Time reversal symmetry breakdown in normal and superconducting states infrustrated three-band systems

Troels Arnfred Bojesen,1, ∗ Egor Babaev,2, 3 and Asle Sudbø1

1Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway2Physics Department, University of Massachusetts, Amherst, Massachusetts 01003, USA

3Department of Theoretical Physics, The Royal Institute of Technology, 10691 Stockholm, Sweden(Dated: June 12, 2013)

We discuss the phase diagram and phase transitions in U(1)×Z2 three-band superconductors withbroken time reversal symmetry. We find that beyond mean field approximation and for sufficientlystrong frustration of interband interactions there appears an unusual metallic state precursory to asuperconducting phase transition. In that state, the system is not superconducting. Nonetheless,it features a spontaneously broken Z2 time reversal symmetry. By contrast, for weak frustrationof interband coupling the energy of a domain wall between different Z2 states is low and thusfluctuations restore broken time reversal symmetry in the superconducting state at low temperatures.

In recent years, the discovery of multiband supercon-ductors such as the Iron Pnictides [1], has generatedmuch interest for multiband superconducting systems.From a theoretical viewpoint, one of the main reasons forthe strong interest is that in contrast to previously knowntwo-band materials, iron-based superconductors may ex-hibit dramatically different physics due to the possibilityof frustrated inter-band Josephson coupling originatingwith more than two bands crossing the Fermi-surface.[2–5]. In two-band superconductors the Josephson couplinglocks the phase differences between the bands to 0 or π.By contrast, if one has three bands and the frustrationof interband coupling is sufficiently strong, the groundstate interband phase difference can be different from 0or π. This leads to a superconducting state which breakstime reversal symmetry (BTRS) [2, 3]. From a symmetryviewpoint such a ground state breaks U(1)× Z2 [4]. Re-cently, such a scenario has received solid theoretical sup-port [5] in connection with hole-doped Ba1− xKxFe2As2.The possibility of this new physics arising also in otherclasses of materials is currently under investigation [6].For other scenarios of time reversal symmetry breakdownin iron-based superconductors discussed in the literature,see [7, 8].

Three band superconductors with frustrated interbandJosephson couplings have a number of properties radi-cally different from their two-band counterparts. Theseinclude (I) the appearance of a massless so-called Leggettmode at the Z2 phase transition [9]; (II) the existence ofnew mixed phase-density collective modes in the statewith broken time-reversal symmetry (BTRS) [4, 5, 10] in

contrast to the “phase-only” Leggett collective mode intwo-band materials [11]; and (III) the existence of (meta-)stable excitations characterized by CP2 topological in-variants [12, 13].

So far the phase diagram of frustrated three-band su-perconductors has been investigated only at the mean-field level [3, 5]. However, the iron-based materials fea-ture relatively high Tc, as well as being far from the type-Iregime. Hence, one may expect fluctuations to be of im-portance.

In this paper, we study the phase diagram of a three-band superconductor in two-dimensions in the Londonlimit, beyond mean-field approximation. The resultsshould apply to relatively thin films of iron-based super-conductors where, owing to low dimensionality, fluctua-tion effects are particularly important. The main find-ings of this work are as follows. (I) When the frustrationis sufficiently strong, the phase diagram acquires an un-usual fluctuation-induced metallic state which is a pre-cursor to the BTRS superconducting phase. This metal-lic state exhibits a broken Z2 time-reversal symmetry. Asalient feature is that, although the state is metallic andnon-superconducting, it nevertheless features a persistentinterband Josephson current in momentum space whichbreaks time reversal symmetry. (II) When the frustrationis weak (i.e. when phase differences are only slightly dif-ferent from 0 or π) we find that the system can undergoa fluctuation driven restoration of the Z2 symmetry atvery low temperatures.

