This is the accepted manuscript made available via CHORUS. The article has been
published as:

Hidden charge-conjugation, parity, and time-reversal
symmetries and massive Goldstone (Higgs) modes insuperconductors
Shunji Tsuchiya, Daisuke Yamamoto, Ryosuke Yoshii, and Muneto Nitta
Phys. Rev. B 98, 094503 — Published 4 September 2018
DOI: 10.1103/PhysRevB.98.094503

Hidden charge-conjugation, parity, time-reversal symmetries and massive Goldstone(Higgs) modes in superconductors
Shunji Tsuchiya,1, 2, ∗ Daisuke Yamamoto,3 Ryosuke Yoshii,1, 2 and Muneto Nitta4, 2
1

Department of Physics, Chuo University, 1-13-27 Kasuga, Bunkyo-ku, Tokyo 112-8551, Japan
2
Research and Education Center for Natural Sciences, Keio University,
Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan3
Department of Physics and Mathematics, Aoyama Gakuin University,
5-10-1 Fuchinobe, Chuo-ku, Sagamihara, Kanagawa 252-5258, Japan
4
Department of Physics, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan(Dated: August 10, 2018)
A massive Goldstone (MG) mode, often referred to as a Higgs amplitude mode, is a collective
excitation that arises in a system involving spontaneous breaking of a continuous symmetry, alongwith a gapless Nambu-Goldstone mode. It has been known in the previous studies that a pureamplitude MG mode emerges in superconductors if the dispersion of fermions exhibits the particlehole (p-h) symmetry. However, clear understanding of the relation between the symmetry of the
Hamiltonian and the MG modes has not been reached. Here we reveal the fundamental connection
between the discrete symmetry of the Hamiltonian and the emergence of pure amplitude MG modes.
To this end, we introduce nontrivial charge-conjugation (C), parity (P), and time-reversal (T )operations that involve the swapping of pairs of wave vectors symmetrical with respect to the Fermi
surface. The product of CPT (or its permutations) represents an exact symmetry analogous to
the CPT theorem in the relativistic field theory. It is shown that a fermionic Hamiltonian with ap-h symmetric dispersion exhibits the discrete symmetries under C, P, T , and CPT . We find thatin the superconducting ground state, T and P are spontaneously broken simultaneously with theU(1) symmetry. Moreover, we rigorously show that amplitude and phase fluctuations of the gap
function are uncoupled due to the unbroken C. In the normal phase, the MG and NG modes become
degenerate, and they have opposite parity under T . Therefore, we conclude that the lifting of thedegeneracy in the superconducting phase and the resulting emergence of the pure amplitude MG
mode can be identified as a consequence of the the spontaneous breaking of T symmetry but not of
P or U(1).

I.

INTRODUCTION

Massive Goldstone (MG) modes, often referred to as
Higgs amplitude modes, and Nambu-Goldstone (NG)modes are ubiquitous in systems that involve spontaneous breaking of continuous symmetries1–4 . In the simplest U(1) symmetry breaking, the former induce amplitude oscillation of a complex order parameter5 and thelatter induce phase oscillation. Whereas NG modes havebeen studied in various condensed matter systems, MG
modes have evaded observations until recently with only
a few exceptions6–8 .
Despite the increasing number of observations,for example, in superconductors6,9–13 , quantum spin
systems14–17 , charge-density-wave materials8,18 , and ultracold atomic gases19–22 , and theoretical studies23–32 ,fundamental aspects of MG modes in condensed matter systems have not been fully understood, in contrast
to NG modes; spontaneous breaking of a continuous
symmetry does not guarantee emergence of MG modes,
while that of NG modes is ensured by the Goldstone
theorem2 . For instance, whereas a MG mode appears in
a Bardeen-Cooper-Schrieffer (BCS) superconductor6,24 ,it does not exist in a Bose-Einstein condensate (BEC)33 ,despite the fact that both of the systems involve U(1)symmetry breaking and furthermore one evolves continuously to the other through the BCS-BEC crossover34–38 .

Varma pointed out that the approximate particle-hole (ph) symmetry, i.e., the linearly approximated fermionicdispersion ξk ≃ vF (k − kF ) (vF is the Fermi velocity and kF is the Fermi wave number), results inthe effective Lorentz invariance of the time-dependent
Ginzburg-Landau equation in the weak-coupling BCS
limit, which yields the decoupled amplitude and phasemodes33 . A pure amplitude MG mode also appears inlattice systems if the energy bands exhibit the rigorous
p-h symmetry26,27 .
It has been thus recognized in the previous studies
that a pure amplitude MG mode emerges in superconductors if the dispersion of fermions ξk exhibits the ph symmetry24–27,33 . However, the p-h symmetry in thecontext of the previous works refers to the characteristic feature of the fermionic dispersion ξk that should
be distinguished from the symmetry of the Hamiltonian.
Meanwhile, clear understanding of the relation between
the symmetry of the Hamiltonian and MG modes has notbeen reached.
In this paper, we reveal the fundamental connection between the discrete symmetry of the Hamiltonian and the emergence of pure amplitude MG modes.
We introduce three discrete operations for general nonrelativistic systems of fermions, which we refer to“charge-conjugation” (C), “parity” (P), and “timereversal” (T ). The product of CPT (or its permutations)represents an exact symmetry analogous to the CPT the-2orem in the relativistic field theory39 . We show that thestandard BCS Hamiltonian with a p-h symmetric dispersion is invariant under C, P, T , and CPT in addition to
the global U(1) gauge invariance. If the U(1) symmetryis spontaneously broken in the superconducting groundstate, the symmetries under P and T are simultaneously
broken while the symmetry under C is unbroken. We rigorously show that amplitude and phase fluctuations of
the gap function are uncoupled due to the unbroken C.
The MG mode thus induces pure amplitude oscillationsof the gap function in a p-h symmetric system. It is also
shown that the MG and NG modes reduce to the degenerate states in the normal phase due to the U(1) symmetry
and they have opposite parity under T . Therefore, thelifting of the degeneracy in the superconducting phaseand the resulting emergence of the pure amplitude MG
mode can be identified as a consequence of the the spontaneous breaking of T symmetry but not of P or U(1).
Thus, the breaking of T proves to be responsible for the
emergence of the pure amplitude MG mode.
This paper is organized as follows: In Sec. II, we
present the model and introduce the pseudospin representation. In Sec. III, we define the three discrete operations C, T , and P to discuss the symmetries of the
Hamiltonian under the operations of C, T , P, and CPT .
In Sec. IV, we study the symmetry of the superconducting ground state. In Sec. V, we discuss collective modes
within the classical spin analysis. In Sec. VI, we givea rigorous proof of the uncoupled amplitude and phasefluctuations of the gap function in a p-h symmetric system due to the unbroken C. In Sec. VII, we give a direct
demonstration of the relation between the emergence ofthe pure amplitude MG mode and the spontaneously broken T symmetry. In Sec. VIII, we summarize. We set~ = kB = 1 throughout the paper.
II.

