Anomalous diamagnetic response in multi-band superconductors with time-reversalbroken symmetry.

Yuriy YerinB. Verkin Institute for Low Temperature Physics and Engineering,

National Academy of Sciences of Ukraine, 47 Lenin Ave., 61103, Kharkiv, Ukraine andInstitute for Physics of Microstructures, Russian Academy of Sciences, 603590 Nizhny Novgorod, GSP 105, Russia

Alexander OmelyanchoukB. Verkin Institute for Low Temperature Physics and Engineering,

National Academy of Sciences of Ukraine, 47 Lenin Ave., 61103, Kharkiv, Ukraine

Stefan-Ludwig Drechsler, Dmitri V. Efremov, and Jeroen van den BrinkInstitute for Theoretical Solid State Physics, Leibniz-Institut fur Festkorper- und Werkstoffforschung IFW-Dresden,

D-01169 Dresden, Helmholtzstrasse 20, Germany(Dated: July 11, 2017)

Within a Ginzburg-Landau formalism we establish analytically the necessary and sufficient con-ditions to realize a doubly degenerate superconducting ground state with broken time-reversal sym-metry (BTRS) in a multi-band superconductor. Using these results we analyze the ground state of athree band superconductor in the cylindrical geometry in an external magnetic field. We show thatdepending on the interband coupling constants, a magnetic flux can induce current density jumpsin such superconducting geometries that are related to adiabatic or non-adiabatic transitions fromBTRS to time-reversal symmetric states and vice versa. This unusual current induced magnetic fluxresponse can in principle be used experimentally to detect superconducting BTRS ground states aswell as corresponding metastable excited states.

I. INTRODUCTION

The phenomenon of superconductivity based is char-acterized by the spontaneous breaking of a gauge sym-metry. But in some cases simultaneously time-reversalsymmetry (TRS) can be broken as well. Because of theirunusual properties such superconductors (SC) with bro-ken time-reversal symmetry (BTRS) are attracting a lotof attention. For instance recently the formation of newcollective modes1,2 (similarly to the occurrence of Leggettmodes in two-band SC) and new topological excitationsin the form of phase kinks, domains and vortices thatcarry fractional magnetic flux values3–12 have been dis-cussed.

So far, BTRS superconductivity has been clearly de-tected in few cases, only. The most frequently citedexample is Sr2RuO4 in which the order parameter wasidentified to be triplet chiral (∆ ∝ px + ipy)13,14. Fur-thermore, evidence for BTRS superconductivity was re-ported for the low-T phase of Th-doped UBe13

15, UPt316,

and in SrPtAs based on muon measurements17. Theo-retically superconductivity with BTRS is also been pro-posed for other compounds such as cuprate SC at low-temperature18,19, transition metal dichalcogenides20,21,NaxCoO2 · yH2O22, strongly doped graphene23, andin some recently discovered Fe-based superconductors(FeSC).

The FeSC are of particular interests as BTRS super-conductivity is anticipated for several dopings as a resultof the multiband electron structure and strong repulsiveinterband couplings. Typically the Fermi-surface of thenon-SC parent compound consists of two or three hole-

like pockets at Γ = (0, 0) point and two electron-likepockets around M = (π, π) point. The resulting nest-ing at the vector Q = (π, π) connecting the Γ and Mpoints drives the system to a spin-density wave (SDW)state. With doping the SDW state melts, giving spacefor superconductivity. The natural symmetry of the or-der parameter in such a situation is given by the so calleds± one, which causes a gap function with opposite signsat the electron and the hole pockets, respectively. InBa1−xKxFe2As2 the hole doping by K-substitution leadsat x close to 1 to the vanishing of the electrons pock-ets and a change of the symmetry of the superconduct-ing order parameter to nodal d-wave symmetry for pureKFe2As2

24–27. A BTRS state was proposed for two dop-ings: for x ∼ 0.7, when the electron pockets vanish, andfor x ∼ 1, when a transition from s to d-wave supercon-ductivity is expected. The proposed intermediate pairingsymmetries are s+ is and s+ id correspondingly28,29.