The London model for a three-band superconductor isgiven by

F =∑

α=1,2,3

|ψα|2

2(∇θα − eA)2 +

∑α,α′>α

ηαα′ |ψα||ψα′ | cos(θα − θα′) +1

2(∇×A)2. (1)

Here, |ψα| eiθα denotes the superconducting condensate components in different bands labeled by α = 1, 2, 3,

arX

iv:1

306.

2313

v1 [

cond

-mat

.sup

r-co

n] 1

0 Ju

n 20

13

2

while the second term represents interband Josephsoncouplings. The field A is the magnetic vector potential

that couples minimally to the charged condensate matterfields. By collecting gradient terms for phase differences,it can also be rewritten as

F =1

2%2

∑α

(|ψα|2∇θα − e|ψα|2A

)2+

1

2(∇×A)2 +

∑α,α′>α

|ψα|2|ψα′ |2

2%2[∇(θα−θα′)]2 +ηαα′ |ψα||ψα′ | cos(θα−θα′), (2)

where %2 =∑α |ψα|2. This shows that the vector poten-

tial is coupled only to the U(1) sector of the model, andnot to phase differences.

When the Josephson couplings ηαα′ are positive, eachJosephson term by itself prefers to lock phase differenceto π, i.e. θα−θα′ = π. Since this is not possible for threephases, the system is frustrated. The system breaks timereversal symmetry when Josephson couplings are mini-mized by two inequivalent phase lockings, shown in Fig.1. The phase lockings are related by complex conjuga-tion of the fields ψα. Thus, by choosing one of thesephase locking patterns the system breaks time reversalsymmetry [2–4].

In this work, we address the phase transitions in atwo dimensional three-band superconductor with bro-ken time-reversal symmetry. A Berezinskii-Kosterlitz-Thouless (BKT) phase transition in U(1) systems isdriven by proliferation of vortex-antivortex pairs, whilean Ising phase transition is driven by proliferation of Z2

domain walls. The nontriviality of the problem of phasetransitions in the three-band model is due to the spec-trum of topological excitations of the model. Firstly, themodel features singly-quantized composite vortices whereall phases wind by 2π, i.e. ∆θ1 ≡

∮∇θ1 = 2π,∆θ2 =

2π,∆θ3 = 2π. We will denote them (1,1,1). As is clearfrom Eq. (2), such a vortex has topological charge onlyin the U(1) sector of the model. It has no phase windingin the phase differences and thus does not carry topolog-ical charge in Z2 sector. In addition, the model featuresother topological defects discussed in detail in Refs. 12and 13. These are Z2 domain walls (several solutionswith different energies), fractional-flux vortices with lin-early divergent energy, as well as CP 2 skyrmions whichare combined vortex-domain wall defects carrying topo-logical charges in both the U(1) and Z2 sectors of themodel. This spectrum of topological excitations distin-guishes this model from other U(1)×Z2 systems, like e.g.XY -Ising model [14]. The model is also principally differ-ent from [U(1)]3 superconductors, since in such systemsfractional vortices have logarithmically divergent energyand thus drive BKT phase transitions [15, 16].

In two dimensions the effective magnetic field penetra-tion length is inversely proportional to the film thickness[17]. We thus begin by discussing the limit of very largepenetration length, in which we may neglect the coupling

to the vector potential. We discuss the phase diagram ofthe model in the case of a finite penetration length in oursummary.

The partition function of the lattice version of themodel (1) reads

Z =∏α,i

[∫ π

−π

dθα,i2π

]exp(−βH), (3)

where the Hamiltonian is given by

H = −∑〈i,j〉,α

cos (θα,i − θα,j)

+∑i,α′>α

gαα′ cos (θα,i − θα′,i) . (4)

i, j ∈ { 1, 2, . . . , N = L× L } denote sites on a lattice ofsize L × L and 〈i, j〉 indicates nearest neighbor latticesites (assuming periodic boundary conditions). β is the(properly rescaled) coupling (“inverse temperature”) andgαα′ are interband Josephson couplings. Here, we con-sider the case of similar prefactors for the three gradientterms.