PSEUDOSPIN REPRESENTATION

We study for simplicity the reduced BCS
Hamiltonian40
X
X † †
H =
ξk c†ks cks − g
ck↑ c−k↓ c−k′ ↓ ck′ ↑ , (1)
k,s

k,k′

where c†ks (cks ) is the creation (annihilation) operator ofa fermion with momentum k and spin s (=↑, ↓), g(> 0)
denotes the attractive interaction between fermions, andξk = εk − µ is the kinetic energy of a fermion measured
from the chemical potential µ. For example, εk = k 2 /2m
in a continuous system (m is the mass of a fermion). We
do not specify the form of εk for generality of argument.
To discuss the symmetries of the Hamiltonian (1), it isconvenient to introduce the pseudospin representation41:
Sµk = 21 Ψ†k τµ Ψk (µ = x, y, z), where τ = (τx , τy , τz )are Pauli matrices and Ψk = (ck↑ , c†−k↓ )t is the Nambu
spinor42 . Note that Szk is related to the fermion number
operator nks = c†ks cks by Szk = 21 (nk↑ + nk↓ − 1). In the

(a)

ξ

(b)

ξ ξ =-2t cosk
k

-̟

O

vF (k-kF)

-ξk

k
k ̟
kF

k=2kF- k
(c)
O

-2t

k
k

kF

ξk

-ξk

k

Г

M

X

Fermi surface

FIG. 1: Illustration of the wave vector k and the dispersion−ξk in (a) a continuous system and (b) the 1D lattice at halffilling (µ = 0). (c) k for the half-filled energy band in thesquare lattice.

pseudospin language, the fermion vacuum is the spindown state (|0ik = | ↓ik ) and the fully occupied state is
the spin-up state (c†k↑ c†−k↓ |0ik = | ↑ik ).
The pseudospin representation of the Hamiltonian (1)is given by
X
XH=
2ξk Szk − g(2)S⊥k · S⊥k′ ,kk,k′where S⊥k = (Sxk , Syk ). The kinetic energy (interaction) term is translated into the Zeeman (ferromagnetic
XY exchange) term in the pseudospin language. The rotational symmetry of the Hamiltonian (2) in the xy-plane
represents the U(1) symmetry of Eq. (1) with respect tothe transformation Ψk → eiτz α Ψk .
III.

HIDDEN DISCRETE SYMMETRIESIn this section, we define three discrete transformations
for fermions and discuss the symmetry of the Hamiltonian (2) under those operations.
A.

Charge-conjugation

Let us consider a unitary transformation for the
Nambu spinor43 :CΨk C = τx Ψk ,CΨ†k C = Ψ†k τx .

(3)Here, k is the mirror reflected wave vector of k withrespect to the Fermi surface, i.e., k and k are on the
opposite side of the Fermi surface and away from it withthe same minimum distance (see Figs. 1 (a)-(c)). For
example, k = (2kF − k)k/|k| in a continuous system.
Note that k = k if k is on the Fermi surface. Since C
transforms a particle (c† ) into a hole (c) and vice versa,
it can be referred to as a “charge conjugation” operation.
C is specifically given by
YY
C=F
σxk , F =fk,k ,
(4)
k

ξk >0

3
where σµk = 2Sµk . The operator fk,k swaps the state ofk and that of k: fk,k |ψik |φik = |φik |ψik . One can showC † = C and C 2 = 1 from Eq. (4).
The pseudospin operators are transformed by C as
CSµk C = (−1)δµ,x +1 Sµk , CSµ C = (−1)δµ,x +1 Sµ , (5)P
where Sµ = k Sµk is the total spin. Equation (5) showsthat C consists of the π rotation of pseudospins about thex-axis and the swapping of k and k.
Transforming Eq. (2) by C, we obtain
X
X
CHC =
2(−ξk )Szk − g(6)S⊥k · S⊥k′ .
k

k,k′

C.

The “parity” operation, denoted by P, is defined to be
the inversion of pseudospins in the xy-plane. It is equivalent to the π rotation about the z-axis and therefore canbe represented as
Y
P=
σzk .
(11)
k

It satisfies P † = P and P 2 = 1. The transformation by
P is given asHence, CHC = H and equivalently [H, C] = 0, if thefermion dispersion satisfies the condition−ξk = ξk .

(7)Equation (7) indicates the invariance of the dispersion
ξk under the successive mirror reflections with respect toξ = 0 and k = kF (see Figs. 1 (a) and 1 (b)), which werefer to particle-hole (p-h) symmetry
P in view of the factthat the density of states N (ξ) = k δ(ξ − ξk ) is even if
Eq. (7) holds.
Figure 1(a) shows that, whereas ξk = k 2 /2m − µ is not
p-h symmetric, the linearized dispersion ξk ≃ vF (k − kF )is p-h symmetric. Therefore, a continuous system has
an approximate p-h symmetry if the interaction is weakenough. On the other hand, Fig. 1 (b) illustrates thatthe tight-binding energy band in the d-dimensional cubicPlattice ξk = −2t di=1 cos(ki ) (t is the hopping matrix
element) exhibits a rigorous p-h symmetry at half-filling(µ = 0).
B.

