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Paramagnetic instability of small topological superconductors

Shu-Ichiro Suzuki1 and Yasuhiro Asano1,21Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan and

2Center for Topological Science & Technology, Hokkaido University, Sapporo 060-8628, Japan

(Dated: May 17, 2018)

The diamagnetism is an essential property of all superconductors. However, we will show thatsmall topological (or unconventional) superconductors can be intrinsically paramagnetic by solv-ing the quasiclassical Eilenberger equation and the Maxwell equation self-consistently on two-dimensional superconducting disks in weak magnetic fields. Because of the topologically nontrivialcharacter of the wave function, the unconventional superconductors host the zero-energy surfaceAndreev bound states, which always accompany so-called odd-frequency Cooper pairs. The param-agnetic property of the odd-frequency pairs explains the paramagnetic response of the disks at lowtemperature.

PACS numbers: 73.20.At, 73.20.Hb

I. INTRODUCTION

The Meissner effect is a fundamental property of su-perconductors as shown in standard textbooks1. Theresponse of superconductors is usually diamagnetic be-cause a superconductor excludes weak enough magneticfields from its interior. The anomalous paramagneticresponse, however, has been observed in small disks ofmetallic superconductor2,3, small high-Tc compounds4–6,and mesoscopic proximity structures7,8. The spatial in-homogeneity of the magnetic property is a key featureto realize the paramagnetic phase. In metallic super-conductors, the inhomogeneous distribution of magneticfields9 and the formation of giant vortex are responsiblefor the paramagnetic Meissner effect PME10. The pres-ence of the π junctions is also pointed out as an originof PME in a network of Josephson junction11. In uncon-ventional superconductors (USs), on the other hand, anexperiment6 has shown the decrease of the pair densitywith decreasing temperature, which suggests a peculiarmechanism of the PME unique to the USs. As a resultof the topological nature in the wave function, the USshave the topologically protected surface Andreev boundstates (ABSs) at the zero-energy12–16. So far theoreti-cal studies have shown that the magnetic response at the(110) surface of high-Tc superconductor is nonlinear17,18

and paramagnetic19–22 due to the ABSs. The paramag-netic response has been mainly explained in terms of theenergetics of the ABSs. Weak magnetic fields shift theenergy of the surface ABSs away from the Fermi leveland decrease the total energy of superconductor, whichleads to the paramagnetic response or the paramagneticinstability. However, there is an important open ques-tion: what carries the large paramagnetic supercurrent?By addressing this issue, we will conclude that the mag-netic properties of USs are intrinsically inhomogeneousand that small USs can be paramagnetic at low temper-ature.

The electric current in equilibrium has two contribu-tions, (i.e., j = jpq + jA). The quasiparticle current jpqdue to the spatial phase gradient of the wave function is

paramagnetic, whereas jA = −ne2A/mc is diamagnetic.In a normal metal, jpq cancels jA because the phase of

an electron is not rigid at all23. In a superconductor, onthe other hand, the phase rigidity of superconductivitydrastically suppress the spatial gradient of phase, whichleads to jqp = 0. As a result, a superconductor shows theperfect diamagnetism. In contrast to excited quasiparti-cles above the superconducting gap, the quasiparticlesbelow the gap have the phase rigidity because they arethe shadow of Cooper pairs. In fact, a normal metal at-taching to a metallic superconductor shows the diamag-netic Meissner effect24. This phenomenon is explained bytwo different but equivalent pictures: the penetration of aCooper pair into the normal metal (proximity effect) andthe Andreev reflection of a quasiparticle below the gap.The appearance of the surface ABSs is a direct result ofthe coherent Andreev reflections of a quasiparticle at theFermi level25. Therefore such phase-rigid quasiparticlesat the ABS cannot carry the large paramagnetic current.

In this paper, we theoretically study the spatial distri-bution of magnetic fields and that of electric currentson small two-dimensional superconducting disks withunconventional pairing symmetry such as spin-singletd wave and spin-triplet p wave. There are several dwave superconductors in organic compounds and heavyfermionic materials in addition to high-Tc cuprates. Re-cently, the effective Hamiltonian for superconductingstates in nanowires26,27 has shown to be unitary equiv-alent to that for px wave superconducting states28. Thesimulation at least two-dimensional system is necessary

FIG. 1. (a) The schematic figure of a superconducting disk.The pair potentials in momentum space are illustrated for thed wave symmetry in (b) and for the p wave symmetry in (c).

