arXiv:1503.07267v3 [cond-mat.supr-con] 25 Dec 2015 ubrmlil fhl u unawr observed were quanta flux half of multiple number eotdt ecnitn ihthe with consistent be to reported i up aebe investigated been have jumps oid fractional with associated functions gap superconducting rvd upr othe to support provide aidb unu fmgei flux magnetic of quantum a by panied auefor nature fmcocpcqatmsaesc ssprudt and superfluidity as such state quantum superconductivity macroscopic of aincreo hnanlrci opsdo Sr of composed coil annular thin a of curve zation superconductors eerpre o MgB for mo reported Leggett were collective the on observations experimental and opsdo badNdFeAsO and Nb of composed opnn uecnutrwsfis icse yLeggett by discussed first was superconductor component odcoswsrie oetm ago time some raised was conductors ei ed eea neetn xmlsaeaalbe A YBa available. superconductor are cuprate examples of interesting ring mag- Several tricrystal external to should field. response the netic symmetry on pairing consequences unique unconventional leave winding, by phase induced the phase ation to intrinsic quantization carrying related the functions intimately gap Since superconductivity is flux supercurrent. magnetic zero of with path closed pnst ihdmninlirdcberpeetto of representation the to irreducible similarly dimensional corre- group high which point field, a magnetic to external sponds without in- even and state functions ing gap superconduc of condensates time-reversal-symmetry-broken(TRSB) phases a duce three in frustrations among generate can repulsions inter-component like bevdt ar afflxquanta flux half a carry to observed rlobtl fF otiuet ut uecnutn con conside a superconducting and multi realistic becomes to possibility contribute this densates, Fe se of where orbitals superconductors iron-pnictide eral of discovery the to h ereo reo frltv hs di phase relative of freedom of degree The nte neetn osblt nmlicmoetsuper multi-component in possibility interesting Another ψ ′∗ j rcinlFu lta nMgeiainCreo Multicom of Curve Magnetization in Plateau Flux Fractional otxwt 2 with Vortex d wv arn symmetry pairing -wave rudtelo nyi ato l odnae.Fractional condensates. all of part Φ a in only loop the around h w avso opacmoaedi accommodate free-ener loop with of halves associated wi two curve states the magnetization two the in by found occupied are halves ch two by with specified superconductor states degenerate two yielding repulsively, ASnmes 42.v 42.p 47.a 74.20.De 74.70.Xa, 74.20.Rp, 74.25.Uv, numbers: PACS pnicitides. iron in that di as phase supercondu such relative broken superconductivity explore TRS to chance of novel evidence a clear vides a is phenomenon This 0 iervra ymty(R)myb rkni superconduct in broken be may (TRS) symmetry Time-reversal 9–16 = 1–4 2 hc rdaeSho fPr n ple cecs nvriyof University Sciences, Applied and Pure of School Graduate .INTRODUCTION I. afvle uodjmsi magneti- in jumps fluxoid Half-valued . nsprodcos otxi accom- is vortex a superconductors, In . / 2 S 2 e + 19,20 lhuhte niiulytk rirr ausdependi values arbitrary take individually they although , π − arn ymtyfriron-pnictide for symmetry pairing hs idn sahallmark a is winding phase hs oiosi n ftetwo the of one in solitons Phase . p wv superconductor -wave 1 ainlIsiuefrMtrasSine skb 305-004 Tsukuba Science, Materials for Institute National nentoa etrfrMtrasNnacietnc (W Nanoarchitectonics Materials for Center International 0 21–26 p . 88 wv arn symmetry pairing -wave F 5 0 narn-hpdsetup ring-shaped a In . . Φ . 