Quantum-geometry-induced anapole superconductivity Taisei Kitamura1Shota Kanasugi1Michiya Chazono1and Youichi Yanase1 1Department of Physics Graduate School of Science Kyoto University Kyoto 606-8502 Japan

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Quantum-geometry-induced anapole superconductivity

Taisei Kitamura,1, ∗Shota Kanasugi,1Michiya Chazono,1and Youichi Yanase1

1Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan

(Dated: March 24, 2023)

Anapole superconductivity recently proposed for multiband superconductors [Commun. Phys. 5, 39 (2022)]

is a key feature of time-reversal (T)-symmetry-broken polar superconductors. The anapole moment was shown

to arise from the asymmetric Bogoliubov spectrum, which induces a ﬁnite center of mass momenta of Cooper

pairs at the zero magnetic ﬁeld. In this paper, we show an alternative mechanism of anapole superconductiv-

ity: the quantum geometry induces the anapole moment when the interband pairing and Berry connection are

ﬁnite. Thus, the anapole superconductivity is a ubiquitous feature of T-broken multiband polar superconduc-

tors. Applying the theory to a minimal model of UTe2, we demonstrate the quantum-geometry-induced anapole

superconductivity. Furthermore, we show the Bogoliubov Fermi surfaces (BFS) in an anapole superconducting

state and predict an unusual temperature dependence of BFS due to the quantum geometry. Experimental veriﬁ-

cation of these phenomena may clarify the superconducting state in UTe2and reveal the ubiquitous importance

of quantum geometry in exotic superconductors.

I. INTRODUCTION

Parity-mixed superconductors, in which even- and odd-

parity pairings coexist, are attracting much attention, as the

parity-mixing phenomena are closely related to the space in-

version (P)-symmetry breaking. Stimulated by the discovery

of noncentrosymmetric superconductivity in heavy fermions

and artiﬁcial heterostructures, time-reversal (T)-symmetric

parity-mixed pairing states such as the s+p-wave state have

been investigated intensively1,2. For a long time, studies fo-

cused on the crystals lacking the P-symmetry allowing an an-

tisymmetric spin-orbit coupling (ASOC). Consequently, the

Rashba superconductor and the Ising superconductor have be-

come fundamental concepts in condensed matter physics1,2.

On the other hand, centrosymmetric crystals were recently

shown to be an intriguing platform of spontaneously P-

symmetry breaking superconductivity3–5. In the absence of

the ASOC, additional T-symmetry breaking is expected3–5

as the ±π/2 phase difference between even- and odd-parity

pairing potentials, such as the s+ip-wave pairing state, is

energetically favored. As a result, both of the P- and T-

symmetry are broken while the combined PT -symmetry is

preserved. The three-dimensional s+ip-wave pairing state

in single-band superconductors was theoretically studied as a

superconducting analog6–8of axion insulators9,10. Such a par-

ing state in Sr2RuO4was theoretically proposed11. Further-

more, recently discovered candidate for spin-triplet supercon-

ductor UTe212,13 is predicted to realize the s+ip-wave pairing

state14, as it is consistent with the experimentally observed

multiple superconducting phases15–21 and multiple magnetic

ﬂuctuations22–27.

Clariﬁcation of the PT -symmetric parity-mixed supercon-

ductivity has been awaited to uncover an exotic state of matter.

However, properties of the PT -symmetric parity-mixed su-

perconductivity are almost unresolved. In particular, theoret-

ical studies of multiband superconductors have not been car-

ried out except for Ref. 28, although it is known that intriguing

superconducting phenomena such as the intrinsic polar Kerr

effect29–31 and Bogoliubov Fermi surfaces (BFS)32,33 may ap-

pear from multiband properties. In Ref. 28, the anapole su-

perconductivity was discussed as an exotic feature of the PT -

symmetric parity-mixed pairing state in multiband supercon-

ductors. If some conditions are satisﬁed, an asymmetric Bo-

goliubov spectrum (BS) arises from the interband pairing28.

When the symmetry of superconductivity has a polar property,

such as in the Ag+iB3u pairing state proposed for UTe214, the

asymmetric BS induces an effective anapole moment, which

is deﬁned as the ﬁrst-order coefﬁcient of the free energy in

terms of the center of mass momenta of Cooper pairs. The

anapole moment characterizes the anapole superconductivity

as it does the anapole order in magnetic materials34–38 and nu-

cleus39.