2

to evaluate the magnetic susceptibility quantitatively be-cause the d and p wave pair potentials are anisotropic inreal space. We solve the Eilenberger equation for thequasiclassical Green function and the Maxwell equationfor magnetic fields self-consistently. The self-consistencyof pair potential and magnetic field is necessary toregularize the nonlinear property in the magnetic re-sponse17,18. The solution of the Green function nearthe disk edge shows the presence of the odd-frequencyCooper pairs. The odd-frequency pairs have paramag-netic property29–31 because of their negative pair den-

sity. The calculated results of the magnetic susceptibil-ity suggest the PME in small USs. We conclude thatthe odd-frequency Cooper pairs carry the large param-agnetic current and causes the paramagnetic response ofsmall superconducting disks.

II. FORMULATION

Let us consider a superconducting disk in two-dimension as shown in Fig. 1, where R is the radius of thedisk. We assume that the disk is in the clean limit andits surface is specular enough. To analyze the supercon-ducting states in equilibrium, we solve the Eilenbergerequation32,

i~vF k ·∇r g +[H, g

]= 0, (1)

H(r,k, iωn) =

[ξ(r,k, iωn) ∆(r,k)

∆˜(r,k) ξ

˜(r,k, iωn)

], (2)

g(r,k, iωn) =

[g(r,k, iωn) f(r,k, iωn)

−f˜(r,k, iωn) −g

˜(r,k, iωn)

], (3)

ξ(r,k, iωn) = iωn + (evF /c)k ·A(r), (4)

where k is the unit vector on the Fermi surface, vF isthe Fermi velocity, ωn = (2n + 1)πT is the Matsubarafrequency, n is an integer number, and T is a tempera-ture. In this paper, the symbol ˆ· · · represents 2×2 matrixstructure in spin space and σj for j = 1-3 are the Paulimatrices. The vector potential is denoted by A and themagnetic field H = ∇×A is in the z direction. We in-troduced a definition X

˜(r,k, iωn) ≡ X∗(r,−k, iωn) for

all functions X . The electric current is given by

j(r) =πevFN0

2iT∑

ωn

∫dk

2πTr

[T3 k g(r,k, ωn)

], (5)

with T3 = diag[σ0,−σ0], where σ0 is the identity matrixin spin space and N0 is the density of state per spin atthe Fermi level. We mainly consider the two equal-timepairing order parameters in two dimension: spin-singletd-wave symmetry ∆(r, θ) = ∆(r) cos(2θ)iσ2 and spin-

triplet p-wave symmetry ∆(r, θ) = ∆(r) cos(θ)σ1, whereθ is a directional angle with kx = cos θ and ky = sin θ.The pair potentials are determined self-consistently from

FIG. 2. (a) The local susceptibility and (b) the current den-sity of the d -wave superconductor, where R = 3 ξ0, λL = 3 ξ0,ωc = 10∆0, and Hext = 0.001Hc1. (c) The d-wave and (d)the p -wave components of the anomalous Green function.

the gap equation

∆(r)iσν σ2 = πN0gT∑

ωn

∫ 2π

0

dθ

2πf(r, θ, iωn)Vx(θ), (6)

where x = s, p and d indicate the pairing symmetry,ν = 0 and 3 for the spin-singlet and the spin-triplet or-der parameters, respectively. The coupling constant gsatisfies {N0g}

−1 = ln(T/Tc)+∑

0≤n<ωc/2πT(n+1/2)−1

with Tc and ωc being the transition temperature and thecut-off energy, respectively. The attractive potentials de-pends on the pairing symmetry Vx(θ) = sxφx(θ) withss = 1 and φs(θ) = 1 for s wave symmetry, sp = 2and φp(θ) = cos θ for p wave symmetry, and sd = 2 andφd(θ) = cos(2θ) for d wave symmetry. The local mag-netic susceptibility is defined by

χm(r) =(H(r)−Hext

)/(4πHext), (7)

where Hext is the uniform external magnetic field in thez direction. The susceptibility of the whole disk is cal-culated to be χ =

∫drχm(r)/(πR2). In the absence of

spin-dependent potential, the spin structure of ∆ and

that of f are always the same with each other. We usethe standard Riccati parametrization33–35 to solve theEilenberger equation Eq. (1). To obtain numerical solu-tions of the Riccati type differential equation in closeddisks, we apply a method discussed in Ref. 36. An initialvalue at a certain place in the closed system is necessaryto solve the Riccati equation. The obtained solution usu-ally depends on the initial condition. However, when wesolve the equation along the long enough classical trajec-tory, the effects of the initial condition is eliminated. In

