H - T Phase Diagram of Multi-component Superconductors with Frustrated Inter-component Couplings
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b H - T Phase Diagram of Multi-component Superconductors with FrustratedInter-component Couplings
Y. Takahashi,
1, 2
Z. Huang,
1, 2 and X. Hu
1, 2 International Center for Materials Nanoarchitectonics (WPI-MANA),National Institute for Materials Science, Tsukuba 305-0044, Japan Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba 305-8571, Japan (Dated: April 11, 2018)Multi-band superconductors in which frustrated inter-band couplings yield a time-reversal-symmetry breaking (TRSB) state are investigated. Stability condition for the TRSB state arederived based on the Bardeen-Cooper-Schrieffer (BCS) theory. With the time-dependent Ginzburg-Landau (GL) method, vortex states are investigated first at the vicinity of critical temperature T c where the GL theory is valid, and the results are extended to compose the H - T phase diagram.When material parameters satisfy the condition that the nucleation field is slightly larger than thethermodynamic field ( H n & H tc ) derived in a previous work (X. Hu and Z. Wang, Phys. Rev. B , 064516 (2012)), an unconventional intermediate state characterized by clustering vortices ap-pears. Calculation of interface energy reveals that the clustering vortices are associated with positiveinterface energy. I. INTRODUCTION
Superconductivity associated with multiple conden-sations was discussed soon after the establishment ofBardeen-Cooper-Schrieffer (BCS) theory more than fiftyyears ago in view of compounds of transition metals.
Later on, possible crucial differences between multibandsuperconductors with frustrated Josephson-like couplingsamong condensates and single-band superconductors at-tracted some attention. Recently, the discovery of iron-based high-temperature superconductors with at leastthree bands contributing to superconductivity, made itimportant to fully understand superconducting phenom-ena associated with the multibandness.A straightforward extension of a single-band theory,as performed for two-band case is not sufficient for caseswith more than three bands, since above three bandsrepulsive inter-band couplings may induce phase frus-trations, and yields a bulk state specified by phase dif-ferences among condensates different from 0 and π andhence time-reversal symmetry breaking (TRSB, a termwe shall use through the present work). A TRSBstate was first proposed based on the BCS theory with s pairing, and later on other pairings such as s + id symmetry or s + is symmetry as well as junctionstructures have also been discussed. Novel propertiesin collective excitations and chiral ground stateswere highlighted. Within the scheme of multi-component Ginzburg-Landau (GL) theory on TRSB superconducting state,the thermodynamic field H tc and the nucleation field H n were analytically derived, and it was revealed that apply-ing external magnetic field to such superconductor wouldgenerate interesting states, which might not be straight-forwardly categorized to type-I or type-II by the GL pa-rameter κ = λ/ξ . As a matter of fact, numerical cal-culations on magnetic properties exposed fractional vor-tices or unconventional vortex states.