However, as stressed above, still there are only fewcompounds where superconductivity with BTRS was un-ambiguously observed. One of the reasons actually is thelack of simple experimental tools to identify it. The mostcommon techniques to establish BTRS in superconduc-tors are µSR and NMR both suffer from restrictions re-lated to the the presence of impurities and other defectseven in high-quality single crystals. In addition NMR re-quires considerable magnetic fields which may themselvessignificantly affect the superconducting GS, especially forlow-temperature SC by paramagnetic pair-breaking ef-fects and field induced coexisting magnetic phases.

Here we first study in Sect. II the nature of possi-ble SC ground states in zero magnetic field and absence

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of currents. Within a Ginzburg-Landau approach weclarify the conditions for BTRS and explicitly show thetwo-fold degeneracy of the corresponding ground states.We adopt the simplest model when the BTRS supercon-ductivity, namely a three-band SC, is described approx-imately within a Ginzburg-Landau approach30–35. Fortwo-band model BTRS superconductivity is possible onlyin special cases like dirty materials in the vicinity of thes± → s++ transition29.

In Sect. III we investigate the magnetic field responseof superconducting cylinders with a BTRS order param-eter and introduce it as a new tool to identify super-conductivity with BTRS. In the three-band frameworkwe investigate the homogeneous current states in sucha mesoscopically one-dimensional system and show thatthe diamagnetic i.e. an orbital dominated response de-pends directly on the nature of the underlying order pa-rameter. Experimental verification of characteristics thatwe predicted here could be used to identify multibandBTRS-superconductivity.

II. GINZBURG-LANDAU APPROACH TOSUPERCONDUCTIVITY WITH BTRS

To describe the multiband superconductors we em-ploy a general Ginzburg-Landau (GL) functional, whichhas been used previously for particular cases (e.g. fortwo-bands or three equivalent bands,36,37 see the re-views Refs. [6], [7] and references therein), only. Wewill provide a rather general solution for three non-equivalent bands with repulsive interband interactionsbeing the most relevant case for BTRS-physics in thesesystems. For the sake of simplicity we address only ho-mogeneous states and isotropic order parameters. How-ever, inhomogeneous states containing different topolog-ical defects and explicit account of spin states can betreated straightforwardly within the same formalism38.The Ginzburg-Landau (GL) Gibbs energy density andthe current density for three-band superconductors canbe written in the following form

∆G =∑3i=1

∫ {1

4mi

∣∣(−i~∇− 2ec A)ψi∣∣2 + ai|ψi|2

+ bi2 |ψi|

4 − Fint}

+ 18π

∫(rotA−H)2 (1)

and

j = −∑i

ie~mi

(ψ∗i∇ψi − ψi∇ψ∗i )− 4e2

cA∑i

|ψi|2

mi. (2)

Here and below we consider the temperature regime be-low Tc for the GL approach is valid, i.e. we ignore theregion of strong fluctuations in the very vicinity of Tc orthe case of very low temperatures. In the first integralin Eq.(1) the integration is performed over the supercon-ductive region whereas in the second integral over thecylinder volume. The term Fint describes the phase sen-

φ

θ

0 1 2 3 4 5 60

1

2

3

4

5

6

−2.5

−2

−1.5

−1

−0.5

0

0.5

1(b)

FIG. 1. The surface (a) and the contour plot (b) of the de-pendence of the interband interaction term Fint of the GLfree energy functional as a function of the phase differencesfor the fixed set G1 = 1, G2 = −1, G3 = 1. Black dashedsquare limits intervals of consideration of phase differences.Crosses indicate global maximum values of Fint within theseintervals. The parameters are φ = φ1 − φ2 and θ = φ1 − φ3.Further details are provided in Supplement, Figs. S4.

sitive Josephson-like interband coupling:

Fint = γ12ψ?1ψ2 + γ23ψ

?2ψ3 + γ31ψ

?1ψ3 + c.c.