Time-reversal

The “time-reversal” operation of the Nambu spinor
and the pseudospin operators are defined to be
T Ψk T −1 = τy Ψk ,T Ψ†k T −1 = Ψ†k τy ,(8)

T Sµk T −1 = −Sµk ,T Sµ T −1 = −Sµ .

(9)The time-reversal T can be written in the formY
T = UT K, UT = F(−iσyk ),(10)

kwhere K is the complex conjugation operator and UT is
the unitary operator that rotates pseudospins π about
the y-axis and swaps k and k. From Eq. (9), the ph symmetric Hamiltonian that satisfies Eq. (7) is timereversal invariant T HT −1 = H. T reverses the time in
the Heisenberg representation as T Sµ (t)T −1 = −Sµ (−t).
It is important to note that T represents “timereversal” in the pseudospin space, which is different fromthe usual time-reversal operation discussed, for example,
in Ref. 44. Although the usual time-reversal symmetry
is not broken in s-wave superconductors44 , T is spontaneously broken simultaneously with the U(1) symmetry
breaking as we shall show later.

ParityPΨk P = τz Ψk ,PΨ†k P = Ψ†k τz , (12)PSµk P = (−1)δµ,z +1 Sµk , PSµ P = (−1)δµ,z +1 Sµ . (13)The Hamiltonian (2) is invariant by P: PHP = H. Since
the π rotation in the xy-plane is an element of U(1), P
is trivially broken in the U(1) broken ground state.

D.

CPT invariance

The transformation by the product Θ = CPT is givenasΘΨk Θ−1 = iΨk ,ΘΨ†k Θ−1 = −iΨ†k , (14)ΘSµk Θ = (−1)δµ,y +1 Sµk , ΘSµ Θ = (−1)δµ,y +1 Sµ . (15)Q
Using Eqs. (4), (10), and (11), we obtain Θ = k (−1)·K
and thus ΘHΘ−1 = H. Since the Lagrangian L =P† ∂−1= L(−t),k iΨk ∂t Ψk − H is transformed as ΘL(t)Θ
the action is invariant and therefore CPT and all otherpermutations of C, P, and T are exact symmetries analogous to the CPT invariance in relativistic systems39 .

IV.

SYMMETRY OF THE GROUND STATE

We study the symmetries of the superconducting
ground state focusing on that of a p-h symmetric system. It is reasonable to expect that all the symmetries of the true groundQstate are realized in the BCS
Here,wave pfunction |Ψi =k + vk | ↑ik ).
k (uk | ↓ipuk = (1 + ξk /Ek )/2 and vk = (1 − ξk /Ek )/2. The
gap function is set positive real
Pin the ground state without losspof generality ∆0 = g k hc−k↓ ck↑ i = ghSx i > 0.
Ek = ξk2 + ∆20 is the dispersion of single-particle excitations (bogolons). |Ψi represents the groundP state of
the mean-field (MF) Hamiltonian HMF = − k Hk0 · Sk ,where Sk = (Sxk , Syk , Szk ). The effective magnetic field
Hk0 = (2∆0 , 0, −2ξk ) lies in the xz-plane with the polar
angle ϕk (see Fig. 2 (b)), where sin ϕk = ∆0 /Ek andcos ϕk = −ξk /Ek . Note that ϕk = π − ϕk , if Eq. (7)holds. The requirement that the average spin Sk0 = hSk i
is in P
parallel with Hk0 leads to the MF gap equation
1 = g k 2E1 k 41 .

4
(a)

z

(b)

Szk

Sk

φk

Sxk

kF

SkF
k

y
φk

Sk

x

FIG. 2: Schematic illustration of the pseudospin distribution
Sk described by the BCS wave function |Ψi for a positive real
gap function (a) as a function of k41 and (b) on the Bloch
sphere. (a) Spins rotate in the xz-plane from up to down
towards the positive x-direction as k increases from below toabove kF . (b) In a p-h symmetric system, Sk is the mirror
reflected image of Sk with respect to the xy-plane.

T ,P

|Ψ̄

|Ψ

−∆0

∆0

FIG. 3: Schematic illustration of the double-well potential
for a real gap function and the spontaneous breaking of thesymmetries under T and P. The operation of either T or P
flips the sign of the gap function and transforms |Ψi to |Ψ̄i.

Figures 2(a) and 2(b) show the pseudospin configuration of the superconducting ground state described by|Ψi. The pseudospins smoothly rotate sidewise in the xzplane from up to down towards the positive x-direction
as k increases41. The spontaneous U(1) symmetry breaking with respect to the phase of the gap function sets the
direction of rotating spins projected in the xy-plane. In
a p-h symmetric system, Sk is the mirror reflected imageof Sk with respect to the xy-plane.
The symmetry under C is unbroken in the ground stateof a p-h symmetric system. In fact, using uk = vk andvk = uk , the BCS wave function is shown to be parity
even (C|Ψi = |Ψi) that reflects the invariance of the MF
Hamiltonian (CHMF C = HMF ). As shown in Fig. 2, thepseudospin configuration is indeed invariant under the πrotation of spins about the x-axis followed by the swapping of k and k. In contrast, the symmetries under T
and P are spontaneously broken accompanied with theU(1) symmetry breaking. The operation of either T or
P flips the sign of the gap asX
T HMF T −1 = PHMF P = −H̄k0 · Sk = H̄MF , (16)SymmetryCTPΘ = CPTU(1)p-h symmetric p-h non-symmetric
H
|Ψi
H
|Ψi
X
X
×
×
X
×
×
×
X
×
X
×
X
X
X
X
X
×
X
×

TABLE I: Symmetry of the Hamiltonian H and the groundstate wave function |Ψi for a p-h symmetric system (ξk =−ξk ) and a p-h non-symmetric system (ξk 6= −ξk ). Xand ×mean presence and absence of the symmetry, respectively.

ing of the symmetry under T and P and their operations on |Ψi. Hereafter, the overline represents the re0
placement
Q ∆0 → −∆0 , e.g., H̄k = (−2∆0 , 0, −2ξk ) and|Ψ̄i = k (uk | ↓ik − vk | ↑ik ).
The symmetries of the Hamiltonian and the groundstate are compared between p-h symmetric and nonsymmetric systems in Table 1. It shows that the broken
symmetry of T and unbroken symmetry of C are characteristic to a p-h symmetric system. Given the fact that apure amplitude MG mode arises only in a p-h symmetricsystem as shown later, Table 1 implies that it results fromthe broken T and C, which we reveal in the following.