However, a systematic study seems lacking at the moment of thiswriting.In the present paper, we investigate magnetic proper-ties of multicomponent superconductors with frustratedintercomponent couplings in the framework of GL theory,under the guidance from analytical results for the way tochoosing parameters in GL theory. The remaining part isorganized as follows. In Sec. II, we first derive the stabil-ity condition of the TRSB state based on multiband BCStheory. In Sec. III, multicomponent GL formalism is in-troduced based on the multiband BCS theory. In Sec.IV, vortex states in a TRSB superconductor are simu-lated with the time-dependent GL (TDGL) method, andwe find a vortex-cluster state. Based on this, we propose H - T phase diagrams. Finally, discussions and a summaryfollow in Sec. V and Sec. VI. II. STABILITY CONDITION OF TRSB STATE
We first discuss the stability condition of TRSB statebased on the BCS theory. The BCS Hamiltonian for asingle band superconductor is extended to the multibandcase, H = X j ≤ N h X k j ,σ ξ k j c † k j σ c k j σ − X k j , k ′ j V jj c † k j ↑ c †− k j ↓ c − k ′ j ↓ c k ′ j ↑ i − X j = l,j,l ≤ N X k j , k l V jl c † k j ↑ c †− k j ↓ c − k ′ l ↓ c k ′ l ↑ , (1)with N ≥
3. The second and third terms correspond tothe intra- and inter-band couplings, respectively. Thecoupled self-consistent BCS gap equations for multi-band superconductors are derived straightforwardly bya mean-field approach, ∆ j = V jj N j ∆ j Z ~ ω D − ~ ω D dξ j E j tanh (cid:16) E j k B T (cid:17) + X j = l V jl N l ∆ l Z ~ ω D − ~ ω D dξ l E l tanh (cid:16) E l k B T (cid:17) , (2)with ∆ j = − P k j V jj h c − k j ↓ c k j ↑ i − P j = l P k l V jl h c − k l ↓ c k l ↑ i and E j = q ξ j + | ∆ j | ;here an identical cut-off energy ~ ω D is taken in allbands for simplicity. The coupled BCS equations for athree-band case are rewritten, g − f ( E , T ) g g g g − f ( E , T ) g g g g − f ( E , T ) × ∆ ∆ ∆ = , (3)where, f j ( E j , T ) = N j Z ~ ω D − ~ ω D dξ j E j tanh (cid:18) E j k B T (cid:19) , (4)and g jl = [ V − ] jl with V = V V V V V V V V V . Let us focus on the solution of Eq. (3) with complexgap functions which specify a TRSB state, where a nec-essary condition g g g > real as alwayspossible, one has the following equation, (cid:20) g − f ( E , T ) g g g − f ( E , T ) (cid:21) (cid:20) Im(∆ )Im(∆ ) (cid:21) = (cid:20) (cid:21) . (5)For nontrivial solution, one obtains the relation on theabsolute values of the gap functions:[ g − f ( E , T )] [ g − f ( E , T )] = g . (6)In the same way, one has two other similar relations. Thethree relations then yield[ g jj − f j ( E j , T )] = g jl g jn g ln , (7)where j = l = n . Noticing that, in the matrix Eq. (3),all diagonal terms g jj − f j take the same sign as seenfrom Eq. (6), and there is only one independent vector, one can show that g jj − f j and g jl g jn /g ln have the samesign. The relations Eq. (7) therefore are rewritten as f j ( E j , T ) = g jj − g jl g jn g ln . (8)Interestingly, they take the same form as a single-bandcase, except for that the intraband coupling is renormal-ized by the interband ones.There is only one independent vector in the matrix inEq. (3), ∆ g + ∆ g + ∆ g = 0 , (9)which actually permits one to have complex solutionsfor Eq. (3). From Eq. (9), it is clear that to search thecomplex solution one needs to form a triangle by usingthe three segments | ∆ j ( T ) /g ln | , and therefore (cid:12)(cid:12)(cid:12)(cid:12) ∆ j ( T ) g ln (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∆ l ( T ) g jn (cid:12)(cid:12)(cid:12)(cid:12) > (cid:12)(cid:12)(cid:12)(cid:12) ∆ n ( T ) g jl (cid:12)(cid:12)(cid:12)(cid:12) . (10)One of these three inequalities may be broken as temper-ature sweeps, and the system transfers to a time-reversalsymmetry reserved (TRSR) state where phase differencesbetween the order parameters (OPs) take trivial values,i.e. 0 or π . The transition from TRSB to TRSR statestakes place where one of the three inequalities is replacedby an equality (cid:12)(cid:12)(cid:12)(cid:12) ∆ j ( T ) g ln (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) ∆ l ( T ) g jn (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ∆ n ( T ) g jl (cid:12)(cid:12)(cid:12)(cid:12) . (11)The critical temperature of a TRSB superconductorcan be derived from Eq. (8) by putting ∆ j = 0, N j ln 2 e γ ~ ω D j πk B T c = g jj − g jl g jn g ln , (12)with the Euler constant γ = 0 . · · · . In the follow-ing discussion, we focus on multiband superconductorswith a TRSB state as an equilibrium bulk state. III. MULTICOMPONENT GL THEORYA. Derivation of GL equations
We concentrate on temperature sufficiently close to thecritical temperature where the GL theory is applicable.By expanding the coupled BCS equations in Eq. (3) near T c , we obtain the GL equations without gradient terms, (cid:18) g jj − N j ln 2 e γ ~ ω D πk B T (cid:19) ∆ j + 7 ζ (3) N j πk B T c ) | ∆ j | ∆ j + g jl ∆ l + g jn ∆ n = 0 , (13)with ζ (3) the Riemann zeta-function. With the conven-tional expression for the GL equations and taking theOPs as ψ j = | ψ j | e iφ j = ∆ j , f = X j =1 , , (cid:20) α j | ψ j | + β j | ψ j | + 12 m j (cid:12)(cid:12)(cid:12) ( ~ i ∇ − ec A ) ψ j (cid:12)(cid:12)(cid:12) (cid:21) − X j The thermodynamic field H tc and nucleation field H n in a TRSB superconductor were analytically derived inRef. [9]. However, an analytic treatment on vortex statesin the TRSB superconductor is very difficult, where spa-tially intertwined amplitudes and phases of OPs are cru-cial.For simplicity, we consider a case that the second andthird bands are same, but different from the first band.In this case, the two typical fields are given by, H tc = 2 r πβ (cid:18) − α − γ γ γ (cid:19)(cid:18) 12 + r α r β (cid:19) / , (16) H n = 2 κ r πβ (cid:18) − α − γ γ γ (cid:19) r α r m + 2 + | r α r m − | /r α r m , (17)with r α = α , /α , r β = β , /β , r m = m , /m and κ = ( m c/ e ~ ) p β / π is a material dependent param-eter for the first component.As the simplest but nontrivial case, we take r α = 1, r β = 1 and γ = γ = γ corresponding to an isotropicbulk state, while sweep the mass ratio r m between 0 and1. In this case, the ratio between nucleation and ther-modynamic fields is given by ρ = H n /H tc = κ r m r m + 1 r . (18)As in a single component superconductor, magnetic re-sponses of multicomponent superconductors change dras-tically across ρ = H n /H tc = 1, which corresponds to acharacteristic value of κ κ ∗ = 2 r m + 13 r m r . (19) Hereafter we perform our numerical study by varying thevalue of κ . For convenience, dimensionless quantities areintroduced. Units of length and energy are given by λ = p m c β / π | α | e and H (0) = 4 π | α | /β ,which are defined in the individual ( γ jk = 0) first-component at T = 0 with α = α ( T = 0). C. Vortex States in TRSB superconductors Vortex states in a TRSB superconductor are studiedbased on the TDGL method adopting the magnetic peri-odic boundary condition which confines fixed number ofvortices N in the simulation box. The results shown below are for N = 8, while we haveconfirmed the main conclusions remain valid using largesystems. As an example we take r m = 0 . κ ∗ ≈ . H n < H tc ( κ < κ ∗ ), H n & H tc ( κ & κ ∗ ) and H n ≫ H tc ( κ ≫ κ ∗ ). InFig. 