27 12 0 / u up nodd- in jumps flux , hrb Josephson- whereby ,wihi h sig- the is which 2, Φ 0 ff rnei two- a in erence = 2 hoHuang Zhao Cu 2 ff hc RuO 3 rn ne-opnn hs ik edn ofiiewinding finite to leading kinks phase inter-component erent Dtd oebr7 2018) 7, November (Dated: 6–8 O / 17 2 7 which , − e 4 Due . δ rable were flux- vari- ff na in was rne hs ikadsltni bqiosmulti-compone ubiquitous in soliton and kink phase erence, 18 17 de v- t- 1 - - , . , 2 n ioHu Xiao and hroyai tblt n aiu oe properties its novel discuss various to and devoted stability been thermodynamic have works subsequent of amount cir yellow red phases the mutual and the parameters to referring order chirality of the The indicates phases chiralities. the opposite denote with time-reversal-symmet occu rows states of are degenerate halves two loop two the a the by where with superconductor, setup (TRSB) Schematic broken 1. FIG. ercwt epc otecretdrcina h conseque the TRS as asym broken direction be of current should the current to respect critical with the metric state, TRSB the supercon- single in multicomponent conventional ductor a a and between superconductor junction component Josephson a in that, naJspsnjnto ewe bnadBaFe and PbIn experimental between junction observed Josephson were a in directions current opposite in cpcaayi hr adsrcue n togycorrela strongly and structures band where mic analysis a in scopic with already consistent realized is been which have superconductors, iron-based may state TRSB the Therefore, hr r w oanwl n Iacmoaiginter-compo accommodating II and kinks. I phase wall th domain of two halves two are the there Between condensates. three among purple rlt fgpfntos ecnie opo such of loop a consider We functions. gap of irality u ltasfr ar eae yteflxquantum flux the by related pairs form plateaus flux ymnm,weetetodmi al between walls domain two the where minima, gy eetyi a enrvae ytepeetauthors present the by revealed been has it Recently hopst hrlte.Fatoa u plateaus flux Fractional chiralities. opposite th r ihtreo oecnestsinteracting condensates more or three with ors tvt,adi eea on fve tpro- it view of point general a in and ctivity, go aeilprmtr n temperature. and parameters material on ng skb,Tuua3587,Japan 305-8571, Tsukuba Tsukuba, 1 , 2 42 samte ffc,ueulciia currents critical unequal fact, of matter a As . oetSprodco Loop Superconductor ponent PI-MANA), ,Japan 4, numbers nt 1 . 8 Co → 0 28–41 . he ar- three 2 blue As loop e pied nent . 2 nce ted ro- ry- cle 43 → ly - - . 2
FIG. 2. Illustration of relation between the phase kinks at domain wall I and II and the flux trapped in the superconductor loop: (a) phase kinks between a same pair of components which can only trap integer multiples of flux quantum; (b) phase kinks between two different pairs of components which can trap a fractional flux FIG. 3. Schematic magnetization curve with fractional flux plateaus (0 <η< 1). Colored curves at the domain walls refer to the phase displayed together with gauge-invariant phase kinks denoted by variations in corresponding condensates, and Dij is for the gauge- [Dik/D jk] in the order of domain walls I and II. invariant phase kink between condensate i and j. The dashed circle denotes the direction for counting phase winding in the loop.