The anapole moment is a polar and T-odd vector34, which

shares the symmetry as the velocity and momentum. There-

fore, it is not surprising that the effective anapole moment

induces a ﬁnite center of mass momenta of Cooper pairs

qeven in the absence of the magnetic ﬁeld. The mecha-

nism of ﬁnite-qpairing is different from the Fulde-Ferrell-

Larkin-Ovchinnikov (FFLO) superconductivity40,41 and heli-

cal superconductivity1,2, which require a ﬁnite magnetic ﬁeld.

In contrast to the FFLO and helical superconductivity, the

anapole superconductivity can be studied with avoiding ex-

perimental difﬁculties due to vortices induced by an external

magnetic ﬁeld. For instance, the anapole domain switching28,

superconducting piezoelectric effect42,43, and Josephson ef-

fect44,45 may uncover intrinsic properties of anapole supercon-

ductivity. Therefore, the anapole superconductivity may be

the key to elucidating the PT -symmetric parity-mixed pair-

ing state, and it may realize and clarify the ﬁnite-qpairing

state which has been searched for a long time1,2,46.

In this paper, we show that the anapole superconductiv-

ity is a ubiquitous feature more than revealed in the previ-

ous paper28, considering the quantum geometry extensively

studied in various ﬁelds37,38,47–58. Recently, an essential role

of the quantum geometry in the superﬂuid weight, namely,

the second-order derivative of the free energy, has been re-

vealed59–62. Thus, it is naturally expected that the quantum

geometry may be essential for the anapole superconductivity.

First, we provide a thorough formulation of the anapole mo-

ment based on the Bardeen-Cooper-Schrieffer (BCS) mean-

ﬁeld theory. The obtained formula contains two terms; one is

arXiv:2210.01399v2 [cond-mat.supr-con] 23 Mar 2023

the geometric term and the other is the group velocity term.

Only a part of the group velocity term was derived in the pre-

vious literature28. Based on the general two-band model with

Kramers degeneracy, the microscopic origin of the geometric

term is revealed to be the interband pairing and the Berry con-

nection, while the group velocity term is induced by the asym-

metric BS. Then, applying the theory to a model of UTe2, we

demonstrate the quantum-geometry-induced anapole super-

conductivity. Moreover, we show unique features of anapole

superconductivity. When the system has a small gap mini-

mum as expected for UTe213, the anapole moment induces the

BFS. The BFS may show a reappearing behavior as decreas-

ing the temperature, causing anomalies in density of states

(DOS) and thermodynamic quantities.

II. GENERAL FORMULA FOR ANAPOLE MOMENT

An order parameter of the anapole superconductivity is the

anapole moment which is deﬁned by the ﬁrst-order coefﬁ-

cient of the free energy with respect to q. In the previous

study28, the anapole moment is derived only when the k-

derivative of normal-state Hamiltonian is proportional to the

identity matrix, namely, ∂µHk∝1. We adopt the notation

∂µ=∂kµ, and Hkis the matrix representation of the single-

particle Hamiltonian. Below, we formulate the anapole mo-

ment in the general case based on the BCS mean-ﬁeld theory.

The normal state is assumed to be P- and T-symmetric, and

therefore, Hk=UTHT

−kU†

Tis satisﬁed, where UT=iσy⊗1

is the unitary part of the Toperator with the Pauli matrix for

the spin space σµ(µ= 0, x, y, z). Thus, the Bogoliubov-de

Gennes (BdG) Hamiltonian for a ﬁnite-qpairing state can be

written as (see Appendix A)

HBdG =1

Ψ†

k,qHBdG

k,qˆ

Ψk,q,(1)

HBdG

k,q=Hk+q∆k

∆†

k−Hk−q,(2)

Ψ†

k,q=ˆ

c†

k+qˆ

−k+qU†

T.(3)

Here, we denote ˆ

c†

k= ( ˆc†

↑1k· · · ˆc†

↑fkˆc†

↓1k· · · ˆc†

↓fk),

where ˆc†

σlkis the creation operator for spin σand the other

internal degree of freedom l. We consider general cases, in-

cluding multi-orbital and multi-sublattice systems, and fis

the total number of degrees of freedom other than spin.