= 2γ12|ψ1||ψ2| cos(φ1 − φ2)

+ 2γ23|ψ2||ψ3| cos(φ2 − φ3) (3)

+ 2γ31|ψ1||ψ3| cos(φ1 − φ3)

Here the order parameter is in general complex, i.e.ψi = |ψi|eiφi . In contrast, the interaction coefficients γijare real and can be positive or negative. It was shownfor two-band superconductors, that the sign of γ12 fullydetermines the symmetry of the order parameter in theclean case. A repulsive interband interaction constantγ12 < 0 leads to unconventional symmetry and a groundstate with π-phase difference between the two bands (de-noted as s± -symmetry), while attractive interband in-teractions γ12 > 0 stabilize a ground state with a zero-phase difference between the their gap functions (denotedas s++-symmetry). We keep the same sign-convention inthe case of three-band SCs considered here.

As a first step we examine the ground state in the ab-sence of external magnetic fields. Then eq. (4) can berewritten as Fint = sgn(γ12γ23γ31)(F 2

x + F 2y − γ2

1 |ψ1|2 −γ2

2 |ψ2|2− γ23 |ψ3|2), where γ1 =

√|γ12γ23γ31|/γ23 and the

other γi are obtained by a cycle permutation. The intro-duced two functions Fx = γ1|ψ1| cosφ1 + γ2|ψ2| cosφ2 +γ3|ψ3| cosφ3 and Fy = γ1|ψ1| sinφ1 + γ2|ψ2| sinφ2 +γ3|ψ3| sinφ3, do absorb the complete phase shift depen-dencies.

Now the minimization of the GL-functional withrespect to the phases is reduced to the maximiza-tion/minimization of F 2

x + F 2y depending on the sign of

(γ12γ23γ31). The geometric meaning of F 2x + F 2

y is theabsolute value of a sum of three vectors in a 2D spacea1 = γ1|ψ1|(cosφ1, sinφ1), a2 = γ2|ψ2|(cosφ2, sinφ2)and a3 = γ3|ψ3|(cosφ3, sinφ3): F 2

x+F 2y = (a1+a2+a3)2.

One can immediately note that BTRS state correspondsto noncollinear vectors ai, while TRS to collinear ones.For (γ12γ23γ31) > 0 the minimum of GL corresponds tothe maximum of the F 2

x + F 2y , which is reached when

3

0

0

π

π

(a)

π

(b)

π

2π 2π

FIG. 2. Regions of existence of a BTRS ground state in termsof the two phase variables for φ (upper panel) and θ (lowerpanel) of a three-band SC . The BTRS ground states are real-ized in the narrow ”tilted x-like” regions with inhomogeneouscolors (θ, φ 6= 0, π), only, distinct from the large homogeneousregions corresponding to TRS ground states. The parametersare φ = φ1−φ2 and θ = φ1−φ3 For more details see Supple-ment, Figs. S2 and S3.

the vectors ai are collinear. It corresponds to the TRSphase. For (γ12γ23γ31) < 0 the minimum of GL corre-sponds to the minimum of the F 2

x + F 2y . The minimum

F 2x + F 2

y = 0 can be reached for noncollinear vectors,satisfying the triangle rule. With this the BTRS GS isrealized. If the length of vectors ai does not satisfy thetriangle rule the minimum is reached for collinear vectorsai, or TRS state. For the BTRS state the free energy∆FBTRS can be rewritten as:

∆FBTRS =∑i

[(ai − γ2i )|ψi|2 +

bi2|ψi|4]. (4)

Its minimum corresponds to the order parameter:

|ψi|2 = (−ai + γ2i )/bi (5)

and

cos θ = cos(φ2 − φ1) =γ2

3 |ψ3|2 − γ21 |ψ1|2 − γ2

2 |ψ2|2

2γ12|ψ1||ψ2|,

cosφ = cos(φ3 − φ1) =γ2

2 |ψ2|2 − γ21 |ψ1|2 − γ2

3 |ψ3|2

2γ13|ψ1||ψ3|.