equivalent condensates with equal mutual repulsion. For effects are taken into account35. Cross checking this novel simplicity all order parameters are taken as s-wave from superconducting phenomenon becomes an important issue. now on. The two states in the loop are given by Ψ = i2π/3 i4π/3 In the present work, we address a new phase-sensitive prop- ψ1, ψ2, ψ3 = ψ 1, e , e and Ψ∗ with opposite chirali- erty of the TRSB superconducting state. As schematically {ties (see Fig.} 1).| |{ Across each of} the two domain walls between shown in Fig. 1, we consider a loop of a multicomponent su- the left and right halves of the superconductor loop, there is perconductorwhere the two halves are occupied by two TRSB a phase kink where the intercomponent phase difference be- states carrying opposite chiralities, accompanied by two do- tween two of the three order parameters shrinks to zero and main walls associated with inter-component phase kinks. We reopens in the opposite way continuously, resulting in a sign reveal explicitly that fractional flux plateaus appear in mag- reversal in the phase difference at the two sides of domain netization curve associated with free-energy minima, where (see Fig. 2). We notice that phase kinks are gauge-invariant the domain walls accommodate phase kinks among different objects, which inevitably appear at the interface between two components leading to 2π phase winding along the loop only bulks of TRSB states with opposite chiralities. in one or two of the three condensates. While the heights of fractional flux plateaus depend on material parameters and When the two domain walls accommodate the phase kink temperature, they form pairs with heights related by the flux between the same two condensates, such as that between con- densate 1 and 2 defined as D12 in Fig. 2(a), the phase rotation quantum Φ0, which is a unique signature of the TRSB super- conducting state and can be used to confirm the state itself. integrated in a counterclockwise manner (indicated by L in In a more general point of view this provides a novel chance Fig. 2) over the two domain walls cancel each other, resulting to explore relative phase difference, phase kink and soliton in in the same phase winding in all the three condensates. In this ubiquitous multicomponent superconductivity. case, the flux trapped in the loop is an integer multiple of flux quantum Φ when the loop is thick enough to fully screen the The remaining part of this paper is organized as follows. 0 magnetic field. The mechanism for fractional flux plateaus in presence of do- main walls is discussed in Sec. II. We then simulate the mag- The situation differs when the two domain walls accommo- netizing process with the time-dependent Ginzburg-Landau date differentphase kinks, such as D12 and D23 in domain wall (TDGL) approach and confirm fractional flux plateaus associ- I and II respectively shown in Fig. 2(b). By inspection one ated with free-energy minima in Sec. III. In Sec. IV, we study sees that ψ2 rotates 4π/3 anticlockwise over the two domain an asymmetric loop where the widths of two domain walls walls, while ψ1 and ψ3 rotate 2π/3. When the external mag- differ. In Sec. V we discuss the stability of the domain-wall netic field provides the additional− phase rotation of 2π/3 in all structure and temperature dependence of the fractional flux condensates, a state with 2π phase winding in ψ2 and 0 in both plateaus. Finally we give a summary in Sec. VI. ψ1 and ψ3 is stabilized. This yields a fractional flux quanta Φ0/3 in the loop. The state with a fractional flux trapped in this loop is expected to be stable in a certain range of exter- II. FRACTIONAL FLUX IN SUPERCONDUCTING LOOP nal magnetic field, which leads to a fractional flux plateau in magnetization curve shown schematically in Fig. 3. In order to reveal the essence of physics we first con- The above discussion can be elucidated by the integration sider an isotropic TRSB state which is generated by three of magnetic flux over the superconductor loop using the GL 3 formalism where the supercurrent is given by44,45