V.

COLLECTIVE MODES

We first discuss collective modes within the classicalspin analysis41 (Details are given in Appendix B). We
study dynamics of thePpseudospins based on the MF′
Hamiltonian HMF= − k Hk · Sk . Here, the magnetic
field Hk = (2Re∆, −2Im∆, −2ξk ) is self-consistently
deP
termined by the gap function ∆ = g k hc−k↓ ck↑ i =g(hSx i − ihSy i), which is allowed to take complex values. The time evolution of Sk (t), which is treated as aclassical spin, is described by the equation of motion
dSk= Sk × Hk .
dt

(18)

Introducing amplitude and phase fluctuations from theground state ∆ = (∆0 + δ∆)eiδθ , one finds that spin fluctuations in the x-direction induce amplitude fluctuationsδ∆ = gδSx and those in the y-direction induce phasefluctuations δθ = −gδSy /∆0 , where δSk = Sk (t) − Sk0 .
Linearizing Eq. (18) by fluctuations δ∆, δθ ∝ e−iωt , weobtain(1 − 2gχxx (ω)) δ∆ − 2gχxy (ω)∆0 δθ = 0,2gχyx(ω)δ∆ − (1 − 2gχyy (ω)) ∆0 δθ = 0,

(19)
(20)

k

T |Ψi = |Ψ̄i,

P|Ψi =()Y(−1) · |Ψ̄i. (17)kFigure 3 schematically illustrates the spontaneous break-where χµν (ω) are theRdynamical spin susceptibilities de∞fined as χµν (ω) = −i 0 h[Sν , Sµ (t)]ie−iωt dt (Sµ (t) is the
Heisenberg representation and h. . . i denotes the average). For example, χxy represents the coupling of amplitude and phase, while χzx represents that of density

5
and amplitude. The susceptibilities are calculated as
X
Ek
ξk2
,(21)
,χ
=
yy
2
2
2
Ek (4Ek − ω )4Ek − ω 2
k
k
iω X
ξk
χxy = −χyx =.
(22)2
Ek (4Ek2 − ω 2 )
χxx =

X

One can analogously show hn|Sz (t)Sx |ni=
hn|Sz Sx (t)|ni = hn|Sx (t)Sz |ni = 0 and therefore
χzx (ω) = χxz (ω) = 0. χxy (ω) = χyx (ω) = 0 can beshown analogously using the opposite parity of Sx and
Sy . Thus, the unbroken symmetry under C is essential
for the pure amplitude character of the MG mode.

k

Using the MF gap equation, one finds that Eqs. (19)and (20) have the NG mode solution (δθ 6= 0, δ∆ = 0)
with ω = 0. They also have a solution for a pure amplitude mode (δ∆ 6= 0 and δθ = 0) with ω = 2∆0 , if phaseand amplitude are uncoupled χxy (2∆0 ) = χyx (2∆0 ) = 0.
From Eq. (22), this leads to the condition
Z
X 1
N (ξ)
= 0.
(23)
= dξ pEk ξkξξ 2 + ∆20kEquation (23) is satisfied if N (ξ) is even. Thus, MG
mode arises as a pure amplitude mode in a p-h symmetricsystem25 .
The p-h symmetry also ensures fermion number conservation (δSz = 0)45 . δSz is represented asδSz = 2χzx (ω)δ∆ + 2χzy (ω)∆0 δθ,(24)

where χs are given by
χzx =

X
k

X
iω∆0 /2∆0 ξk,χzy =.(25)
2
2
Ek (4Ek − ω )Ek (4Ek2 − ω 2 )k

The MG mode solution (δ∆ 6= 0, δθ = 0, and ω = 2∆0 )satisfies δSz = 0, if χzx (2∆0 ) = 0, which reduces to
Eq. (23). Hence, the MG mode does not induce density
fluctuation P
and indeed conserves total fermion number
N = 2Sz + k 1.
If the p-h symmetry is absent, due to χzx (ω) 6= 0 andχzy (ω) 6= 0, Eq. (24) indicates that δ∆ and δθ must be
finite in order to satisfy δSz = 0. As a result, δ∆ isinevitably coupled with δθ and therefore the MG mode
induces both amplitude and phase fluctuations. The energy of the MG mode becomes greater than 2∆0 26,27 .
VI.

RIGOROUS PROOF OF χxy = χzx = 0

The arguments in the last section are based on the
MF approximation restricted to zero temperature (T =0). We rigorously show that amplitude is decoupledfrom phase and density in a p-h symmetric system at
any temperature.
P We focus on χzx (ω) and evaluateh[Sx , Sz (t)]i ∝ n e−En /T hn|[Sx , Sz (t)]|ni. Here, |ni denotes an exact eigenstate of H with energy En . Since C
is not broken, |ni is parity either even or odd under C.
Using the fact that Sx and Sz have opposite parity underC, we obtainhn|Sx Sz (t)|ni = (hn|C)(CSx C)(CSz (t)C)(C|ni)= −hn|Sx Sz (t)|ni = 0.
(26)

VII.

EMERGENCE OF THE MG MODE BY
THE BROKEN T SYMMETRY

We show that the spontaneous breaking of T is responsible for the emergence of the MG mode. The creation†
operator of the MG mode βH
and that of the NG mode†βNG derived by the Holstein-Primakoff theory are givenby (see Appendix C for details)†βH=AX ξk 
Ek
X 1

k

†
βNG
= A′

k

Ek


Sk′−
Sk′+
(27)
,
+
2|∆0 | − 2Ek
2|∆0 | + 2Ek(Sk′+ + Sk′− ).