The dependence of the interband interaction term Fintof the GL free energy functional as a function of the phasedifferences for a fixed set Gi is given in Fig. 1. Here wehave introduced convenient new variables:

G1 =γ12

|γ23||ψ1||ψ2|

, G2 = sign(γ23) and G3 =γ13

|γ23||ψ3||ψ2|

.

A BTRS state exists for zero external magnetic field onlywithin a relative small volume in the six-dimensional pa-rameter space (|ψj |,γij). The corresponding projectedregions onto the planes G1-G3 are shown in the form of”tilted X-like” regions in Fig. 2. The richness of these andother figures shown in the Supplement is a consequenceof the high-dimensionality of the parameter space that isgeneric for multi-band superconductors. We note that itresembles mathematically to some extent the richness of

L

d = R - R R = 2

R + R1 22 1

H

z

FIG. 3. (Color online) Sketch of a tube made from a three-band superconductor with a BTRS ground state in a parallelexternal magnetic field.

the 11-dimensional superstring theory manifested in thesix-dimensional Calabi-Yau manifolds39,40.

Noteworthy we have found that even in the case of anodd number of repulsive interband interactions, the de-generacy of ground states can be removed and TRS statecan be stable. This means that the presence of one orthree repulsive interband interactions in three-band su-perconductors does not provide a necessary and sufficientcondition for the occurrence of a BTRS GS, contrary tosome statements found in the literature31,41,42. To get adeeper insight into the nature of the interband frustra-tion responsible for the appearance of BTRS in three-band SCs, a rigorous and straightforward mathematicalapproach is necessary (for details see the calculationsand results presented graphically in the Figs. S1 andS2 of the Supplement.) Correspondingly, for higher n-band frustrated superconductors one is confronted withn (n− 1) /2 − 1 mutual phase differences, which can beconsidered within the proposed geometrical interpreta-tion. However, in contrast to the three band case thebilinear interaction between the superconducting bandorder parameters is not enough for unambiguous deter-mination of the phase differences and higher order termshave to be considered.

III. BTRS AND TRS SUPERCONDUCTORS ONA CYLINDER AND IN MAGNETIC FIELD

In order to distinguish readily a three-band SC witha BTRS ground state from those traditional SCs witha TRS ground state, we propose to apply a magneticflux to a (topologically) doubly-connected system. Inparticular, we consider a long and thin tube approxi-mated by two concentric cylinders with the inner andouter radii R1 and R2, respectively, (see Fig. 3). Itssymmetry axis is denoted as the z axis in cylindrical co-ordinates (r, ϑ, z). An external constant magnetic fieldH is thought to be applied along the symmetry axis ofsuch a cylinder:H = (0, 0, H), where the vector-potentialgauge is chosen as A = (0, Aϑ (r) , 0), Aϑ (r) = Hr

2 . We

4

assume that the radius of the tube R and the thickness dsatisfy the following conditions: R � λ, ξ and d � λ, ξ,where λ(T ) and ξ(T ) are the London penetration depthand the coherence length respectively. The first condi-tion precludes the formation of magnetic vortices or anydomains in the cylinder, while due the second one theself-induced magnetic fields are small and can thereforebe ignored in our calculations. It means that we studyonly homogeneous solutions |ψi| = const, while the phasedepends on the polar angle ϑ, only. In the considered ge-ometry the phase must fulfil the quantization condition∮

Γ∇φi = 2πni, where the integral is taken over an arbi-

trary closed continuous contour Γ lying inside the cylin-der and n = 0,±1,±2 are the phase winding (topological)numbers. Here we assume these winding numbers to beequal: n1 = n2 = n3.

It is interesting to note that a similar experimentalsetup was proposed with the aim to detect a fractionalflux plateau in the magnetization curve of a supercon-ducting loop that is topologically the same as the one

considered here43. These authors investigate numericallymetastable phase kinks protected by a large energy bar-rier within a GL functional adopting certain special pa-rameter values. The excited states with BTRS discussedin the following differ significantly from those solitonicstates.