2e 2 2π J = ψ j ~ ϕ j A , (1) m | | ∇ − Φ ! j=X1,2,3 j 0 with m j and ϕ j the effective mass and phase of component- j. For a thick loop, the supercurrent is zero deep inside the superconductor. In this case the magnetic flux trapped in the loop is given by the line integration of phase differences as FIG. 4. Amplitudes and phases of order parameters in two TRSB can be seen from Eq. (1) states with parameters α1′ = 0.012, α2′ = 0.013, α3′ = 0.011, γ12′ = γ23′ = 0.24, γ13′ = 0.25, m1′ = m3′ = 1, m2′ = 1.1, β1′ = β2′ = Φ0 p1 ϕ1 + p2 ϕ2 + p3 ϕ3 β = 1− and κ = 1.5. See− text for definitions of the dimensionless GL Φ= ∇ ∇ ∇ dl 3′ 1 2π "IC p1 + p2 + p3 # parameters. Φ p ϕ + p ϕ = 0 ϕ dl + 2∇ 12 3∇ 13 dl , (2) 1 + + 2π "IC ∇ ZDW p1 p2 p3 # the coefficients in Eq. (3) satisfy conditions revealed in a pre- 33 2 vious work . To be specific, we put γ12, γ13, γ23 < 0, namely with p j = ψ j /m j and ϕi j = ϕ j ϕi for i, j = 1, 2, 3, where ”C” is a closed| | path along the loop− with zero supercurrent all repulsive Josephson-like couplings, since it is easy to see everywhere (see Fig. 1), and the ”DW” denotes domain-wall that other TRSB states can be generated from this case by a regimes (gray parts in Fig. 2) with phase kinks. In the second simple gauge transformation. 46 line, we divide the integrand into two terms, indicating two We adopt dimensionless quantities given by contributions to the total magnetic flux. The first contribution ~ 2 2e ψ10(0) x = λ1(0)x′, A = λ1(0)Htc1(0) √2A′, J = J′, should be an integer multiple of 2π due to the single-valued m1ξ1(0) wave function in the loop. The integrand in the second contri- ψ = ψ (0)ψ , α = α (0)α , β = β β , γ = α (0)γ , j 10 ′j j − 1 ′j j 1 ′j jk − 1 ′jk bution is nonzero only on domain walls. This contribution is m = m m , κ = λ (0)/ξ (0), F = G F nonzero when two different phase kinks are realized at domain j 1 j′ 1 1 1 0 ′ wall I and II, with the value depending also on the quantities 2 2 2 2 with ψ10(0) = α1(0)/β1, λ1(0) = m1c /[4πψ10(0)(2e) ], p j. − q Presuming the same length of domain walls I and II, the ~2 2 ξ1(0) = /[2m1α1(0)], Htc1 = 4πα1(0)ψ10(0) and two configurations [D /D ]and[D /D ] at the domain walls − q− ik jk jk ik G = H2p(0)/4π. In the dimensionless units, the GL free [I/II] take the same free energy. However, integrating phase 0 tc1 energy is rewritten as differences for these two configurations along the closed path 2 4 2 in the counterclockwise manner (see Fig. 2) results in op- β′j 1 1 F′ = α′ ψ′ + ψ′ + A′ ψ′ posite fractional values of 2π in the second term in Eq. (2). j j 2 j m j′ iκ1 j j=1,2,3 ∇− Therefore, these two configurations give two fractional fluxes P 2 Φ and Φ related by the flux quantum Φ . Fractional flux γ′jk ψ∗j′ψk′ + c.c. + ( A′) . (4) 1 2 0 − j,k=1,2,3; j 2 ∂ψ′ j 2 1 1 = α′ ψ′ β′ ψ′ ψ′ A′ ψ′ + γ′ ψ′ III. TDGL APPROACH ∂t − j j− j| j| j−m iκ ∇− ! j jk k j′ 1 k=1X,2,3;k, j (5) Here we adopt the GL formalism to check the thermody- with j = 1, 2, 3 and namic stability of states carrying fractional fluxes. The GL free-energy functional of a three-band superconductor with ∂A′ 1 2 1 33,45 σ = ψ′ ϕ′ A′ A′ (6) Josephson-like inter-component couplings is given by ∂t m | j| iκ ∇ j − ! −∇×∇× j=X1,2,3 ′j 1 2 2 β j 4 ~2 2π F = α j ψ j + ψ j + ∇ A ψ j 2 2m j i Φ0 with σ the coefficient of normal conductivity. At equilibrium j=1,2,3 − P the left-hand sides of Eqs. (5) and (6) are zero, which gives 1 2 γ jk ψ∗jψk + c.c. + 8π ( A) , (3) the GL equations. By solving three GL equations in Eq. (5) − j,k=1,2,3; j 1 (a) 0.8 ] 0.6 0 Φ [ Φ 0.4 0.2 0 -38.8 (b) FIG. 6. (color online) Magnetization curve with two fractional flux -39.0 plateaus for an asymmetric superconductor loop, where the width of ] 0 side including domain II is enlarged to 12λ1(0) from