where is the polar angle of , as follows

 D±m=dxm±idym (24) D0m=dzm. (25)

Moreover, as the gap function is real, modes transforming differently under charge conjugation again decouple. That is to say, we can further separate degrees of freedom. We first look at the modes governed by the equation

 (ω2−4Δ2)d(−)+4(Δ⋅d(−))Δ=2ωΔδε(+). (26)

Organizing the dynamical equations using the basis , we could find

 (ω2−4Δ2)D(−)0±=0 (27a) (ω2−2Δ2)D(−)±±=2ωΔδε(+)±2 (27b) (ω2−4Δ2)(D(−)+−−D(−)+−)=0 (27c) ω2(D(−)+−+D(−)−+):=ω2D(−){+−}=4ωΔδε(+)0. (27d)

Consequently, has 2 sub-gap massive modes of the same mass , and they are sourced by the quadrupolar molecular field.

Next we look at , which obeys

 [ω2d(+)−4Δ(Δ⋅d(+))]=−2iωΔ×δε(+). (28)

Following the same procedure to project each component to different sectors, we would obtain

 ω2d(+)0=−2iω(Δ×δε(+))⋅^z (29a) (ω2−2Δ2)D(+)±±=∓2ωΔδε(+)±2⋅^z (29b) ω2(D(+)−+−D(+)+−)=4Δωδε(+)0⋅^z (29c) (ω2−4Δ2)(D(+)+−+D(+)+−)=0. (29d)

It can be seen that the sums of the mass squares of and modes with the same quantum numbers all equal , which serves an example of Nambu’s sum rule.

In spite of the pure phenomenological view adopted in this work, if the underlying UV physics possesses lattice symmetry, there are other permitted unitary and fully gapped real ground states, , , and . They differ from the B-phase by either a rotation, spatial reflection, or a combination of these two operations. It is straightforward to check and find unsurprisingly all of them have the same mass spectrum characterized by different quantum numbers.

#### A-phase

2 dimensional A-phase has a different symmetry breaking pattern SO(3)SO(2)U(1) U(1)U(1). Consequently, there are 3 Goldstone bosons and the bosonic fluctuations should be characterized by eigenvalues .

Let us now consider a ground state described by

 Δ=px+ipypFΔ^z=eiθΔ^z, Δ∈R. (30)

The dynamic equations for and () are now coupled and given as follows.

 ⟨eiℓθ[(ω2−2Δ2)d+2Δ2d′e2iθ−4Δ2^z(^z⋅d′)ei2θ]⟩ = ωΔ⟨eiℓθeiθ[^zδε(+)−i^z×δε(+)]⟩. (31a) ⟨eiℓθ[(ω2−2Δ2)d′+2Δ2de−2iθ−4Δ2^z(^z⋅d)e−i2θ]⟩ = −ωΔ⟨eiℓθe−iθ[^zδε(+)+i^z×δε(+)]⟩. (31b)

We first look at and . They obey the equations

 (ω2−2Δ2)d1=ωΔ⟨^zδε(+)2−i^z×δε(+)2⟩ (32a) (ω2−2Δ2)d′−1=−ωΔ⟨^zδε(+)−2+i^z×δε(+)−2⟩ (32b)

and have the same mass . These six modes are the massive Nambu partners of one another in A-phase. On the other hand, equations for and are coupled. Solving them, one can find 3 massless modes and 3 modes of mass .

Quite intuitively, the spectrum would be the same if even we have considered the ground state aligned in other directions, e.g. . Microscopically, a different choice of ground state polarization direction corresponds to a different choice of orientation in which the spins are equally aligned. For instance, in the ground state (30), which is conjectured to be the ground state of SrRuO, the orbital angular momentum of the Cooper pair points in direction, whereas the spins of constituent fermions lies in plane.