3

FIG. 3. (a) The local susceptibility and (b) the current den-sity of the p -wave superconductor, where R = 3 ξ0, λL = 3 ξ0.(c) The p -wave and (d) the s -wave component of the anoma-lous Green function.

numerical simulation, we increase the length of the tra-jectory until solutions do not depend on the initial condi-tions. The vector potential A is obtained by solving theMaxwell equation ∇ × H = (4π/c)j with Eq. (5). Wecalculate self-consistent solutions of the vector potentialand pair potential by solving the Maxwell equation andthe Eilenberger equation simultaneously. The anomalous

Green function f(r, θ, iωn) is originally defined by thetwo annihilation operators of two electrons consisting of

a Cooper pair. Therefore f(r, θ, iωn) must be antisym-metric under the interchange of the two electrons, whichstems from the Fermi-Dirac statistics of electrons. Suchfundamental relation is represented by

f(r, θ, iωn) = −[f(r, θ + π,−iωn)]T, (8)

where T represents the transpose of matrices.

III. RESULTS

The external magnetic field and the cut-off energy arefixed at Hext = 0.001Hc1 and ωc = 10∆0, respectively.Here Hc1 = ~c/|e|ξ20 is the first critical magnetic field.The length is measured in units of ξ0 = ~vF /∆0 with∆0 being the amplitude of the pair potential at T = 0.The current density is normalized to J0 = ~c/|e|ξ30 . Thecharacteristic length scale of the Maxwell equation isλL = (4πne2/mc2)−1/2 and is a parameter in the numer-ical simulation. Throughout this paper, we use a unit ofkB = 1.In Fig. 2, we first show the calculated results of the lo-

cal susceptibility (a) and the current density (b) for the

d -wave superconducting disk, where we fix the parame-ters as R = 3 ξ0, λL = 3 ξ0, and T = 0.3Tc. We set +xand +y axes to be identical to (100) and (010) directionsof the high-Tc crystal. The central region of the disk isdiamagnetic as usual, whereas the surfaces in the (110)and (110) directions are paramagnetic as shown in (a).The current density has the complex structure near thesurface as shown in (b), where the arrow indicates the di-rection of current and its length represents the amplitudeof current. Here we present the picture only for x > 0and y > 0 in (b) because the results are fourfold symmet-ric due to the d -wave character of order parameter. Thediamagnetic current flows at the edges in the (100) and(010) directions, whereas the paramagnetic current flowsat the edges in the (110) direction. The vortex-like cur-rent profile can be seen near the surfaces because the twocurrents flow the opposite directions to each other. Atthe central region, on the other hand, only the diamag-netic current flows. Such magnetic properties are uniqueto unconventional superconductors. In s -wave case, thesusceptibility is diamagnetic everywhere in the disk asshow in the Appendix A.The anomalous paramagnetic response is well ex-

plained by appearing the odd-frequency Cooper pairs.The anomalous Green function can be decomposed intos -, p -, and d -wave components by

fx(r, iωn)iσν σ2 =

∫ 2π

0

dθ

2πVx(θ)f(r, θ, iωn), (9)

for x = s, p, and d. Figure 2(c) shows the amplitude ofthe d -wave component at ω0 = πT . The spatial profileof the order parameter is almost similar to that of (c).The d -wave component drastically suppresses in (110)and (110) directions, which has been well known as a re-sult of appearing of topologically protected Andreev sur-face bound states at the zero-energy14,15. At the sametime, the p -wave component of the anomalous Greenfunction grows at the corresponding edges as shown in(d). The spin-singlet p -wave Cooper pairs must have theodd-frequency symmetry to satisfy Eq. (8). The break-down of the translational symmetry at the surface mixesthe even- and odd-parity components. The appearanceof the Andreev surface bound states and that of the odd-frequency pairs are the two different faces of the samephenomenon. To have the zero-energy peak in the den-sity of states, the frequency symmetry of Cooper pairmust be odd29,37,38. The odd-frequency pairs have socalled negative pair density29, which leads to the para-magnetic instability as shown in Appendix B. Thereforewe conclude that the paramagnetic current is carried bythe induced odd-frequency Cooper pairs. Comparing theFigs. 2(b) with 2(d), the paramagnetic current flows atthe regions where the odd-frequency Cooper pairs stay.We have also obtained qualitatively the same results