(a) +1 (b) −1

FIG. 1. (Color online) Examples of phase configurations forthe two Z2 symmetry classes of the ground states on sitesof 2 × 2 lattice. Here g12 > g23 > g13 > 0. The arrows(−→,−→,−→) corresponds to (θ1, θ2, θ3).

Algebraically decaying correlations and frustrationeffects typically render two dimensional U(1) × Z2-symmetric models difficult to investigate numericallythrough equilibrium Monte Carlo (EMC) simulations[18]. In this work we used a non-equilibrium approach,namely that of short time critical dynamics (STCD). See

3

e.g. the review articles 19 and 20, and references therein.See online supplementary material for details.

First, we consider the case g12 = g23 = g31 = g,which is shown in Fig. 3. The phase transitions for theZ2 and U(1) symmetries are close, but clearly separatedfor all values of g. This means that beyond the mean-field approximation there appears a new phase. As thetemperature increases from the low-temperature maxi-mally ordered phase, an unbinding of vortex-antivortexpairs of composite vortices first takes place. In theresulting state the free composite vortices (1, 1, 1) and(−1,−1,−1) do not further decompose into fractionalvortices (1, 0, 0), (0, 1, 0), (0, 0, 1) because Josephson cou-pling provides linear confinement of the constituent frac-tional vortices [13]). Due to this confinement, the pro-liferation of (1, 1, 1) and (−1,−1,−1) vortex-antivortexpairs disorders only the U(1) sector of the model de-scribed by the first term in Eq. (2). However, thesedefects do not restore Z2 symmetry. The resulting stateis non-superconducting with broken time-reversal sym-metry. This sector is described by the last terms in Eq.(2),

FZ2=

∑α,α′>α

{ |ψα|2|ψα′ |2

2%2[∇(θα − θα′)]2

+ ηαα′ |ψα||ψα′ | cos(θα − θα′)}

(5)

Secondly, at higher temperatures the Z2 domain wallsproliferate and restore the symmetry completely. Thephysical interpretation of this precursor normal statewith broken time reversal symmetry is as follows. Inthe BTRS superconducting state there is a ground statephase difference other than 0 or π between components.This implies the existence of persistent interband Joseph-son currents. Two different Z2 phase locking patternsmean that there are two inequivalent interband Joseph-son current “loops in k-space”. Namely, one loop is of thetype band 1 → band 2 → band 3 → band 1, the otheris of the type band 1 → band 3 → band 2 → band 1.The non-superconducting Z2-ordered phase correspondsto the situation where superconducting phases exhibitexponentially decaying correlations due to proliferationof vortex-antivortex pairs. What sets this state apartfrom the situation found in conventional superconduc-tors is that the three-band system retains a persistentinterband Josephson current in k-space which breaks thetime reversal symmetry. The phase configurations in thisstate are illustrated in Fig. 2.

Next, we consider the case of a more general modelwhere the Josephson couplings are different. By tuningsome of the Josephson couplings one can make the dif-ference between two out of three phases arbitrarily smallin the BTRS ground state. This also implies that theenergy of Z2 domain walls can be made arbitrarily small.Thus, one can interchange critical temperatures of U(1)

(a) A Z2 broken, U(1)symmetric configuration

with +1 chirality.

(b) A Z2 and U(1)symmetric configuration.

FIG. 2. (Colors online) A schematic illustration of phaseconfigurations in the normal state which break time rever-sal symmetry vs the normal state which does not. Here,g12 > g23 > g13 > 0. The arrows (−→,−→,−→) correspondto (θ1, θ2, θ3).

0 5 10 15 20 250.5

0.55

0.6

0.65

0.7

0.75

Z2 symmetry. Ordi-nary U(1) symmetry.

Z2 symmetry bro-ken. Algebraically

decaying U(1)-phase.