1(a-c) where κ < κ ∗ , we observe a typical phaseseparation between superconductivity and normal stateswhich represents a type-I superconductor. This resultis the same as the conventional single-component super-conductors. As seen in Fig. 1(a), the OPs show differentrecovery lengths from normal to bulk superconductivityregion due to anisotropic mass ratio r m .For κ & κ ∗ , we find an unconventional vortex state.Figures 1(d-g) show a vortex cluster, where multiple vor-tices ( N = 8) form a domain of vortex bundle. As seenin Fig. 1(d), the OPs have finite values inside the domainregion which is clearly different from Fig. 1(a). As seenin Fig. 1(g), phase differences inside the cluster are eitherzero or π indicating a TRSR state, while the bulk regionkeeps a TRSB state ( φ jk ≡ φ k − φ j = 2 π/ κ ≫ κ ∗ , Figs. 1(h-k) show typical triangular vor-tex lattice configurations which represent a type-II su-perconductor. In this parameter regime, OP phase dif-ferences at the vortex core are slightly modulated frombulk values, and no TRSR domain can be observed.It is noted that when the system is totally isotropic,namely r m = 1, the vortex-cluster state does not appear,where modulation in amplitudes and phases in OPs aredecoupled as discussed in Ref.[9]. D. Interface energy in TRSB superconductors In a single component superconductor, the GL param-eter κ = 1 / √ H n = H tc coin-cides with that where sign change in interface energytakes place, which dichotomizes a superconductor x / λ y / λ -20 -10 0 10 20-20-1001020 00.010.020.030.040.05 x / λ y / λ -20 -10 0 10 20-20-1001020 00.050.10.150.2 -30 -15 0 15 3000.050.10.150.2 x / λ | ψ j | , B z | Ψ || Ψ || Ψ | B z |(cid:2) (cid:3) |(cid:4) (cid:5) -30 -15 0 15 3000.050.10.150.2 x / λ | ψ j | , B z | Ψ || Ψ || Ψ | B z x / λ y / λ -20 -10 0 10 20-20-1001020 00.010.020.030.040.05 x / λ y / λ -20 -10 0 10 20-20-1001020 x / λ y / λ -20 -10 0 10 20-20-1001020 00.20.40.60.81 x / λ y / λ -10 0 10-10010 00.010.020.030.040.05 x / λ y / λ -10 0 10-10010 00.050.10.150.2 x / λ y / λ -10 0 10-10010 00.20.40.60.81 (cid:2) (cid:3) (cid:2) (cid:6) (cid:2) (cid:7) |(cid:2) (cid:8) |, (cid:4) (cid:5) (cid:10) (cid:6)(cid:7) (cid:11) (cid:12) < (cid:11) (cid:14)(cid:15) (cid:11) (cid:12) ≳ (cid:11) (cid:14)(cid:15) (cid:11) (cid:12) ≫ (cid:11) (cid:14)(cid:15) AC B A AC B -15 -10 -5 0 5 10 1500.050.10.150.2 x / λ | ψ j | | Ψ || Ψ || Ψ | B z |(cid:2) (cid:6) ||(cid:2) (cid:7) | (a) (d) (h)(b) (e) (i)(c) (f) (j)(g) (k) FIG. 1. (Color online) Typical vortex configurations solved by the TDGL method for (a-c) H n < H tc , (d-g) H n & H tc and(h-k) H n ≫ H tc , for κ = 2 . κ = 2 . 25 and κ = 6 . 0, respectively, with the mass ratio r m = 0 . 3, interband couplings γ = γ = γ = − . | α | and temperature T = 0 . T c . Panels (a, d, h) are spatial profiles of | ψ | , | ψ | , | ψ | and B z along y = 0 (red line) on the other panels. Panels (b, e, i) denote magnetic induction B z . Panels (c, f, j) denote the amplitude oforder parameter of the first-component | ψ | . Panels (g, k) denote phase difference between the second and third components φ ≡ φ − φ . into type-I or type-II. Therefore, it is interesting to eval-uate interface energy in the TRSB superconductor wherethe vortex-cluster state appears, and thus the simple clas-sification of type-I and type-II superconductor does notapply. For this purpose, we calculate the excess Gibbsfree energy in a one-dimensional system, Γ = Z + ∞ ( g sH − g s0 ) dx, (20) where, g sH and g s0 are