for a spin-triplet p -wave superconducting disk at T =0.2Tc as shown in Fig. 3, where the local magnetic sus-ceptibility (a), the current density (b), the p -wave com-

ponent of f (c), and s -wave component of f (d) are

4

FIG. 4. (a) The diamagnetic-paramagnetic phase diagram ofsuperconducting disks for the d -wave (square) and p -wave(circle) pairing symmetry, where λL = 3 ξ0, ωc = 10∆0. (b)The penetration length dependencies of the paramagnetic-diamagnetic crossover temperatures, where the square andcircle symbols are the results for the d -wave and p -wave pair-ings, and the open and closed symbols are the results for theR = 3 ξ0 and R = 5 ξ0 superconducting disks, respectively.

presented in the same manner as Fig. 2. The resultsin Fig. 3 show the twofold symmetry reflecting the p -wave order parameter. The surface bound states areappear at the (100) surfaces at which the p -wave com-ponent of the anomalous Green function is suppressed.Correspondingly, the s -wave component becomes largeat the surfaces of (100) directions. The spin-triplet s -wave component belongs to the odd-frequency symmetryclass according to Eq. (8). The main difference betweenFigs. 2 and 3 is the property of the surface ABS at thezero-energy. In the spin-triplet p -wave disk, Majoranafermions appear at the surface28. From Figs. 2 and 3,we conclude that the magnetic property of unconven-tional superconductors are intrinsically inhomogeneousand can be paramagnetic because of the odd-frequencyCooper pairs at the surface.

The magnetic properties of superconductors stronglydepends on the disk size because the odd-frequency pairsspatially localize near the surface limited by ξ0 from theedge. Next, therefore, we discuss the relation betweenthe magnetic property and the disk size. Figure 4 is theparamagnetic-diamagnetic phase diagram of the d- andp -wave superconducting disks, where the vertical axis isthe paramagnetic-diamagnetic crossover temperature Tp

and the horizontal one is the radius of superconductingdisk R. The disk is paramagnetic χ > 0 at the tem-peratures below Tp. The results show that Tp decreasewith increasing the radius of the superconductor. Asshown in Figs. 2 and 3, the paramagnetic area is lim-ited to ξ0 from the surface because odd-frequency pairsare confined there. On the other hand, the bulk areaare diamagnetic because even-frequency pairs stay there.Roughly speaking, the relative area of staying the odd-

frequency pairs to the whole area of disk qualitativelydetermines the magnetic response of the disk. Thereforethe paramagnetic phase disappears in large disks withR ≫ ξ0. because the contribution from the surface isnegligible in large enough disks. This argument is sup-ported by the λL dependence of Tp shown in the inset ofFig. 4, where open (filled) symbols represent the resultsfor R/ξ0 = 3 (5) and the circles (squares) are the resultsfor p (d) wave disks. The crossover temperature is totallyinsensitive to λL. To be paramagnetic, the larger disksrequire the stronger contribution from the odd-frequencyCooper pairs. The odd-frequency Cooper pairs energeti-cally localize around the zero-energy29. The temperaturesmears effects of them on the magnetic response. There-fore Tp decreases with increasing the disk size as shownin Fig. 4.Finally, we discuss the susceptibility of whole super-

conducting disk as a function of temperature as shownin Fig. 5, where we fix the penetration depth at λL = 3 ξ0.The results for d - and p - wave symmetries are presentedin (a) and (b), respectively. The magnetic susceptibil-ity just below Tc is negative as usual. With decreasingtemperature, the paramagnetic current due to the odd-frequency Cooper pairs increases. As a consequence, thesusceptibility upturns at low temperature, which is qual-itatively different from the susceptibility in the s wavecase as shown in Appendix A. Below Tp, the param-agnetic odd-frequency Cooper pairs dominate the mag-netic response of the superconductor. Therefore the de-pendence of the susceptibility on temperature shows thereentrant behavior as demonstrated in Fig. 5. In exper-iments, it is possible to measure the susceptibility as afunction of temperature.

IV. DISCUSSION

Our theoretical results may correlate to the measure-ment of the pair density at low temperature6. They mea-sured the penetration depth λL = (4πnse

2/mc2)−1/2 ofa YBCO film on which (110) oriented internal surfacesare introduced by heavy-ion bombardment. They foundthat λ first decreases with decreasing temperature fromTc then increases at very low temperature. This results

FIG. 5. The temperature dependencies of the magnetic sus-ceptibility for the (a) d -wave and (b) p -wave superconductingdisks.