Z2 symmetry bro-ken. OrdinaryU(1) symmetry.

g

β

βZ2 βU(1)

FIG. 3. (Color online) Phase diagram for the three-bandmodel with g12 = g23 = g31 = g. g ∈ [1, . . . , 25]. The βU(1)

line lies above the βZ2 line for the investigated values of g.Error bars are smaller than symbol sizes. Lines are guides tothe eye.

and Z2 phase transitions. Moreover, inclusion of fluctu-ations can in a certain limit dramatically suppress thecritical temperature of the Z2 phase transition. Resultsfrom Monte-Carlo simulations shown in Fig. 4 displaysuch behavior.

Finally, consider the effect of a finite penetrationlength. As can be seen from Eq. (2), the gauge fieldcouples only to the U(1) sector of the model, makingthe U(1) symmetry local. It also makes the energy of(1, 1, 1) and (−1,−1,−1) finite [13]. As a result, at anyfinite temperature, there is a finite probability of excit-ing such topological defects, which from a formal view-point suppresses superconductivity at finite temperaturein the thermodynamic limit. In a real experiment on a fi-nite system, with large but finite penetration length, this

4

physics manifests itself as a conversion of a BKT transi-tion to a broad crossover which takes place at lower char-acteristic temperatures than the U(1) phase transition inthe global U(1)×Z2 model. Since, on the other hand Z2

phase transition is not directly affected by this coupling,the Z2 ordered non-superconducting state persists. Thus,in the thermodynamic limit a system with finite penetra-tion length features U(1)× Z2 superconductivity at zerotemperature, while at any nonzero temperature it residesin a Z2 metallic state, up to the temperature where theZ2-symmetry is restored.

0.45

0.5

0.55

0.6

0.65

0.7

(a)

β

βZ2

βU(1)

7.5 10 12.5 15

0

π3

2π3

(b)

θ023

g23

θ0 23

FIG. 4. (a) The phase diagram for the three-band model withunequal Josephson couplings. We set g12 = g13 = 15 andvaried g23. Error bars are smaller than symbol sizes. Linesare guides to the eye. (b) The phase difference θ023 betweenband 2 and 3 in the ground state, as defined in the phasevector inset. θ023 = 0 for g23 < 7.5. θ012 = θ013 = (2π − θ023)/2for all g23.

In conclusion, we have studied the phase diagramof three band superconductors with spontaneously bro-ken time reversal symmetry due to frustrated interbandJosephson-couplings, beyond mean field approximation.We have found that there is a new fluctuation-inducedU(1)-symmetric state, which is a precursor to the su-perconducting state. However, it exhibits a broken Z2

(Ising) symmetry. This state is therefore distinct froman ordinary metallic state, due to spontaneously brokentime reversal symmetry associated with persistent inter-band Josephson currents in k-space. Experimentally, itcan be distinguished from superconducting and ordinarynormal states by a combination of local (e.g. tunnel-

ing) and transport measurements (with later showing nosuperconductivity). Another way of possibly detectingthis state would be by observing an Onsager anomalyin the specific heat in the normal state. Besides thatthe existence of broken Z2 symmetry in the normal stateimplies the existence of mixed “phase difference-density”collective modes previously discussed in the supercon-ducting state [4, 5, 10]. For interband coupling valueswhere these modes exist, they may thus be observed ina similar way as the Leggett mode [21]. On the otherhand, in the regime where a ground state phase differ-ences are very close to 0 or π, we find that fluctuationscan restore the Z2 symmetry at substantially lower tem-peratures than those which are predicted by mean-fieldtheory. These predictions could be used also to verify ifBa1− xKxFe2As2 breaks time reversal symmetry at cer-tain doping (which as recently discussed in [5] is a strongcandidate for a BTRS superconductor).

We thank J. Carlstrom and E. V. Herland for fruit-ful discussions and feedback. T.A.B. thanks NTNUfor financial support, and the Norwegian consortiumfor high-performance computing (NOTUR) for com-puter time and technical support. A.S. was supportedby the Research Council of Norway, through Grants205591/V20 and 216700/F20. E.B. was supported byKnut and Alice Wallenberg Foundation through theRoyal Swedish Academy of Sciences Fellowship, SwedishResearch Council and by the National Science Founda-tion CAREER Award No. DMR-0955902.