To illustrate the principle of identifying BTRS for agiven three-band SC, we consider a simple case and as-sume firstly that the equilibrium values of the order pa-rameters are given but without adopting thereby theequality for the moduli of the interband interactions. Sec-ondly, the strengths of the interband interactions coincidebut, for instance, at least one of these interactions is re-pulsive. Since we are interested in a three-band supercon-ductor with initial BTRS state we control the selection ofparameters of interband interactions numerically in orderto avoid the possible occurrence of non-frustrated groundstates even for an odd number of repulsive interband in-teraction (see Figs. S1-S8 in the Supplemental materials).Then we can write the GL free-energy of the system inthe momentum space (see the Supplemental material)

∆F

πR2L=∑i

(ai|ψi|2 +

1

2bi|ψi|4 + κi|ψi|2q2

)− 2γ12|ψ1||ψ2| cosφ− 2γ13|ψ1||ψ3| cos θ − 2γ23|ψ2||ψ3| cos (θ − φ) , (6)

and the current density is:

j =∑i

κi|ψi|2q. (7)

Further we will use κi = κi/κ1. The superfluid momen-tum q depends on the winding number n and the mag-netic flux Φ as q = 1

R (n− Φ/Φ0) with Φ0 = π~c/e beingthe flux quantum. One sees that Eq. (6) can be ob-tained from the corresponding equation in zero magneticfield by substituting ai → ai + κiq

2, i.e. an increase of qacts in the same way as an increase of temperature. Withthis remark we can apply the considered above results forzero magnetic field.

To demonstrate the induced transition from a BTRSto a TRS state by an external magnetic field we considera particular case of equal ai = a, bi = b, but keeping theκi different and γ12 = γ23 = −γ13 = γ. At zero q a dou-bly degenerate BTRS state with φ = 2π − θ = 5π/3 andφ = 2π − θ = π/3 is realized. With increase q first at qc,which is the solution of the equation cosφc, θc = 1, we gettransitions to TRS states (see Supplement). In the TRSstate the minimization of the GL energy can not be doneanalytically. In this case a numerical procedure must beapplied. The dependence of the GL free energy Eq.(6) onthe applied magnetic flux for different ratios of κi is pre-sented in Fig. 5. We track the evolution with magneticflux of one of the ground states, namely for φ = 5π/3,θ = π/3. The full procedure can be found in the Supple-ment. We find that for a given value of γ the ground state

0 0.5 1 1.5 2 2.5−6

−5

−4

−3

−2

−1

Φ/Φ0

G/π

R2 L

(a)

l0

l1 l2l2’l1’ l3

0 0.5 1 1.5 2 2.5−6

−5

−4

−3

−2

−1

Φ/Φ0

G/π

R2 L

(b)

l0l1

l1’

l2 l3l2’

FIG. 4. Dependencies of the GL free energy on the appliedmagnetic flux for a three-band superconductor with γ = 1for attractive interband interactions between the first and thesecond, the first and the third bands and a repulsive one be-tween the second and the third bands, and for κ2 = 4 andκ3 = 2 (a) and κ2 = 0.25 and κ3 = 0.5 (b). Solid and dashedlines: GL free energies of a three-band superconductor withBTRS and without one, respectively, for different windingnumbers n = 0 (brown), n = 1 (red), n = 2 (blue), n = 3(green) etc. The filled circle markers with captions determinepossible ways of evolution for the three-band superconductinglong tube setup shown in Fig. 2 in an external magnetic field(for explanations, see text).

of a three-band superconductor under consideration ex-hibits always a BTRS. This means that despite the valueof the trapped flux, by increasing the flux we will movealong the bottom part of the solid curves (Fig.4), follow-ing the ”route” l0− l1− l2− ... But for an non-adiabatic,fast switched on magnetic flux, the three-band supercon-ducting system can be excited and can be flipped to a