5

can be interpreted as a result of decreasing the pair den-sity ns at low temperature. The odd-frequency pairshave the negative pair density. Thus the decrease of ns

may suggest the increase of odd-frequency pair fraction.The experimental results on a high-Tc superconductorare consistent with our theoretical results.In real materials, the inelastic scatterings dephase the

Cooper pairs and broaden the energy profile of the pair-ing functions. The inelastic mean free path also limitsthe size of disks in the phase diagram shown in Fig. 4. Ind -wave superconductors, it has been shown that the sur-face roughness also broadens the zero-energy peak at thesurface. In such situation, we infer that the roughnesswould suppress the paramagnetic effect. On the otherhand in p -wave superconductors, the surface zero-energypeak is robust under the disordered potential. Thereforeeffects of surface roughness on the paramagnetic effectwould be different in the two pairing symmetries. Thisis an important future issue.The diamagnetism of superconductor is a result of

gaining the condensation energy below the transitiontemperature. Therefore the paramagnetic superconduct-ing states may be impossible in uniform thermodynamiclimit. The paramagnetic phase in Fig. 4 can be consid-ered as an unstable state and should disappear for largeR/ξ0. As shown in Fig. 2. and 3, the magnetic inho-mogeneity is an intrinsic feature of unconventional su-perconductors. Such inhomogeneous property assists theappearance of the paramagnetic phase in small disks. In-deed we confirm that the paramagnetic phase appears intwo cooling processes: field cool and zero-field cool.The spontaneously time-reversal symmetry (TRS)

breaking states has been discussed in high-Tc grains42.The subdominant component of order parameter nearthe surface breaks TRS. The results in Fig. 2 also indi-cates the TRS breaking superconducting state even whenwe simply assume the pure d wave order parameter. Weare thinking that the symmetry crossover from d waveto TRS breaking d + is might be possible in small sam-ples. To prove this, however, we need to compare thefree-energy among possible symmetry states. This issuegoes beyond the scope of this paper.Odd-frequency pairs appear also in superconduc-

tor/ferromagnet proximity structures38. When odd-frequency pairs are dominant in the ferromagnet39–41, theparamagnetic instability may lead to spontaneous cur-rent there31.

V. CONCLUSION

In conclusion, we have theoretically studied the mag-netic response of small unconventional superconductingdisks by using the quasiclassical Green function method.We conclude that small unconventional superconductorscan be paramagnetic at low temperature due to the ap-pearance of odd-frequency Cooper pairs at their surface.The magnetic properties of unconventional superconduc-

tors are intrinsically inhomogeneous as a result of theirtopologically nontrivial nature. Our results show up suchuniversal property of unconventional superconductivity.

ACKNOWLEDGMENTS

The authors are grateful to Y. Tanaka and S. Hi-gashitani for useful discussion. This work was sup-ported by the ”Topological Quantum Phenomena” (No.22103002) Grant-in Aid for Scientific Research on Inno-vative Areas from the Ministry of Education, Culture,Sports, Science and Technology (MEXT) of Japan.

Appendix A: Results for s -wave disk

We supply the calculated results for conventional spin-singlet s -wave superconductors. Fig. 6 shows the calcu-lated results of the local susceptibility (a) and the cur-rent density (b) for the s wave superconducting disk,where we fix the parameters as R = 3 ξ0, λL = 3 ξ0,and T = 0.2Tc. Because of the isotropic property in thes wave pair potential, the results are also isotropic in realspace. Therefore we plot the results as a function of x aty = 0. The results in (a) show that the response is dia-magnetic everywhere in the disk. Correspondingly thecurrent profile in (b) suggests the usual Meissner screen-ing current. The amplitude of the s -wave component ofthe anomalous Green function is almost uniform because

-0.0010

-0.0005

0.0000

0.0005

0.0010

-3 -2 -1 0 1 2 3-0.3

-0.2

-0.1

0.0

0.1

-3 -2 -1 0 1 2 3

0.00

0.25

0.50

0.75

1.00

-3 -2 -1 0 1 2 3

FIG. 6. (a) The local susceptibility and (b) the current den-sity of the s wave superconductor, where R = 3 ξ0, λL = 3 ξ0,ωc = 10∆0, Hext = 0.001Hc1 and T = 0.2Tc. (c) The s

wave component of the anomalous Green function. (d) Thesusceptibility vs temperature.