that given in Fig. 5. The parameters are the same as Fig. 4 except for α′ = 0.025, G[G 1 -39.2 α2′ = 0.028 and α3′ = 0.022. deviate from Φ /3 and 2Φ /3, because of the three inequiv- -39.4 0 0 0 0.009 0.018 0.027 0.036 alent condensates in the system. Nevertheless, it is clear that H[Htc1(0)] the relation Φ1 +Φ2 =Φ0 is satisfied as revealed in Sec. II. For even larger magnetic fields, the stable state permits one flux FIG. 5. (a) Magnetization curve of thermodynamically stable state quantum Φ0 inside the loop. Note that the magnetic fields for and (b) Gibbs free energies of several competing states in the su- the fractional flux plateaus are very small as compared with perconductor loop shown in Fig. 1 upon sweeping external magnetic the typical field Htc1(0) and thus there is no vortex inside the field. Parameters are the same as Fig. 4. The superconductor loop body of the superconductor. is of square shape with outer frame of 24λ1(0) and inner frame of We check the phase kinks at the two domain walls I and II 8λ1(0). and the phase windings along the loop in the three conden- sates at the fractional flux plateaus. At Φ1 = 0.26Φ0, phase kinks D12 and D23 are realized at regions I and II respectively, are positive and small. In Fig. 4 we display the two TRSB and ψ2 rotates 2π along the loop leaving ψ1 and ψ3 unwinding. states with opposite chiralities, which are used for the follow- At Φ1 = 0.74Φ0, the phase kinks D12 and D23 are at domain ing study on the magnetization process. walls II and I, in contrast to the case of Φ1 = 0.26Φ0, and ψ1 We take a square loop with external dimension 24λ1(0) and ψ3 rotate 2π with ψ2 unwinding. At integer flux quanta × Φ = Φ=Φ 24λ1(0) and the wall thickness 8λ1(0) to investigate the mag- 0 and 0, the phase kink is D12 at both domain netic response. From Eq. (1) we obtain the the penetration walls I and II. All these are in accordance with the discus- 2 2 1 2 sion in Sec. II. In general, there are at most six fractional flux length λ = ( j ψ′j /m′j)− λ (0) in the dimensionless form. | | 1 plateaus between the integer flux quanta 0 and Φ . For the For the statesP given in Fig. 4, λ = 1.22λ1(0) is much smaller 0 than the thickness of loop wall. Therefore, one can take the GL parameters given in Fig. 4 we can only see two plateaus closed path ”C” as the middle line of the superconductor loop in Fig. 5(a) because the free energies of domain walls satisfy where supercurrent is negligibly small (see the Appendix). At F(D12) . F(D23) < F(D13) such that only the phase-kink pair the edge of superconductor, we presume that no supercurrent D12 and D23 is stabilized. flows out of the superconductor,and the B field at the external edge of superconductor loop is fixed to the value of applied magnetic field. IV. ASYMMETRIC SUPERCONDUCTOR LOOP The magnetization curve derived from TDGL equations (5) and (6) is shown in Fig. 5, where fractional flux plateaus cor- Up to this point, the superconductor loop is presumed to responding to states with free-energy minima are obtained. have the same width at domain wall I and II, which guaran- When the magnetic field is small, there is no flux pene- tees the degeneracy of domain-wall energy between phase- trating into the loop. As the magnetic field increases to kink pairs [Dik/D jk] and [D jk/Dik]. In this case, it is easy to H/Htc1(0) = 0.009, the stable state takes a fractional flux see that the magnetization curve in Fig. 3 should be symmet- Φ1 = 0.26Φ0. This state remains stable until the magnetic ric with respect to the direction of magnetic field. In general, field H/Htc1(0) = 0.019, yielding a plateau in the magneti- the two widths can be