energy density with and withoutapplied fields, respectively.The boundary conditions are given, | ψ j | = 0 and B ( x ) = H tc for x → | ψ j | = | ψ j | and B ( x ) = 0 for x → + ∞ where | ψ j | ’s are bulk values of OPs in each component,and H tc is the thermodynamic field of TRSB supercon-ductor in Eq. (16).Typical interface structure is shown in Fig. 2(a) for -3 κ Γ / ( α λ / β ) r m = 1 r m = 0.5 r m = 0.3 r m = 0.2 (cid:1) (cid:2)∗∗ (a)(b) x / λ | ψ j | , B z , φ B z | ψ || ψ || ψ | φ FIG. 2. (Color online) (a) TDGL calculation results of spatialvariation for | ψ j | , φ and B z , which are normalized by bulkvalues | ψ j | , φ = 2 π/ H tc ,respectively. (b) κ dependence of the interface energy Γ inTRSB superconductors. Γ is normalized by the thermody-namic field for the first component without inter-componentcoupling. r m = 0 . κ = 3. Figure 2(b) shows κ depen-dence of the interface energy Γ for several typical valuesof mass ratio r m . Numerical errors are negligible in theseplots. The interface energy decreases monotonically withincrease of κ and changes its sign at κ ∗∗ . Therefore, ina TRSB superconductor, there are two threshold valuesof κ , for example, κ ∗∗ ≈ . κ ∗ ≈ . 18 for r m = 0 . κ ∗ and κ ∗∗ which separate the Meissner phase, vortex-cluster phaseand vortex lattice phase. The two phase boundaries over-lap at r m = 1, which is consistent with that no vortex-cluster state can be found in the isotropic system. E. Vortex State at H . H n Here we consider the field dependence of vortex statesin the regime κ ∗ < κ < κ ∗∗ where a vortex cluster isobserved. Variations of the system upon sweeping ap-plied magnetic fields is simulated by changing the num-ber of vortices N with fixed system size. Figure 4 showsthe vortex configuration with the same material param- r m κ (cid:1) (cid:2)∗∗ (cid:1) (cid:2)∗ Meissner Vortex LatticeVortexCluster FIG. 3. (Color online) Phase diagram for vortex state in theTRSB superconductor in terms of r m and κ including theMeissner, vortex cluster and vortex lattice phases. See textfor definitions. x / λ y / λ -20 -10 0 10 20-20-1001020 00.050.10.150.2 x / λ y / λ -20 -10 0 10 20-20-1001020 00.20.40.60.81 (a) |(cid:2) (cid:3) | (b) (cid:4) (cid:5)(cid:6) FIG. 4. (Color online) Vortex configurations with magneticflux number N = 36. (a) Amplitude of ψ . (b) OP phasedifference φ . Material parameters are same as in Fig. 1(d-g). eters and system size as those in Fig. 1(d-g) but with N = 36. A typical vortex lattice is observed, and phasedifference are either 0 or π as displayed in Fig. 4(b) as-sociated with a TRSR state in the whole system. OPsare suppressed by the magnetic field in different waysin accordance with effective masses m j , which results ina breaking of the stability condition of TRSB state inEq. (10). The magnetic-field-induced TRSB to TRSRtransition is seen for κ > κ ∗ . F. H - T Phase Diagrams of TRSB superconductor In this section, we construct H - T phase diagramsof multicomponent TRSB superconductor in the threeregimes (a) H n < H tc ( κ < κ ∗ ), (b) H n & H tc ( κ ∗ < κ < κ ∗∗ ) and (c) H n ≫ H tc ( κ > κ ∗∗ ).In Fig. 5(a), the TRSB superconductor shows simplytypical type-I property. At high magnetic fields, super-conductivity with a TRSB state is totally suppressed,and transfers to a normal state ( | ψ j | = 0). This is es-sentially the same H - T phase diagram as a conventionalsingle-component superconductor. The phase transitionbetween Meissner and normal states is unambiguouslyfirst order.Figure 5(b) shows the novel H - T phase diagram whichincludes the vortex-cluster