(28)

′
′
Here, Sk′± = Sxk± iSyk, which creates and annihilates apair of bogolons, are the raising and lowering operators of′′′
the pseudospins for bogolons Sk′ = (Sxk
, Syk
, Szk
). Sk′±
are transformed as (see Appendix A)

CSk′± C = −Sk′± , PSk′± P = −S̄k′± , T Sk′± T −1 = S̄k′± . (29)
Using Eq. (29), one can show that the MG mode is evenand the NG mode is odd under C:††
CβH
C = βH,††CβNG
C = −βNG.

(30)

Their opposite parity under C is consistent with the uncoupled phase and amplitude. A single MG mode is thus
prohibited to decay into odd number of NG modes by theselection rule. Moreover, since the excited states of energy 2∆0 with a pair of bogolons are odd under C (see Appendix A), a MG mode with energy 2∆0 is stable against
decay into independent bogolons.
The MG and NG modes thus have definite parity under
C due to the unbroken C, while the discrete symmetries
under T and P are broken. From Eq. (29), we obtain
† −1
†
†
T βH
T
= PβH
P = −β̄H
,

(31)

†
†
T βNG
T −1 = β̄NG
,

(32)

†
†
PβNG
P = −β̄NG
,

†
†
†
†
where βH→ β̄Hand βNG→ β̄NGby the replacement∆0 → −∆0 . Note that using Eqs. (30), (31), and (32),
† −1
†
†
†
ΘβH
Θ = βHand ΘβNG
Θ−1 = βNG
are indeed satisfied.
Denoting the vacuum state for βH and βNG (β̄H andβ̄NG ) as |vaci (|vaci), we have the relation T |vaci =P|vaci = |vaci, since either T or P flips the sign of the

6
gap function46 . From Eqs. (30), (31) and (32), one obtains††
C(βH|vaci) = βH|vaci,

(33)

†
†
†
T (βH
|vaci) = P(βH
|vaci) = −β̄H|vaci,††C(βNG |vaci) = −βNG |vaci,††
T (βNG|vaci) = β̄NG|vaci,††P(βNG |vaci) = −β̄NG |vaci.

(34)
(35)
(36)
(37)

†
In the normal phase, setting ∆0 = 0, βH|vaci and†trivially reduce to the same state βH0|FSi ≡†††|φH i, while βNG |vaci and β̄NG |vaci reduce to βNG0 |FSi ≡|φNG i. Here, |FSi denotes the vacuum in the normal††
phase. βH0and βNG0
are given by

†
β̄H
|vaci

†
†
βH0
≡ βH

∆0 =0

†
†
βNG0
≡ βNG

∆0 =0

∝

X 1
k

∝

ξk

X 1
k

ξk

Syk ,

(38)

Sxk .

(39)

†
†
Since βNG0can be transformed to βH0by the π/2 rotation about the z-axis in the pseudospin space, Eqs. (38)and (39) indicate that the |φH i and |φNG i states are degenerate in the normal phase before breaking the U(1)symmetry47,48 . Setting ∆0 = 0 in Eqs. (33), (34), (35),(36), and (37), we obtain49

C|φH i = |φH i, T |φH i = P|φH i = −|φH i, (40)
T |φNG i = |φNG i, C|φNG i = P|φNG i = −|φNG i. (41)The above equations show that |φH i is odd and |φNG i
is even under T . On the other hand, both |φH i and|φNG i are odd under P. From these facts, we can conclude that the lifting of the degeneracy of |φH i and |φNG iin the superconducting phase should be induced by the
spontaneous breaking of T symmetry, not by the breaking of P or U(1) symmetry. Consequently, the breaking of T proves to be responsible for the emergence ofthe pure amplitude MG mode. The spontaneously induced magnetic field that breaks the T symmetry is0given by Hxk= 2∆0 . Therefore, the energy splitting
between the MG and NG modes should be of the order0of |Hxk| = 2∆0 . This is consistent with the fact that theenergy gap of the MG mode is 2∆0 .

non-relativistic Hamiltonian for fermions with a p-h symmetric dispersion exhibits nontrivial discrete symmetries
under C, P, T , and CPT . In the U(1) broken superconducting ground state of such a p-h symmetric system,
T and P are spontaneously broken, while C is unbroken. We have shown that the spontaneous breaking of
the discrete T symmetry leads to the emergence of the
MG mode that induces pure amplitude oscillation of thegap function due to the unbroken C. It may be possible toshow a similar relation between the discrete symmetry of
the Hamiltonian and the emergence of the MG modes inother non-relativistic systems, such as ultracold bosons
in optical lattices20,32 and quantum spin systems15,16 .

Acknowledgments

ST is grateful to C. A. R. Sá de Melo, T. Nikuni, and
N. Tsuji for inspiring discussions. DY thanks the support of CREST, JST No. JPMJCR1673, and of JSPS
Grant-in-Aid for Scientific Research (KAKENHI Grant
No. 18K03525). The work of RY and MN is supportedby the Ministry of Education, Culture, Sports, Science(MEXT)-Supported Program for the Strategic Research
Foundation at Private Universities “Topological Science”(Grant No. S1511006). The work of MN is also supported in part by JSPS Grant-in-Aid for Scientific Research (KAKENHI Grant No. 16H03984 and 18H01217),
and by a Grant-in-Aid for Scientific Research on Innovative Areas “Topological Materials Science” (KAKENHI
Grant No. 15H05855) from the MEXT of Japan.

Appendix A: Pseudospin representation for bogolons

In this Appendix, we introduce a pseudospin representation for bogolons and examine the symmetries of thestates involving excited bogolons. The pseudospins for′′′ 41bogolons Sk′ = (Sxk
, Syk
, Szk
) are defined as
 


′
Szk
− cos ϕk − sin ϕk 0
Szk
′ 
 Sxk
=  sin ϕk − cos ϕk 0   Sxk  . (A1)
′
Syk
0
0
1
Syk


Using Eq. (A1), the MF Hamiltonian is represented as
X
′
HMF =
2Ek Szk
.
(A2)
k

VIII.