different. In the latter case [Dik/D jk] zation curve. As the magnetic field increases further, another may be unstable even though [D jk/Dik] is stable and associ- state appears with fractional flux Φ2 = 0.74Φ0 in the regime ated with the free-energyminimum. As shown in Fig. 6 for an 0.019 H/H (0) 0.029. We can see that both Φ and Φ asymmetric superconductor loop, the fractional flux plateau at ≤ tc1 ≤ 1 2 5 provided by widening the left and right arms of the loop. The increase in free energyin states with fractional fluxes and inte- ger flux quanta upon application of external magnetic field is smaller by one order of magnitudethan the free-energybarrier as seen in Fig. 7, which justifies the discussion on fractional flux plateaus in the present work. At this point we notice that the free-energy barrier in Fig. 7 generated by the two TRSB states at the two halves of su- perconductor loop is crucially important for the thermody- namic stability of fractional flux plateaus. From Eq. (2) one might think that a two-component superconductor or a three- component one with preserved TRS can also accommodate fractional fluxes. However, in these cases there is no free- energy barrier like that in Fig. 7, and states with fractional fluxes are unstable. FIG. 7. (color online) Gibbs free energy for the system in Fig. 1 with The amplitudes and inter-component phase differences of two halves occupied by the two TRSB states with opposite chiralities order parameters change with temperature, leading to varia- as a function of the location of domain walls x defined in the inset. tion in stable domain-wall structures. As a result, both height Parameters and sample size are the same as Fig. 5. and width of a fractional flux plateau should change as tem- perature is swept. During field-cooling or field-heating pro- cesses, jumps between states with fractional fluxes and/or in- Φ = 0.74Φ0 remains stable for 0.017 H/Htc1(0) 0.023, teger multiples of flux quantum can take place. Φ= . Φ ≤ ≤ while that at 0 26 0 disappears in contrast to Fig. 5, since Half-valued fluxoid jumps in magnetization curve were re- ff they are associated with di erent phase-kink configurations at ported in a small multicomponent superconducting sample domain wall I and II. It is worth noticing that even in this comparable with penetration length17. This results in incom- Φ= Φ asymmetric loop the plateau at 0.26 0 is still stable for plete screening of magnetic field, and thus the magnetization . / . − 0 020 H Htc1(0) 0 014, since it is associated with the curve exhibits a finite slope for any external magnetic field. − ≤ ≤ − Φ = Φ same phase-kink configuration with that at 0.74 0 and Namely, there is no fractional flux plateau in their system. In ff Φ a di erence of flux quantum 0 coming from the first term in previous studies on vortex states of TRSB superconductor it Eq. (2). In this asymmetric loop, the magnetization curve is was discussed that vortex cores of different condensates can asymmetric with respect to the direction of the magnetic field deviate from each other in space39,49–52. However, without as in Fig. 6. The property that fractional flux plateaus in posi- a closed path along which supercurrent is zero everywhere, ff tive and negative magnetic fields are paired with the di erence there is no fractional flux plateau in the magnetization curve. of flux quantum Φ0 is robust, and can be taken as a crosscheck for fractional flux plateaus originated from the TRSB state. VI. CONCLUSION V. DISCUSSIONS To summarize, we have studied the magnetic response of a loop of three-component superconductor with two degenerate In the present work we study the case