### iii.3 d-wave pairing

The -wave gap fluctuation is captured by the complex field with irreducible complex degrees of freedom , represented by the modes In this work, we consider the ground state

 Δp2F(px+ipy)2=Δei2θ, Δ∈R. (33)

Equations (18a), and (18b) then become

 (ω2−2Δ2)d2=ωΔδε(+)4 (34a) (ω2−2Δ2)d′−2=−ωΔε(+)−4 (34b) (ω2−4Δ2)(d−2+d′2)=0 (34c) ω2(d−2−d′2)=2ωΔδε(+)0. (34d)

Clearly and have masses .

### iii.4 Higher L chiral ground states

Extending the analyses for equations (31), (31), (34a), and (34b), we could actually consider a more general ground state

 singlet:ΔeiLsθ, Ls=even (35a) triplet:^zΔeiLtθ, Lt=odd. (35b)

Modes , , and would automatically satisfy

 (ω2−2Δ2)dLs=ωΔδε(+)2Ls (36a) (ω2−2Δ2)d−Ls=−ωΔδε(+)−2Ls (36b) (ω2−2Δ2)^z⋅dLt=ωΔδε(+)2Lt (36c) (ω2−2Δ2)^z⋅d′−Lt=−ωΔδε(+)−2Lt. (36d)

In this sense is a universal order parameter collective mode for any chiral ground state of angular momentum , each of which is sourced by quasi-particle self-energy .

As one can notice from either the dynamical equations or the arguments presented, the eigenfrequency can be universal because all chiral ground states follow similar symmetry breaking patterns. In 2 dimensions, all ground state considered in the present work are equivalently left with a SO(2) or U(1) rotational symmetry. For the B-phase of superfluid, it refers to the total angular momentum , whereas for other chiral superconductors, it is a combination of SO(2) orbital rotation and residual U(1) gauge transformation and thus can be labeled formally by an angular momentum number. These massive sub-gap modes are those with the highest angular momenta in their pairing channels. While other modes of smaller angular momenta are able to interfere and break their degeneracies into Goldstone and Higgs bosons, they decouple from each other simply because of selection rules imposed by the rules of angular momenta addition.

## Iv Fermi Liquid Corrections

In the previous section we found for chiral ground states of given , bosonic modes and have finite mass in mean field approximation. In this section we compute the Fermi liquid corrections to the mass spectra. Before presenting quantitative details, we point out some general features. Those modes with mass in general are not sourced by fermionic self-energy, and consequently these modes are not renormalized. On the other hand, for those massless modes, short-range fermionic self-energy can at most renormalize the sound speed and the magnitude of external source fields instead of generating a gap. We will demonstrate this with some specific examples and in the rest of the section we will focus on the sub-gap mode .

### iv.1 Massless Modes

Let us first look at (22b). The right hand side consists of pure external perturbation and the renormalization coming form the integral part of (10a). In long wavelength limit,

 δε(+)(θ)=δu(+)ext +∫dθ′2πFs(θ,θ′)[−λδε(+)+ωλ2Δd(−)]. (37)

Projecting out component, we obtain

 (1+λ(ω)Fs0)δε(+)0=δu(+)0+ωλ2ΔF0d(−), (38)

plugging which back into (22b) yields

 ω2d(−)=2ωΔδu0. (39)

It entails that remains massless. To demonstrate a triplet-pairing example, we look at B-phase (23) and (27c). For triplet-pairing states, the diagonal term of (10a) reads

 δε(+)(θ)=δu(+)ext + ∫dθ′2πFs(θ,θ′)[−λ(ω)δε(+)+12ω¯λΔ⋅d(−)], (40)

whose projection to th mode is

 δε(+)ℓ=δu(+)ℓ+12¯λωFsℓ(Δ⋅d(−))ℓ1+λ(ω)Fsℓ. (41)

For ,

 B:δε(+)0=δu0+λω4ΔFs0D(−){+−}1+λFs0 (42)

and we again find

 ω2D(−){+−}=4ωΔδu0. (43)

In these 2 examples, we see the dynamical equations for and are not even modified by . By dimension counting, it could be seen that the renormalization of self-energy are given by the following relation

 (1+c1Fλ)δε=δu+c2ω2λFD, (44)

where and are dimensionless numbers of order 1. It can be implied gapless modes cannot be gapped by short-ranged interactions parametrized by Landau parameters.