The off-diagonal part ∆k=∆g

k+∆u

kis the gap func-

tion in the matrix representation, where ∆g(u)

kis the P-even

(odd) component of the pair potential. Coexistence of Cooper

pairs with different parities, i.e. parity-mixed state, leads to

broken P-symmetry. Furthermore, the T-symmetry break-

ing is theoretically predicted3–5, when the normal state pre-

serves the P-symmetry, Thus, we assume the ±π/2phase dif-

ference between ∆g

kand ∆u

k, consistent with the theoretical

prediction3–5. As a result the P- and T-symmetry are broken

by the parity-mixed gap function while the PT -symmetry is

preserved. In addition, to make the anapole moment ﬁnite,

throughout the paper, we assume the gap function ∆kbelongs

to polar irreducible representation.

Expanding the free energy by qas Fq=T·q+· · · , we

obtain the anapole moment as,

Tµ=1

f(Eak)hψak|∂µH+

k|ψaki,(4)

H+

k=Hk0

0Hk.(5)

Here, we use the eigenvalue equation HBdG

k|ψaki=

Eak|ψakiwith HBdG

k≡HBdG

k,0and the Fermi-distribution

function f(E). The derivation of Eq. (4) is shown in Ap-

pendix A. When the anapole moment in superconductors Tµ

is ﬁnite, a superconducting state due to Cooper pairs with ﬁ-

nite center of mass momenta becomes most stable.

To obtain further insights, using the Bloch wave function

which follows Hk|unχki=nk|unχki, we expand |ψa(k)i

as |ψaki=Pn,χ φa+

nχk|unχkiPn,χ φa−

nχk|unχkiT.

Because of Kramers degeneracy, we distinguish two degen-

erate bands by the helicity χ=↑↓. Here, φa±

nχkis the matrix

element of the unitary matrix which diagonalizes the band rep-

resentation of the BdG Hamiltonian. After calculations, the

anapole moment Eq. (4) is divided into two parts,

Tµ=Tvelo

µ+Tgeom

µ,(6)

where

Tvelo

µ=X

n,χ

Cnχnχk∂µnk,(7)

Tgeom

µ=X

n6=m,χχ0

Cnχmχ0k

×(mk−nk)hunχk|∂µumχ0ki,(8)

Cnχmχ0k=1

f(Eak)φa+∗

nχkφa+

mχ0k+φa−∗

nχkφa−

mχ0k.

(9)

Tvelo

µin Eq. (7) is called the group velocity term as it con-

tains the group velocity ∂µnk. In the next section, using

the general two-band model, we show that this term arises

from the asymmetric BS. Equation (8) for Tgeom

µis named

the geometric term because it contains the Berry connection

hunχk|∂µumχ0ki. Through the Berry connection in the geo-

metric term, the geometric properties of Bloch electrons may

contribute to the anapole moment. Some conditions have to

be satisﬁed for a ﬁnite group velocity term, which vanishes in

simple models28. On the the other hand, the geometic term

has been overlooked in the previous study. Owing to the ge-

ometric term, the anapole superconductivity becomes recog-

nized as a ubiquitous feature of the PT -symmetric mixed-

parity pairing state in multiband superconductors.

III. ORIGIN OF ANAPOLE SUPERCONDUCTIVITY

A. General discussion

Before demonstrating the anapole superconductivity due to

quantum geometry, we discuss the physical origin and the mi-

croscopic process of the anapole moment using the Ginzburg-

Landau (GL) theory. We also discuss their relation to the

group velocity and geometric terms. Up to the second-order

of the gap function ∆k, the anapole moment is given by,

TGL

µ=1

βX

kωn

tr hGp

kωn∂µHkGp

kωn

∆g

kGh

kωn

∆u†

− Gp

kωn∂µHkGp

kωn

∆u†

kGh

kωn

∆g

ki+ (g ↔u),(10)

where tr represents the trace over normal state degrees of free-

dom. Here, Gp(h)

kωn= (iωn∓Hk)−1is the Green function for

the particle (hole) part. The derivation of the formula (10) is

shown in Appendix B. From this formula, we see that P- and

T-symmetry breaking is needed for the anapole superconduc-

tivity (see also Appendix B).

We can rewrite the formula in the Bloch band basis,

TGL

µ=1

βX

kωnX

nmp X

χnχmχp

CGL

nmpkωntr [Pnχnk∂µHkPmχmk

×∆g

kPpχpk∆u†

k−∆u†

kPpχpk∆g

ki+ (g ↔u),

(11)

where CGL

nmpkωn= (iωn−nk)−1(iωn−mk)−1(iωn+

pk)−1, and Pnχnk=|unχnki hunχnk|is the projection op-

erator. For n=m=p, the summand of Eq. (11) vanishes

(see Appendix Cfor details). Therefore, at least two pairs of

n,mand pmust be nonequivalent for a ﬁnite contribution to

the anapole moment. In other words, two interband processes

are necessary for the anapole superconductivity.