6

s wave superconductors are topologically trivial and donot host any surface states. The amplitude of the p wavecomponent is much smaller than that of s wave one. Thesusceptibility of disk is plotted as a function of temper-ature in (d). The susceptibility decreases monotonicallywith decreasing temperature, which is usually observedin experiments. Since the disk size is not much largerthan λL, the perfect diamagnetism (i.e., 4πχ = −1) isnot archived.

Appendix B: Paramagnetic current due to

odd-frequency pairs

We discuss the contribution of odd-frequency pairs tothe paramagnetic current within the linear response the-ory. Here we consider that the pair potential has a singlecomponent in spin space ∆(r,k) = ∆(r,k)iσν σ2, whereν is one of 0-3. In such case, the Eilenberger equation isreduces to a 2× 2 matrix equation

i~vFk ·∇r g +[H, g

]= 0, (B1)

H =

[iωn + evF

c k ·A i∆(r,k)i∆(r,k) −iωn − evF

c k ·A

], (B2)

g(r,k, ωn) =

[g(r,k, ωn) f(r,k, ωn)

sp f˜(r,k, ωn) −g(r,k, ωn)

], (B3)

where sp is 1 for even-parity order parameter and -1 forodd-parity one. The electric current is given by

j(r) = −2ievFπN0T∑

ωn

∫dk

Sdkg(r,k, ωn). (B4)

Here g is the normal Green function in the presence of thevector potential. In what follows, we estimate g withinthe linear response of A. In the Eilenberger equation,the vector potential formally shifts the energy. Thus theGreen function can be expressed as

g = g0 + ∂ωng0(−ievF /c)k ·A (B5)

within the linear response, where g0 is the Green func-tion at A = 0. In what follows, we omitted ”0” fromthe subscript of the Green function for simplicity. Bysubstituting the expression to Eq. (B3), we obtain

j(r) =−ne2πA

2mcT∑

ωn

∂ωn〈g(r, ωn)〉k (B6)

=−nse

2A

mc, (B7)

ns

n=πT

∑

ωn

〈∂ωng(r, ωn)〉k (B8)

=1

2

∫ ∞

−∞

dω 〈∂ωg(r, ω)〉k , (B9)

where n is the density of electrons in the normal stateand N0 is the density of states per spin at the Fermilevel. The ωn derivative of the Green function can bedefined only at T = 0. To discuss magnetic property ofthe anomalous Green function, we write the derivative ofg to

∂ωg(r,k, ω) =1

2

(f2E − f2

O

)∂ω ln

(1 + g

1− g

), (B10)

fE(r,k, ωn) =1

2

(f + spf

˜

)∣∣∣∣(r,k,ωn)

, (B11)

fO(r,k, ωn) =1

2

(f − spf

˜

)∣∣∣∣(r,k,ωn)

, (B12)

where we have used the normalization condition in theMatsubara representation g2 + spff

˜= 143.

At A = 0, the Eilenberger equation is decomposed intothree equations for the three components in the matrixstructure,

~vFk ·∇r g =2∆fO, (B13)

~vFk ·∇r fE =− 2ωnfO, (B14)

~vFk ·∇r fO =2(∆g − ωnfE). (B15)

Here we note that the all functions are real when wedelete the superconducting phase. For the uniform case,we obtain the solution

g =ωn√

ω2n +∆2

, f = spf˜= fE =

∆√ω2n +∆2

, (B16)

and fO = 0. In the uniform bulk region, fE is thesource of the order parameter ∆. Thus parity, spin,and frequency symmetries of fE and those of ∆ shouldbe identical to each other. The contribution of fE tothe pair density must be positive in Eq. (B10) becausethe uniform superconductor is diamagnetic. The sur-face and the interface are source of the inhomogeneityin superconductor and mix the two orbital symmetry:even parity and odd parity. Nonzero spatial derivative inEqs. (B13)−(B15) allows fO component.

The component fO is an odd function of ωn as shown inEq. (B13) because g is always an odd function of ωn dueto a symmetry relationship g(r,k, ωn) = −g∗(r,k,−ωn).Equation (B14) indicates that the frequency symmetryof fO are always opposite to those in fE. Therefore in-homogeneity induces fO component which has the odd-frequency symmetry. Eq. (B10) tells us that the pairdensity of odd-frequency component is negative, whichleads to the paramagnetic response.

7

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