### iv.2 Massive Sub-gap Modes

In this section let us continue to examine how Landau parameters renormalize massive modes. We start with equation (27b). Take component of (41).

 B:δε(+)±2=δu±2+ωλ4ΔFs2D(−)±±1+λFs2. (45)

Plugging this back into (27b) renormalizes the solutions as

 D(−)±±=2ωΔδu±2(ω2−2Δ2)+12λFs2(ω2−4Δ2). (46)

The new mass is given by the zero of the denominator. In the limit ,

 ω2≃2Δ2(1+12λFs2). (47)

is a positive number of order 1. From this expression we can see modes get heavier for repulsive interactions and soften for attractive interactions .

Next let us look at the amplitude mode (29b) sourced by spin-dependent quasi-particle energy.

 δ ε(+)z(θ)=δhz +∫dθ′2πFa(θ,θ′)[−λδε(+)z−iω2¯λ(Δ×d(+))z]. (48)

Projecting it to modes,

 δε(+)z,±2=δh±2∓Fa2ωλ4ΔD(+)±±1+λFa2. (49)

Substituting this back into (29b) yields

 D(+)±±=±2ωΔδhz,±2(ω2−2Δ2)+12λFa2(ω2−4Δ2). (50)

Therefore, the mass correction is given by the same transcendental equation with the replacement .

We are now ready to repeat the above computation for general chiral ground states. As it can be inferred from the previous analyses, the equations for singlet-pairing states are identical to ones for the longitudinal components of the triplet-pairing states. Hence, we will concentrate on triplet-pairing states and take without loss of generality since higher states have the same algebraic forms.

The main difference between the preceding analyses and the one for chiral states is that the gap function can no longer be chosen real by a gauge transformation. Consequently, the scalar self-energy would satisfy the equation

 δε(+)=δu+ext+∫dθ′2π F(θ,θ′)[−λδε(+) +12ω¯λ(Δ⋅d−Δ⋅d′)]. (51)

Let us take the component of (32a) and (32b)

 (ω2−2Δ2)d1z=ωΔδε(+)2 (52) (ω2−2Δ2)d′−1z=−ωΔδε(+)−2. (53)

Renormalizing with (IV.2), we find

 d1z=ωΔδu(+)2(ω2−2Δ2)+12λFs2(ω2−4Δ2) (54a) d′−1z=ωΔδu(+)−2(ω2−2Δ2)+12λFs2(ω2−4Δ2). (54b)

Finally we look at the transverse fluctuation by looking at the component.

 (ω2−2Δ2)d1x=−iωΔ(^z×δε(+)2)x (55a) (ω2−2Δ2)d′−1x=−iωΔ(^z×δε(+)−2)x. (55b)

The spin-dependent self-energy now takes the form

 ^z×δε(+)=^z× δh(+)+∫dθ′2πFa(θ,θ′)[−λ^z×δε(+) −iω2¯λ^z×[(Δ∗×d)+(Δ×d′)]]. (56)

Projecting it to allows to solve

 d1x=−iωΔ(^z×δh(+))2(ω−2Δ2)+12λFa2(ω2−4Δ2). (57)

To sum up, the analyses in this section have shown the following: (i) The Goldstone modes are not gapped by short-range interaction parametrized by Landau parameters. (ii) For -wave superconductors in both B-phase and A-phase, the sub-gap modes receives renormalization from quadrupolar Landau parameters or . (iii) For all chiral ground states of finite orbital momenta , the sub-gap modes longitudinal to their ground states receive mass renormalization from the channel