state as an unconventional in-termediate phase. The vortex-cluster phase is located H tc Meissner Normal T c H c1 Normal T c MeissnerH n Vortex Cluster(TRSB+TRSR)Vortex Lattice(TRSR) H c1 Normal T c MeissnerH n Vortex Lattice(TRSB)Vortex Lattice(TRSR) (a) (b) (c) FIG. 5. (Color online) H - T phase diagrams for multicomponent superconductors with frustrated intercomponent couplings.Three diagrams are characterized by conditions: (a) H n < H tc ( κ < κ ∗ ), (b) H n & H tc ( κ ∗ < κ < κ ∗∗ ) and (c) H n ≫ H tc ( κ > κ ∗∗ ). The double and single lines represent first- and second-order transition, respectively. The dashed line representsthe TRSB-TRSR transition as discussed in Sec. III E. above the lower critical field H c where vortices startto penetrate into a superconductor. For the strongermagnetic field slightly below the nucleation field H n , aconventional vortex lattice with TRSR state appears asshown in Fig. 4.The phase transition between the vortex cluster andthe Meissner states is likely not continuous. The second-order transition at H c1 in a type-II superconductor isunderstood by additional Gibbs free energy ∆ G due tovortex penetration represented by ∆ G = G s (cid:12)(cid:12) first flux − G s (cid:12)(cid:12) no flux = B Φ ǫ + X F ij − BH π , where ǫ is vortex line energy and F ij = Φ π λ K ( r ij λ ) isvortex interaction energy with the zeroth Bessel function K . For conventional case with repulsive vortex inter-action, the energy cost is ∆ G ≈ B is muchsmall inside the superconductor ( B ≈ F ij ≈ 0) as inter-vortexdistance is large enough. However when vortices form acluster as observed in the TRSB superconductor, vorticespenetrate into a superconductor feeling finite interactionenergy F ij , and consequently the system will see a finiteenergy jump ∆ G which corresponds naturally to a first-order transition.Finally, Fig. 5(c) shows the H - T phase diagram for H n ≫ H tc ( κ > κ ∗∗ ). Since the vortex lattice state isobserved at magnetic fields H c1 ≤ H ≤ H n , the phase di-agram is essentially same as the single component case.However, it is remarked that there are two regimes interms of OP phase configurations. For a low magneticfield regime, OP phases are locally modulated only in a vortex core, and the overall system preserves a TRSBstate. For a high magnetic field slightly below H n , thesystem transfers to a TRSR state as seen in Fig. 4. Be-tween the two states, vortex configurations do not showobvious differences and inter-vortex distance changes pro-portionally to strength of applied magnetic field. IV. DISCUSSIONS Using a numerical approach based on multicompo-nent GL theory, we have revealed that, in a multicom-ponent superconductor with frustrated intercomponentcouplings, a vortex-cluster state appears at an interme-diate magnetic field regime between Meissner and vor-tex lattice states when the material parameters satisfy H n & H tc . While numerical results are shown explicitlyfor the case where the material parameter of the firstcomponent κ and mass ratios between the components r m are varied whereas the other parameters relevant toa bulk value of OP are put the same, the appearance ofa vortex-cluster state is general for all possible param-eters as far as the stability conditions of a TRSB statediscussed in Sec. II are satisfied except the isotropic case.The vortex-cluster state is expected to be observableby conventional vortex imaging methods. It is also inter-esting to examine the behavior of magnetization around H c1 . The magnetization curve will be different from ei-ther that of type-I or that