5

κ2

κ 3

2 4 6 8 10 12 14

2

4

6

8

10

12

14

0 0.5 1 1.5 2 2.5−6

−5

−4

−3

−2

Φ/Φ0

G/π

R2 L

adiabatictransitions from

BTRS to non−BTRS

non−adiabatictransitions from

BTRS to non−BTRS

FIG. 5. Phase diagram of a three-band SC, where transitionsfrom BTRS to TRS can occur due to an excitation by anexternal magnetic flux in the experimental setup shown inFig. 2] and without any excitation along an adiabatic path ofchanging the GS (pink region). The inset shows the evolutionof the GL free energy in dependence on the applied magneticflux for a three-band superconductor with γ = 1 for attractiveinterband interactions between the first and the second bands,the first and the third bands and a repulsive one between thesecond and the third bands, and for κ2 = 15 and κ3 = 1.5.Black circles: critical (final) points for BTRS states (solidlines).

metastable states with TRS. For instance, the previousground state ”route” l0− l1− l2− l3− ... can be replacedby the path l0 − l1 − l1′ − l2′ − l2 − l3 − ..., where thedashed part l1′− l2′ corresponds to the mentioned abovemetastable state with TRS of the three-band supercon-ducting tube, or to a more complicated “route”, whichwill involve more excited states with TRS. Also we foundthat if qc < 1/2 then the transitions between BTRS andTRS states can occur without any excitation by an ex-ternal magnetic flux. This means that solid (BTRS) anddashed (TRS) lines cross before Φ/Φ0 = 1/2 (see inset inFig. 5). The phase diagram for a three-band supercon-ductor, which determines the intervals of the parametersκi for the transitions from a BTRS to a TRS state withan excitation and without one is given in Fig. 5.

Further numerical examination give that, if κi > 1(i ≥ 2) , a non-adiabatic switching on the increase of themagnetic flux can lead to a transformation of a three-band SC with a BTRS GS into an excited state with TRSand an s+++ order parameter (see Fig. 6a and 6c) andthen it relaxes again to a BTRS GS. If one or both κi < 1(i ≥ 2), then the increasing magnetic flux can transforma three-band SC with BTRS into an s+± three-band SCand finally again to a BTRS state (see Fig. 6b).

From the experimental point of view transitions fromBTRS state to TRS ones and vice versa can be detectedfollowing the response of the current density on an ap-plied magnetic flux (see Fig. 7). We revealed appropri-ate jumps on the j (Φ/Φ0) dependencies (see Figs. 7b,7d and 7e) induced by these transitions. Since in thepresent geometry changing of the magnetic field is equiv-alent to changing temperature, we expect also specialfeatures for the specific heat related to these BTRS to

FIG. 6. (Color online) Possible evolution of the phase differ-ences φ (blue) and θ (green) for a three-band superconductorwith non-adiabatic (a, b) and adiabatic (c) transitions be-tween BTRS and TRS states. The parameters are κ2 = 4and κ3 = 2 (a), κ2 = 0.25 and κ3 = 0.5 (b) and κ2 = 15 andκ3 = 1.5 (c).

0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Φ/Φ0

| j |

(a)

0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Φ/Φ0

| j |

(b)

0 0.5 1 1.5 2 2.5

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

Φ/Φ0

| j |

(c)

0 0.5 1 1.5 2 2.5

0.5

1

1.5

2

Φ/Φ0

| j |

(d)

0 0.5 1 1.5 2 2.5

1

2

3

4

5

Φ/Φ0

| j |

(e)

FIG. 7. (Color online) Current densities vs. the applied mag-netic flux in the setup shown in Fig. 3 made employing athree-band SC with γ12 = 1, γ23 = −1, γ13 = 1 (black line)and with γ12 = 1.1, γ23 = −1, γ13 = 1.2 (blue line) and forκ2 = 4 and κ3 = 2 (a, b) κ2 = 0.25 and κ3 = 0.5 (c, d) andκ2 = 15 and κ3 = 1.5 (e). The curves (a) and (c) correspondto a three-band SC without transitions between BTRS andTRS states. The plots (b), (d) and (e) are for a three-bandSC with transitions between BTRS and TRS states with andwithout excitation by an external magnetic field, respectively.