CONCLUSIONS

Extending the previous understanding of the emergence of the MG mode in the presence of the p-h symmetric fermionic dispersion, we have revealed the fundamental connection between the emergence of the pureamplitude MG mode and the discrete symmetry of the
Hamiltonian in superconductors, which has not beenclarified in the previous works. We have shown that a′
as | ↑′ ik and | ↓′ ik ,Denoting the eigenstates of Szk
they can be written as| ↑′ ik = uk | ↑ik − vk | ↓ik ,| ↓′ ik = uk | ↓ik + vk | ↑ik ,

(A3)
(A4)

where | ↓′ ik represents the vacuum of bogolons and | ↑′ ik
the excited state of energy 2Ek , in which a pair of bogolons are excited. Since Sk is rotated about the angle

7z

π − ϕk in the xz-plane in Eq. (A1), all the rotated pseudospins Sk′ are aligned downward in the z direction in
the ground state. In fact, the BCS wave function can be
written asY|Ψi =| ↓′ i.
(A5)

y
φk

φk

k

The raising and lowering operators, which creates andannihilates a pair of bogolons, are given by′′± iSykSk′± = Sxk∆0ξk
Sxk ± iSyk +
Szk .
=
Ek
Ek

FIG. 4: Illustration of the unit vectors φ̂k and ŷ.

(A6)Equation (29) can be derived from the above equation.
If we denote the excited state with a single pair ofbogolons asY|ek i = Sk′+ |Ψi = | ↑′ ik(A7)| ↓′ ik′ ,k′ 6=kboth |ek i and |ek i have excitation energy 2Ek and degenerate in a p-h symmetric system. Using Eq. (29), we
obtainC|ek i = −|ek i.

(A8)

From Eq. (A8), it can be easily shown that |ek i − |ek i
is parity even, while |ek i + |ek i is parity odd under C.
The even parity states vanish at k = k = kF because of|ek i = |ek i. It means that the lower edge of the singleparticle continuum with energy 2∆0 consists of parity
odd states.
Appendix B: Classical spin analysis

In this Appendix, we give details of the classical spin
analysis. Linearizing Eq. (18) with respect to fluctuation
δSk = Sk (t) − Sk0 (δH(t) = Hk (t) − Hk0 ), we obtain1dδSkk = − δHy + Hk0 δSyk ,
(B1)
dt
2
1
d
δSyk = δHx cos ϕk − Hk0 δSkk .
(B2)
dt
2
Here, we decompose the spin fluctuation into the two
orthogonal directions as δSk = δSyk ŷ + δSkk ϕ̂k . ŷ is theunit vector in the y-direction and ϕ̂k = cos ϕk x̂−sin ϕk ẑ
is the unit vector illustrated in Fig. 4. We note thatX
δHx = 2gδSx = 2g
δSkk cos ϕk ,(B3)kδHy = 2gδSy = 2g

X

δSyk .

(B4)k

Using Eqs (B2) and (B1), we obtain
X ddδSz = −
δSkk sin ϕk
dt
dt
k
X
X
1
= δHy
sin ϕk − 2∆0δSyk = 0.
2
k

k

x

(B5)

Since δSz = 0 at the initial moment, Eq. (B5) shows thatthe fermion number is conserved (δSz = 0) through the
dynamics.
If the gap function is constant in time, setting δHx =δHy = 0 in Eqs. (B1) and (B2), each pseudospin undergoes precession independently with frequency ω = 2Ek .
It represents a pair of bogolons arising from a broken
Cooper pair.
We consider collective dynamics of pseudospins involving nonzero δHx and/or δHy . Assuming δSk (t), δH(t) ∝e−iωt in Eqs. (B2) and (B1), we obtain
ξk
iω/2
δHx +
δHy , (B6)
4Ek2 − ω 2
4Ek2 − ω 2
Ekiω cos ϕk /2
δHx +δHy . (B7)
δSyk = −4Ek2 − ω 2
4Ek2 − ω 2
δSkk = −Substituting the above equations into Eqs. (B3) and(B4), we obtain the coupled equations for δHx and δHyas(1 − 2gχxx (ω)) δHx + 2gχxy (ω)δHy = 0, (B8)2gχyx (ω)δHx + (1 − 2gχyy (ω)) δHy = 0, (B9)δSz = χzx (ω)δHx − χzy (ω)δHy ,(B10)
where χs are given by Eqs. (21), (22), and (25). Equations (19), (20), and (24) can be readily derived from
Eqs. (B8), (B9), and (B10) by rewriting them in termsof δ∆ and δθ.
If we set ω = 0 in Eqs. (B8) and (B9), since χxy (0) =χyx (0) = 0, δHx and δHy are uncoupled. Using 1 −2gχyy (0) = 0 that reduces to the MF gap equation and
1−2gχxx(0) 6= 0, we obtain the solution for the NG mode:δHx = 0 and δHy 6= 0 (δ∆ = 0 and δθ 6= 0). Equation (B10) indicates that the NG mode solution fulfills
the number conservation δSz = 0, because χzy (0) = 0.
From Eq. (B6) and (B7), one obtains
δSkk = 0,

δSyk =

1
δHy .
4Ek

(B11)

The NG mode thus induces oscillations of pseudospins in
the y-direction as illustrated in Fig. 5 (a). Since δSyk =δSyk , the NG mode induces in-phase oscillation of δSykand δSyk .
In a p-h symmetric system, if we set ω = 2∆0 in
Eqs. (B8) and (B9), since χxy (2∆0 ) = χyx (2∆0 ) = 0,

8
z

(a)

z

(b)

δSyk

δSyk

δSǁk
0
k

0
k

S

S

Sk0

x

δSyk

fluctuation. They satisfy the usual commutation relations [αk , α†k′ ] = δk,k′ and [αk , αk′ ] = [α†k , α†k′ ] = 0.
When fluctuation is small α†k αk ≪ 1, Sk′+ ≃ α†k andSk′− ≃ αk and therefore αk and α†k reduce to the annihilation and creation operators of a pair of bogolons,
respectively.

y

x

y

Sk0

We expand Eq. (C1) in terms of αk and α†k . The zerothand first order terms readδSǁk δSyk

FIG. 5: Illustration of pseudospin oscillation induced by the
NG mode (a) and the MG mode (b) in a system with p-hsymmetric ξk . (a) The NG mode induces in-phase oscillationof δSyk and δSyk . (b) The MG mode induces out-of-phaseoscillation of δSkk and δSkk , as well as δSyk and δSyk .