of type-II superconductor.Careful experiments are required and such unique mag-netic behavior will also support the novelty of a TRSBsuperconductor.It is appropriate to make some discussions on the na-ture of transition between the vortex cluster and vortexlattice (see discussions in Sec. III E and Fig. 5(b)). Sincethe spatial symmetry is different between the two states,a weak first-order transition is expected. However, inthe present work we could not find clear evidences forit since no thermodynamic quantity has been calculateddirectly. On the other hand, the nature of the TRSB-TRSR transition in Fig. 5(c) is a more subtle issue. Thistransition has only been discussed in absence of magneticfield (and thus without any vortex). While it was ar-gued to be first order, a numerical analysis indicated acontinuous transition. Therefore, the nature of TRSB-TRSR transition remains as an issue to be addressed infuture works.Similar vortex-cluster states have been also reportedin numerical studies based on the three-component GLtheory. It is mentioned in these studies that vortexcores of individual components do not overlap at thedomain boundary, suggesting the existence of fractionalvortices . In our study, similar vortex-cluster structure isobserved as indicated on the panels in Fig. 1(f), whereblue lines denote orientation of the vortex cluster. How-ever, separation of the cores is still unclear, and possibil-ity of fractional vortices should be studied further.While fractional vortices which appear at the bound-ary between two chiral TRSB superconductors were stud-ied well, those at a boundary between TRSB andTRSR states are also interesting, and deserve furtherstudy. V. CONCLUSION Magnetic properties of multiband superconductorswith frustrated interband couplings are investigated.The stability condition of the time-reversal-symmetrybreaking state is derived based on Bardeen-Cooper-Schrieffer (BCS) theory for zero magnetic field. Derivingmulticomponent Ginzburg-Landau (GL) theory fromthe BCS theory, we have investigated response of thenovel superconducting state to an external magneticfield. When parameters satisfy the condition H n & H tc ,with H n the nucleation field and H tc the thermodynamicfield, we have revealed the novel H - T phase diagramincluding the unconventional vortex state, namely vortexcluster , which cannot be categorized to either type-Ior type-II. The vortex cluster is associated with localdomain separation between time-reversal symmetrybroken and time-reversal symmetry reserved states, andit is expected to appear via a first order transition fromthe Meissner state. We have studied the interface energyin a time-reversal-symmetry broken superconductor,and found that material parameters for sign changein the interface energy do not coincide with those for H tc = H n . Note added. After completion of this work, we becameaware of Ref. 39 which contains similar discussions for the stability condition of a time-reversal-symmetry breakingstate based on BCS theory. ACKNOWLEDGMENTS This work was supported by the WPI initiative on Ma-terials Nanoarchitectonics, and partially by the Grant-in-Aid for Scientific Research (No. 25400385), MEXT ofJapan. Appendix A: Magnetic properties of multibandTRSR superconductors We here discuss a multicomponent superconduc-tor with intercomponent couplings unfrustrated , i.e. γ γ γ > 0, and analytically derive that it is similarto a single-component case close to T c .The multicomponent GL equations are derived fromthe free energy density functional in Eq. (14) by a varia-tional method, α