that the left and right time-reversal symmetry broken states at two halves. When halves of the superconductor loop take the two TRSB states the two domain walls between the two halves accommodate with opposite chiralities. This situation can be realized in different intercomponent phase kinks, fractional flux plateaus experiments by cooling the whole system from temperature 48 appear in the magnetization curve which form pairs related to above Tc with laser heat pulse irradiated on regions I and II . each other by the flux quantum. These properties are expected The two halves condensate independently and by chance ar- to be helpful for detecting experimentally the time-reversal rive at the different TRSB superconducting states, leading to symmetry broken superconductingstate which can be realized the two domain walls at region I and II after releasing the ir- in iron-pnictide superconductors. In general, this endeavour radiation. In order to check the stability of this configura- provides a novel chance to explore relative phase difference, tion, we estimate the free energy of the whole system in terms phase kink and soliton in ubiquitous multi-component super- of the TDGL approach. As shown in Fig. 7, the state with conductivity. two domain walls located at the middle of the top and bottom sides of the loop corresponds to a free-energy minimum. The domain walls once generated should be stable since moving them outside the loop is prohibited by a large free-energy bar- ACKNOWLEDGEMENTS rier which is produced by an elongated, single domain wall during the process of domain-wall relocation (see that at x2 in The authors are grateful to Masashi Tachiki, Zhi Wang, the inset of Fig. 7). The stability of the present setup against Yusuke Kato and Atsutaka Maeda for stimulating discussions. relocating one of the two domain walls along the loop can be This work was supported by the WPI initiative on Materials 6 Nanoarchitectonics, and partially by the Grant-in-Aid for Sci- entific Research (No. 25400385), MEXT of Japan. APPENDIX: DISTRIBUTION OF SUPERCURRENT IN THE SUPERCONDUCTOR LOOP An example of the distribution of supercurrent density in the superconductor loop and order parameters on domain walls is shown in Fig. A1, where the superconductivity sur- vives in all components and the Meissner effect screens the magnetic field and thus the total supercurrent to zero. The important feature here is that, although the supercurrents in individual components are not zero due to the phase shifts, FIG. A1. (color online) Distribution of supercurrent density J′ in the they flow in opposite directions and cancel each other, lead- superconductor loop for the fractional flux plateau of Φ= 0.26Φ0 in ing to zero total supercurrent. The situation of a domain wall Fig. 5, and that of order parameters on domain wall II. The directions as discussed in the present work differs considerably from a and lengths of arrows refer to phases and amplitudes of order param- vortex, where the phase winding induces a divergent kinetic eters, where ψ2 and ψ3 are suppressed to 0.63 and 0.77 of the bulk energy, which necessitates total suppression of the amplitude values while| ψ | remains| | almost unchanged. | 1| of the superconducting order parameter at the vortex core. 1 F. London, Superfluids, Vol. I., Wiley, New York, (1950); 24 A. Gurevich and V. M. Vinokur, Phys. Rev. Lett. 90 047004 2 L. Onsager, Phys. Rev. Lett. 7, 50 (1961). (2003). 3 N. Byers and C. N. Yang, Phys. Rev. Lett. 7, 46 (1961). 25 H. Bluhm, N. C. Koshnick, M. E. Huber, and K. A. Moler, 4 A. A. Abrikosov, Sov. Phys. JETP 5, 1174 (1957). Phys. Rev. Lett. 97, 237002 (2006). 