δHx and δHy are uncoupled. Using 1 − 2gχxx(2∆0 ) = 0
that reduces to the MF gap equation and 1 − 2gχyy (0) 6=
0, we obtain the solution for the MG mode: δHx 6= 0 andδHy = 0 (δ∆ 6= 0 and δθ = 0). Equation (B10) indicatesthat the MG mode solution fulfills the number conservation δSz = 0, because χzx (2∆0 ) = 0 if ξk satisfies Eq. (7).
From Eq. (B6) and (B7), one obtains−1
δSkk =
δHx ,
4ξk

i∆0
δSyk =δHx .
4Ek ξk(B12)

The MG mode thus induces oscillations of pseudospins
both in the y-direction and the direction of φ̂k as illustrated in Fig. 5 (b). Since δSkk = −δSkk and δSyk =−δSyk , the MG mode induces out-of-phase oscillation ofδSkk and δSyk .

H0 = −

X ξ2

k

Ek

k

H1 =

X

−

∆20
,
g

(C4)

(ξk sin ϕk + ∆0 cos ϕk )(αk + α†k ). (C5)k

The first order term vanishes in the ground state H1 =0 using sin ϕk = ∆0 /Ek and cos ϕk = −ξk /Ek . The
second order term reads
H2 = 2X

Ek α†k αk +kgX{(1 − cos ϕk cos ϕk′ )4 ′k,k× (αk αk′ + α†k α†k′ ) − (1 + cos ϕk cos ϕk′ )o× (αk α†k′ + α†k αk′ ) .

(C6)

We diagonalize H2 by a Bogoliubov transformation
βλ =

X

∗
∗ †
(Xλk
αk + Yλk
αk ),

(C7)

(Xλk α†k + Yλk αk ),

(C8)

k

βλ† =

X
k

Appendix C: Holstein-Primakoff theory

In this Appendix, we develop the Holstein-Primakoff
theory for the pseudospin Hamiltonian (2) to derive thecreation and annihilation operators of the MG and NG
modes.
Substituting Eq. (A1) into Eq. (2), one obtains
X′′H=2ξk (− cos ϕk Szk+ sin ϕk Sxk)

where λ labels the excited states. The bosonic operator
βλ satisfies the commutation relations[βλ , βλ† ′ ] =

X

∗
∗
(Xλk
Xλ′ k − Yλk
Yλ′ k ) = δλ,λ′ , (C9)k[βλ† , βλ† ′ ] =X(−Xλk Yλ′ k + Yλk Xλ′ k ) = 0. (C10)

k

k

−g

X

From Eqs. (C7) and (C8), one can easily derive the in′′′
Sxk(cos ϕk cos ϕk′ Sxk′ + sin ϕk cos ϕk′ {Szk , Sxk′ } verse transformation

k,k′

′
′
′
′
Szk
+ sin ϕk sin ϕk′ Szk′ + Syk Syk′ ).

(C1)Spin fluctuation can be quantized by the HolsteinPrimakoff transformation50 :qSk′+ = α†k 1 − α†k αk , Sk′− = (Sk′+ )† ,
(C2)


1
′
(C3)
− α†k αk ,Szk=−
2where α†k and αk denote, respectively, the creation andannihilation operators of a boson that represents spin

αk =

X

∗ †
(Xλk βλ − Yλk
βλ ),

(C11)

∗
(Xλk
βλ† − Yλk βλ ).

(C12)

λ

α†k =

X
λ

Assuming that the second order term is diagonalizedP
as H2 = λ ωλ βλ† βλ + const., we obtain[αk , H2 ] =

X
λ

∗ †
ωλ (Xλk βλ + Yλk
βλ ).

(C13)

9
Setting Xke = Yke = 0, we obtain Xk = Xko and Yk =On the other hand, using Eq. (C6), one obtains(
Yko . The condition b + d = 0, which is obtained fromX
gX
[αk , H2 ] =2Ek Xλk −(cos ϕk cos ϕk′ + 1)Xλk′ Eqs. (C19) and (C20), reduces to
2 ′
X
λ
k
ωξk
!
= 0.
(C27)
X
g
Ek (4Ek2 − ω 2 )k(cos ϕk cos ϕk′ − 1)Yλk′ βλ+2 ′kSince Eq. (C27) is equivalent to χxy (ω) = 0, it ensuresgX
the uncoupled phase and amplitude fluctuations.
∗∗+ −2Ek Yλk −(cos ϕk cos ϕk′ − 1)Xλk′
Substituting Eq. (C26) into Eq. (C25), we obtain
2 ′
k
! )
X ξ2
Ek
gX
k
∗
1 − 2g
= 0.
(C28)
+
βλ† .
(C14)
(cos ϕk cos ϕk′ + 1)Yλk′
Ek2 4Ek2 − ω 2
2 ′
k

k

Comparing Eqs. (C13) and (C14), one obtains sets ofequations for Xλk and Yλk asg
2Ek Xλk − {(aλ − cλ ) cos ϕk + (bλ + dλ )}2= ωλ Xλk ,(C15)
g−2Ek Yλk − {(aλ − cλ ) cos ϕk − (bλ + dλ )}2= ωλ Yλk ,(C16)where the coefficients aλ , bλ , cλ , and dλ are given byXXaλ =cos ϕk Xλk , bλ =
Xλk ,
(C17)
cλ =

k

k

X

X

cos ϕk Yλk ,

dλ =

k

Yλk .