Online supplementary material: Details about thesimulations in the paper “Frustration, time-reversalsymmetry breaking, and unusual metallic states in a

three-band superconductor”.

We used short time critical dynamics (STCD) to in-vestigate the three band model. We briefly describe theprocedure here, and refer to the review articles 19 and 20and references therein for more details on this method.

The STCD method is based on the observation thatwhen a system, initially prepared in either the groundstate or in a random state, is released under a MonteCarlo (MC) dynamics of local updates, the MC time tevolution of the mean of a local observable will be verysensitive to the coupling constant of the dynamics. Typ-ically (and this is what we have done here) one looksat some kind of magnetization m(t) when heating theground state, as this tend to give more accurate resultsthan cooling down a random (“infinite temperature”) ini-tial state. When the coupling differs from the criticalcoupling βc (which turns out to be the same as the equi-librium critical coupling), the magnetization goes expo-nentially to either zero or a constant value as t → ∞,depending on whether the coupling is sub- or supercrit-ical. If the coupling is critical, m(t) ∼ ta, where a is a

5

constant which can be related to equilibrium critical con-stants. Thus, by examining the behavior of m(t) aroundβc, it is relatively straightforward to determine the criti-cal point to great precision.

In practise, the time, tmicro, after which the t → ∞behavior starts to emerge, is rather short: For simplenon-frustrated systems it is typically in the order of tensof MC sweeps; in frustrated systems somewhat longer, acouple of orders of magnitudes more. The advantage ofSTCD simulations lies in the short time scale (comparedto EMC). Thermalization sweeps are not necessary, and,given sufficiently large system sizes, the finite size correla-tion effects are exponentially damped within the simula-tion time, effectively making the system behave as in thethermodynamic limit. The last phenomenon can be un-derstood from the fact that in the short time regime, cor-relations have not had sufficient time to expand throughthe entire system. Furthermore, the dynamics being farfrom equilibrium means that the equilibrium problem ofcritical slowing down is avoided, rendering the big systemsizes needed for the longer simulation times of frustratedmodels unproblematic.

In the simulations of the three band model we haveused two “magnetizations”, one for each symmetry, aslocal order parameters. For the Z2 symmetry, we choosethe mean Ising magnetization, mZ2

≡ N−1∑i σi, as or-

der parameter. Here σi ≡ 1 if the local phases are dis-tributed as 1→ 2→ 3→ 1 (going counterclockwise), andσi ≡ −1 if they are distributed as 1 → 3 → 2 → 1. TheU(1) symmetry order parameter is chosen to be the aver-age of the absolute value of the mean global XY vectorsum of each component, mU(1) ≡ (3N)−1

∑α |∑i sα,i|,

where sα,i ≡ [cos(θα,i), sin(θα,i)]. It should be noted thatthe magnetization curves used to obtain the results areaverage values of several, independent runs.

The procedure of finding the critical coupling of Z2

symmetry, called βZ2, is rather straightforward: For a

given set of Josephson couplings, we start looking for theapproximate critical coupling by performing some testsimulations for a wide range of β-values, manually search-ing for the best power-law behavior by log-log plottingthe magnetization curves. From this it is easy to ob-tain a relatively small interval containing βZ2

. Then weperform heavier simulations (bigger system size, largermaximal MC time, tmax, more runs) at β values spreadout in this interval. Finally we use quadratic splines tointerpolate between the obtained magnetization curvesand search for the β-value which corresponds to the bestpower-law fit using nonlinear regression. We have usedthe standard deviation of the residuals as a goodness-of-fit parameter for this optimization problem.