6

TRS transition. Hence accompanying thermodynamicmeasurements might provide further support for identi-fication of the BTRS states.

IV. DISCUSSIONS AND CONCLUSIONS

Based on these analytical and numerical calculationsit is natural to suggest that such a behavior remains ona qualitative level the same also for other possible setsof interband interaction coefficients which admit the ex-istence of frustrated states in the equilibrium state (seesupplemental materials).44 In other words jumps in thecurrent dependencies in the magnetic flux driven regimecan be expected for any three-band superconductor witha primordial (before switching on a magnetic field) BTRSstate. Moreover with some restrictions it is reasonable toexpect the same behavior also for other BTRS multi-band superconductors, whose electronic structure andphysical properties are described by more than three or-der parameters and where frustrated states are globalground states. Restrictions of the application of suchmethod are connected with the special case of multi-bandsuperconductors with even number of bands and all equalrepulsive interband interactions, where BTRS and TRSstates have the same energy.45–47 We believe that thepresence of such jumps can be considered as an experi-mental proof for the detection of BTRS and frustrationin unconventional three- and multi-band superconduc-tors. It is important to note that the detection methodproposed here compares favorably with surface-sensitivetechniques (interference or proximity based contacts) forthe detection of properties related to the symmetry ofthe order parameter, because it probes the entire volumeof the superconductor under examination. Based on ourresults we propose to detect the presence of frustrationand BTRS in experiments with mesoscopic thin rings ortubes made from unconventional three-band supercon-ductors, by measuring a generic current response on theapplied magnetic flux.

It should be noted that currently the exact location ofthe soliton states on the energetic scale of a three-bandsuperconductor is not known. Knowledge of all possibletopological defects and their energies in case of three-and other multi-band superconductors is very importantfor the detection of the BTRS phenomenon in order todistinguish the jumps, connected with the presence ofBTRS to TRS transitions and from the transitions from

a BTRS ground state to excited soliton states. If theenergy of phase-inhomogeneous solutions is higher thanthe BTRS and TRS states then during the excitation onecan in principle observe additional jumps on the current-magnetic flux dependencies due to relaxation processesfrom higher energetic levels (soliton states) to the groundstate via metastable TRS states. Another situation is re-alized for solitons, whose energy is within the intervalbetween BTRS and TRS states. In this case during theexcitation process a three-band superconductor can bepromoted to a TRS state as an intermediate state andthen relax to the ground state via other intermediatestates of solitonic nature. So also in this case additionaljumps will also appear on the experimental dependen-cies. The last possibility can occur if the BTRS stateis not a globally stable state and the ground state ofa three-band superconductor already contains solitons.The realization of such a scenario was predicted recentlyfor a three-band superconductor48 based on non-rigorousstability considerations of phase kinks for an infinitelyextended superconducting system. To the best of ourknowledge a study of topological defects in three-bandsuperconductors for a doubly-connected finite supercon-ducting system as considered here is still lacking. And itis not clear whether such solitons in a restricted geome-try can also occur as globally stable phenomena. We willstudy this interesting but complex problem in more detailin the future. Also the cases of imperfect three-band su-perconductors with impurities as well as inhomogeneousstates due to the presence of solitonic nonlinear excita-tions mentioned above requires a special analysis outsideof the scope of the present paper and be left for futurestudy.

V. ACKNOWLEDGMENT

We acknowledge M. Kiselev, O. Dolgov, M. Zhito-mirsky, N.M. Plakida J. Schmalian, and A. Chubukovfor helpful and critical discussions. Y.Y. thanks the ITFat the IFW-Dresden for hospitality and financial supportwhere large parts of the present work has been performed.D.E. and S.-L.D. acknowledge the VW-foundation forpartial support within the trilateral grant ”Synthesis,theoretical examination and experimental investigation ofemergent iron-based superconductors”. This work wassupported by the DFG-RSF grant and SFB1143. Y.Y. isgrateful for support by the Russian Scientific FoundationGrant No 15-12-10020.

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