(C18)

k

Equations (C15) and (C16) can be formally solved asg (aλ − cλ ) cos ϕk + (bλ + dλ ),22Ek − ωλg (aλ − cλ ) cos ϕk − (bλ + dλ ).
Yλk = −
2
2Ek + ωλXλk =The above equation is equivalent to 1−2gχxx(ω) = 0 andtherefore has the MG mode solution ω = 2|∆0 | for which
it reduces to the MF gap equation. We thus obtain
Xko = −

Yke = (Yk + Yk )/2,
Yko = (Yk − Yk )/2,

A = qPg (a − c) cos ϕk,
2 2Ek − ω

(C21)
(C22)

k

Yko =

1

|∆0 |
2
k Ek ξk.

(C30)

The creation operator of the MG mode is thus obtainedasX ξk 
1
1††.
βH = Aα +αk(C31)
Ek 2|∆0 | − 2Ek k 2|∆0 | + 2Ek
The equations for even components readg
2Ek Xke − (b + d) = ωXke ,2
g−2Ek Yke + (b + d) = ωYke ,
2
X
X
e
b=
Xk , d =
Yke .

(C20)

If a − c 6= 0, the formal solutions of Eqs. (C23) and (C24)
are given by
Xko =A cos ϕk(,C29)
2|∆0 | + 2Ek

k

(C19)

where the former two are even as Xke = Xke and Yke = Yke ,while the latter two are odd as Xko = −Xko and Yko =−Yko .
The equations for odd components readg
2Ek Xko − (a − c) cos ϕk = ωXko ,
(C23)
2
g
−2Ek Yko − (a − c) cos ϕk = ωYko ,
(C24)
2
X
X
a=
cos ϕk Xko , c =cos ϕk Yko . (C25)k

Yko = −

where A is the normalization constant. A is determinedby the normalization condition (C9) as

We omit λ below.
If the p-h symmetric condition (7) is satisfied,
Eqs. (C15) and (C16) can be decoupled by introducing
the even and odd components as
Xke = (Xk + Xk )/2,
Xko = (Xk − Xk )/2,

A cos ϕk,2|∆0 | − 2Ek

g (c − a) cos ϕk.(C26)
2 2Ek + ω

k

(C32)
(C33)
(C34)

k

If b + d 6= 0, the formal solutions of Eqs. (C32) and (C33)
are given by
Xke =

g b+d
,
2 2Ek − ω

Yke =

g b+d
.
2 2Ek + ω

(C35)

Setting Xko = Yko = 0, we obtain Xk = Xke and Yk =Yke . The condition a − c = 0, which is obtained from
Eqs. (C19) and (C20), reduces to Eq. (C27).
Substituting Eq. (C35) into Eq. (C34), we obtain1 − 2g

X
k

Ek
= 0.
4Ek2 − ω 2(C36)

The above equation is equivalent to 1−2gχyy (ω) = 0 andtherefore has the NG mode solution ω = 0, for which it
reduces to the MF gap equation. We thus obtain
Xke = Yke = A′ /Ek ,

(C37)

10
where A′ is the normalization constant.
However,
Eq. (C37) does not fulfill the normalization condition(C9). This anomaly is typical for zero energy modes.
It can be avoided by introducing a small fictitious external field in the Hamiltonian (2)51 . The creation operatorof the NG mode is thus obtained as
X 1
†
βNG
= A′
(α† + αk ).
(C38)
Ek k

In the limit of small fluctuation α†k αk ≪ 1, using α†k ≃′+Sk and αk ≃ Sk′− , the creation operators for the Higgs
mode and the NG mode can be obtained as Eqs. (27) and
(28).

k

∗

Electronic address: tsuchiya@phys.chuo-u.ac.jp
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We remark that the full Hamiltonian H̃ = k,σ ξk c†kσ ckσ −
18

11
P
g k,k′ ,q c†k+q↑ c†−k+q↓ c−k′ +q↓ ck′ +q↑ , is invariant by all thediscrete transformations C, P, T , and Θ. The extension of
the argument in this paper to more general cases for H̃ isstraightforward.
41
P. W. Anderson, Phys. Rev. 112, 1900 (1958).
42
Y. Nambu, Phys. Rev. 117, 648 (1960).
43
A similar transformation Ψk → iτy Ψ−k was introduced in
the study of superfluid 3 He in J. W. Serene, in Quantum
Fluids and Solids-1983 (AIP, New York, 1983), Vol. 103,
p. 305. (See also R. S. Fishman and J. A. Sauls, Phys. Rev.
B 31, 251 (1985).) The ground state is, however, a priori
assumed to be invariant under the transformation in thepaper.
44
M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991).
45
δSz = 0 is derived in Eq. (B5) in Appendix B.
46
For the BCS wave function |Ψi, P|Ψi = |Ψ̄i holds ifQ
k (−1) = 1.
47
|φH i and |φNG i form the bound states inside the bandgap, which are referred to as “Cooperons”, if the fermionicdispersion has a band gap or Dirac cones. Evolution of
Cooperons into the MG and NG modes in a Dirac fermion
system across the quantum critical point was studied inRef. 26.
48
1 − 2gχxx (ω) = 0 and 1 − 2gχyy (ω) = 0 in the normal phase (∆0 = 0) give the poles that correspond to
the excitation energy of |φH i and |φNG i, respectively.
χxx (ω) = χyy (ω) in the normal phase due to the U(1)symmetry therefore indicates the degeneracy of |φH i and|φNG i. In fact, within
P the MF theory, Eq. (21) leads toχxx (ω) = χyy (ω) = k |ξk |/(4ξk2 − ω 2 ).
49
Note that [C, T ] = [C, P] = [T , P]Q= 0 and simultaneous
eigenstates of C, T , and P exist if k (−1) = 1.
50
T. Holstein and H. Primakoff, Phys. Rev. 58, 1098 (1940).
51
S. Tsuchiya and D. Yamamoto, unpublished.