The above necessary condition for n,mand pcan be sat-

isﬁed in three cases. The ﬁrst case, n=m6=p, corresponds

to the group velocity term, and the odd- and even-parity in-

terband pairings play the role of two interband processes. In

the following subsection, it is shown that the group velocity

term is closely related to the asymmetric BS. The effect of

asymmetric BS on the group velocity term is also discussed in

Appendix D.

The remaining two cases correspond to the geometric term

since the Berry connection is necessary. In the second case,

n6=mand n=p(or m=p), the Berry connection of

Bloch electrons and either odd-parity or even-parity interband

pairing play the role of two interband processes. Finally, in

the third case, n6=m6=p6=n, all of the even-parity in-

terband pairing, odd-parity interband pairing, and the Berry

connection appear in the contribution to the anapole moment.

In both cases, via the Berry connection, the Bloch electrons

undergo an interband transition from the initial band to the

different band, which is coupled to the initial band through

the interband Cooper pairs. Thus, the two or more interband

processes, due to the Berry connection and interband Cooper

pairs, induce the anapole moment. which is a physical picture

of quantum-geometry-induced anapole superconductivity.

B. General two-band model

Next, for a more transparent understanding, we derive the

anapole moment in general two-band superconductors with

Kramers degeneracy. Although we here adopt a two-band

model, the following results can be applied to any multi-

band model with multiple bands near the Fermi surface. The

normal-state Hamiltonian is written as,

Hk=h0k1+hk·γ,(12)

by using the 4×4gamma matrices γ= ( γ1· · · γ5)

that anti-commute each other. Here, h0kand hk=

(h1k· · · h5k)depend on the details of the model. The en-

ergy dispersion is given by ±,k=h0k± |hk|. Note that

the 4 by 4 normal-state Hamiltonian has two bands due to the

Kramers degeneracy.

The PT -symmetric parity-mixed pair potential is ex-

pressed as28,

∆k=∆g

k+∆u

k,(13)

∆g

k=η0k1+ηk·γ,∆u

k=i

2Pij ˜ηijkγiγj,(14)

where η0k,ηk= ( η1k· · · η5k)and ˜ηijk=−˜ηjikare

the complex valued order parameters for even- and odd-parity

pairing channels. Here, taking appropriate U(1) gauge, η0k

and ηikare real while ˜ηijkbecomes pure imaginary.

Because of the Kramers degeneracy, the particle Green

function can be projected to the two degenerate bands as63

kωn=akωn1+bkωn˜

Hk,(15)

akωn=1

(iωn−h0k± |hk|)−1,(16)

bkωn=1

∓(iωn−h0k± |hk|)−1,(17)

with ˜

Hk= (hk·γ)/|hk|=ˆ

hk·γ. The hole Green function

is also given by,

kωn=ckωn1+dkωn˜

Hk,(18)

where akωn=−ck−ωnand bkωn=−dk−ωn. Hereafter, we

omit the (k, ωn)dependence for simplicity. Inserting these

expressions of Green functions into Eq. (10), we get

TGL

µ=1

βX

ωna2ctr h∂HM(1)

−i+a2dtr h∂HM(2)

−i

+abctr hn∂H, ˜

HoM(1)

−i+abdtr hn∂H, ˜

HoM(2)

−i

+b2ctr h˜

H∂H ˜

HM(1)

−i+b2dtr h˜

H∂H ˜

HM(2)

−i.(19)

Here, we introduce the P- and T-odd bilinear prod-

ucts28,33,63,64,

M(1)

−=∆g,∆u†+ (g ↔u),(20)

M(2)

−=h∆g˜

H∆u†−∆u†˜

H∆gi+ (g ↔u).(21)

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Quantum-geometry-inducedanapolesuperconductivityTaiseiKitamura,1,ShotaKanasugi,1MichiyaChazono,1andYouichiYanase11DepartmentofPhysics,GraduateSchoolofScience,KyotoUniversity,Kyoto606-8502,Japan(Dated:March24,2023)Anapolesuperconductivityrecentlyproposedformultibandsuperconductors[Commun.Phys.5,39(2...

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Quantum-geometry-induced anapole superconductivity Taisei Kitamura1Shota Kanasugi1Michiya Chazono1and Youichi Yanase1 1Department of Physics Graduate School of Science Kyoto University Kyoto 606-8502 Japan

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