5 C. C. Tsuei and J. R. Kirtley, Rev. Mod. Phys. 72, 969 (2000). 26 V. Vakaryuk, V. Stanev, W. C. Lee, and A. Levchenko, 6 C. T. Chen, C. C. Tsuei, M. B. Ketchen, Z. A. Ren, and Z. X. Zhao, Phys. Rev. Lett. 109, 227003 (2012). Nature Phys. 6, 260 (2010). 27 D. F. Agterberg, V. Barzykin, and L. P. Gor’kov, Phys. Rev. B 60, 7 W. Q. Chen, F. Ma, Z. Y. Liu, and F. C. Zhang, Phys. Rev. Lett. 14868 (1999). 103, 207001 (2009). 28 V. Stanev and Z. Tesanovic, Phys. Rev. B 81, 134522 (2010). 8 D. Parker and I. I. Mazin, Phys. Rev. Lett. 102, 227007 (2009). 29 Y. Tanaka and T. Yanagisawa, J. Phys. Soc. Jpn. 79, 114706 9 I. I. Mazin, D. J. Singh, M. D. Johannes, and M. H. Du, (2010). Phys. Rev. Lett. 101, 057003 (2008). 30 J. Carlstrom, J. Garaud, and E. Babaev, Phys. Rev. B 84, 134518 10 K. Kuroki, S. Onari, R. Arita, H. Usui, Y. Tanaka, H. Kontani, and (2011). H. Aoki, Phys. Rev. Lett. 101, 087004 (2008). 31 R. G. Dias and A. M. Marques, Supercond. Sci. Technol. 24, 11 W. Q. Chen, K. Y. Yang, Y. Zhou and F. C. Zhang, Phys. Rev. Lett. 085009 (2011). 102, 047006 (2008). 32 T. Yanagisawa, Y. Takana, I. Hase, and K. Yamaji, 12 S. Graser, T. A. Maier, P. J. Hirschfeld, and D. J. Scalpino, J. Phys. Soc. Jpn. 81, 024712 (2012). New J. Phys. 11, 025016 (2009). 33 X. Hu and Z. Wang, Phys. Rev. B 85, 064516 (2012). 13 F. Wang, H. Zhai, Y. Ran, A. Vishwanath, and D. H. Lee, 34 S. Z. Lin and X. Hu, Phys. Rev. Lett. 108, 177005 (2012). Phys. Rev. Lett. 102, 047005 (2009). 35 S. Maiti and A. V. Chubukov, Phys. Rev. B 87, 144511 (2013). 14 K. Seo, B. A. Bernevig, and J. Hu, Phys. Rev. Lett. 101, 206404 36 B. J. Wilson and M. P. Das, J. Phys.: Condens. Matter 25, 425702 (2008). (2013). 15 K. Ishida, Y. Nakai, and H. Hosono, J. Phys. Soc. Jpn. 78, 062001 37 N. V. Orlova, A. A. Shanenko, M. V. Milosevic, and F. M. Peeters, (2009). Phys. Rev. B 87, 134510 (2013). 16 P. C. Dai, J. P. Hu, and E. Dagotto, Nature Phys. 8, 709 (2012) 38 M. Marciani, L. Fanfarillo, C. Castellani, and L. Benfatto, 17 J. Jang, D. G. Ferguson, V. Vakaryuk, R. Budakian, S. B. Chung, Phys. Rev. B 88, 214508 (2013). P. M. Goldbart, and Y. Maeno, Science 331, 186 (2011). 39 Y. Takahashi, Z. Huang, and X. Hu, J. Phys. Soc. Jpn. 83, 034701 18 A. J. Leggett, Prog. Theor. Phys. 36, 901 (1966). (2014). 19 G. Blumberg, A. Mialitsin, B. S. Dennis, M. V. Klein, N. D. Zhi- 40 A. Hinojosa, R. M. Fernandes, and A. V. Chubukov, gadlo, and J. Karpinski, Phys. Rev. Lett. 99, 227002 (2007). Phys. Rev. Lett. 113, 167001 (2014). 20 J. Nagamatsu, N. Nakagawa, T. Muranaka, Y. Zenitani, and 41 Y. Tanaka, Supercond. Sci. Technol. 28, 034002 (2015). J. Akimitsu, Nature 410, 63 (2001). 42 Z. Huang and X. Hu, Appl. Phys. Lett. 104, 162602 (2014). 21 M. Sigrist and D. F. Agterberg, Prog. Theor. Phys. 102, 965 43 S. Schmidt, S. Doring, F. Schmidl, V. Grosse, and P. Seidel, (1999). Appl. Phys. Lett. 97, 172504 (2010). 22 Y. Tanaka, Phys. Rev. Lett. 88, 017002 (2002). 44 M. Tinkham, Introduction to Superconductivity (McGraw-Hill, 23 E. Babaev, Phys. Rev. Lett. 89, 067001 (2002). Inc., New York, 1996). 45 A. Gurevich, Physica C 456, 160 (2007). 7 46 S. Z. Lin and X. Hu, Phys. Rev. B 84, 214505 (2011). 49 J. Garaud, J. Carlstrom, and E. Babaev, Phys. Rev. Lett. 107, 47 W. D. Gropp, H. G. Kaper, G. K. Leaf, D. M. Levine, M. Palumbo, 197001 (2011). and V. M. Vinokur, J. Comput. Phys. 123, 254 (1996). 50 J. Garaud and E. Babaev, Phys. Rev. Lett. 112, 017003 (2014). 48 J. Tate, B. Cabrera, S. B. Felch, and J. T. Anderson, 51 S. Gillis, J. Jaykka, and M. V. Milosevic, Phys. Rev. B 89 024512 Phys. Rev. Lett. 62, 845 (1989). (2014). 52 Z. Huang and X. Hu, J. Supercond. Nov. Magn. (in press, arXiv:1510.05890).