j ψ j + β j | ψ j | ψ j + 12 m j (cid:18) ~ i ∇ − ec A (cid:19) ψ j − γ jl ψ l − γ jn ψ n = 0 , (A1)and for supercurrents, c π ∇ × ∇ × A = X j e ~ m j | ψ j | (cid:18) ∇ φ j − π Φ A (cid:19) , (A2)with j, l, n for indices of components and φ j for phase ofthe OP.Around the critical temperature, the order parametersare given by the linearized version of Eq. (A1). α − γ − γ − γ α − γ − γ − γ α ψ ψ ψ = X · Ψ = . (A3)The critical temperature T c is given when the determi-nant of X becomes zero.To satisfy the condition that X is positive definiteat T > T c , all determinants of principal minors takenon-negative values according to the Sylvester’s criterion,namely α j ≥ α j α l − γ jl ≥ 0. For the case where X has a single zero eigenvalue at T = T c (in contrast tothe case of two zero eigenvalues for the TRSB state ),one has P j = l ǫ j ǫ l = P j = l ( α j α l − γ jl ) > ǫ j theeigenvalues of X , which indicates that at least one termin the second summation should be positive (equivalentlya single zero eigenvalue), for example α α − γ > X has two independent vectors at T = T c , theratios among order parameters for T ≤ T c are given bythe Cramer’s rules from Eq. (A3), ψ ψ = α α − γ γ α + γ γ = γ α + γ γ α α − γ ,ψ ψ = α α − γ γ α + γ γ = γ α + γ γ α α − γ , (A4)where α j γ ln + γ jl γ jn = 0 since α j ≥ γ ln γ jl γ jn > α α − γ > α α − γ > 0. One then arrives at ψ ψ = α α − γ α α − γ ,ψ ψ = α α − γ α α − γ . (A5)For T ≤ T c , the OPs follow the coupled GL equations, α + β ψ − γ − γ − γ α + β ψ − γ − γ − γ α + β ψ ψ ψ ψ = X ′ · Ψ = . (A6)Since the determinant of X ′ should be zero,one has thefollowing OPs taking into account Eq. (A5), ψ ≈ − K det X β K + β K + β K ,ψ ≈ − K det X β K + β K + β K ,ψ ≈ − K det X β K + β K + β K , (A7)where K jl = α j α k − γ jl up to O (1 − T /T c ).Other quantities for a TRSR superconductor arestraightforwardly available with the conventional approach. The thermodynamic magnetic field isderived by free energy difference between superconduc-tivity and normal state in absence of magnetic fields,namely f n − f sc = P j =1 , , α j | ψ j | + β j | ψ j | , H π = 12 (det X ) β K + β K + β K . (A8)In order to calculate the coherence length, we considera one-dimensional system with the boundary conditionthat the order parameters recover from normal to bulkvalue, | ψ j | → | ψ j | as x → + ∞ , ~ m ∂ ( ψ − ψ ) ∂x = α ( ψ − ψ ) + 3 β ψ ( ψ − ψ ) − γ ( ψ − ψ ) − γ ( ψ − ψ ) , ~ m ∂ ( ψ − ψ ) ∂x = α ( ψ − ψ ) + 3 β ψ ( ψ − ψ ) − γ ( ψ − ψ ) − γ ( ψ − ψ ) , ~ m ∂ ( ψ − ψ ) ∂x = α ( ψ − ψ ) + 3 β ψ ( ψ − ψ ) − γ ( ψ − ψ ) − γ ( ψ − ψ ) . (A9)Taking ψ j − ψ j = a j exp( −√ x/ξ ), the equations arerewritten, α + 3 β ψ − ~ ξ − m − γ − γ − γ α + 3 β ψ − ~ ξ − m − γ − γ − γ α + 3 β ψ − ~ ξ − m · a a a = Y · a = 0 . The coherence length is then obtained from det Y = 0, ξ − ≈ ~ − det X K /m + K /m + K /m . (A10)The London penetration depth λ is straightforwardlyobtained from the GL equation for supercurrents inEq. (A2), λ − = 4 π (2 e ) c (cid:18) | ψ | m + | ψ | m + | ψ | m (cid:19) ≈ π (2 e ) c (cid:18) K m + K m + K m (cid:19) × − det X β K + β K + β K . (A11)Finally, the nucleation field is derived from the lin-earized GL equations in the presence of fields H . Takingthe gauge A y = Hx, A x = 0 , A z = 0, the OPs can beexpressed ψ j = e ik y y e ik z z f ( x ), which yield similar formsto the Schr¨odinger equation, − ~ m j f ′′ j + ~ m j (cid:18) πH Φ (cid:19) ( x − x ) f j = − (cid:18) α j + ~ k z m j (cid:19) f j + γ jl f l + γ jm f m , (A12)where x = k y Φ / πH . 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