The time correlation c∆t(t) ≡ 〈m(t)m(t+ ∆t)〉 is atthe critical point expected to follow a power law as well,c∆t ∼ tb. To counteract this, we let the time intervals be-tween the simulation samplings follow a power too, keep-ing all the points of the regression approximately equally

100 101 102 103 104

10−0.4

10−0.2

100

t / sweeps

mZ 2

FIG. 5. log-log plot of some mZ2(t)-curves for g = 1 at, frombelow, β = 0.748, 0.749, . . . , 0.754. The critical coupling isdetermined to be βZ2 ≈ 0.750 39. The error is smaller thanthe line width. tmicro ∼ O(103).

correlated.An example of mZ2

(t)-curves of a STCD simulation isshown in Fig. 5.

Finding βU(1), the BKT transition point of the U(1)symmetry, is in many aspects similar to the Z2 procedure.The main difference is that we now have to deal withthe entire low temperature phase displaying power lawbehavior in mU(1)(t), making the search for the transitioncoupling more challenging. In this work we have used atechnique from Ref. 22. We noted that in this case themethod was less numerical stable than the Z2 criticalpoint search explained above, requiring more relying onmanual initial parameter guesses. Therefore the errorbars for the U(1) line in the phase diagrams of this workare probably a bit too small, but not so much that anyqualitative conclusion could be changed.

An example of mU(1)(t)-curves of a STCD simulationis shown in Fig. 6.

A system size of L = 2048 (grid parallelized and dis-tributed over several CPUs) was used for all results; testsagainst L = 1024 and L = 4096 simulations confirmedthat this system size was sufficiently large for any finitesize effects to be negligible at the Monte Carlo time scaleused in the simulations.

In the determination of βZ2and βU(1) only data for t >

5000 sweeps was used in order to avoid the non-universalmicroscopic time regime. tmax = 30 000 sweeps was usedfor the determination of all βZ2 values, βU(1) values forthe g12 = g23 = g13 simulations and for g23 > 12.5 in theg12 = g23 6= g13 simulations. tmax = 50 000 was used forg23 < 12.5 in the g12 = g23 6= g13 simulations.

Pseudorandom numbers were generated by theMersenne-Twister algorithm [23]. Errors were deter-

6

100 101 102 103 104

10−0.6

10−0.4

10−0.2

100

t / sweeps

mU(1

)

FIG. 6. log-log plot of some mU(1)(t)-curves for g = 1 at,from below, β = 0.748, 0.749, . . . , 0.756. βU(1) ≈ 0.752 20.

The error is smaller than the line width. tmicro ∼ O(103).

mined using the jackknife method.

∗ troels.bojesen@ntnu.no[1] Y. Kamihara, T. Watanabe, M. Hirano, and H. Hosono,

J. Am. Chem. Soc. 130, 3296 (2008).[2] T. K. Ng and N. Nagaosa, Europhys. Lett. 87, 17003

(2009).[3] V. Stanev and Z. Tesanovic, Phys. Rev. B 81, 134522

(2010).[4] J. Carlstrom, J. Garaud, and E. Babaev, Phys. Rev. B

84, 134518 (2011).[5] S. Maiti and A. V. Chubukov, Phys. Rev. B 87, 144511

(2013), arXiv:1302.2964 [cond-mat.supr-con].[6] S. Mukherjee and D. F. Agterberg, Phys. Rev. B 84,

134520 (2011).

[7] W.-C. Lee, S.-C. Zhang, and C. Wu, Physical Re-view Letters 102, 217002 (2009), arXiv:0810.0887 [cond-mat.supr-con].

[8] C. Platt, R. Thomale, C. Honerkamp, S.-C. Zhang,and W. Hanke, Phys. Rev. B 85, 180502 (2012),arXiv:1106.5964 [cond-mat.supr-con].

[9] S.-Z. Lin and X. Hu, Physical Review Letters 108, 177005(2012), arXiv:1107.0814 [cond-mat.supr-con].

[10] V. Stanev, Phys. Rev. B 85, 174520 (2012),arXiv:1108.2501 [cond-mat.supr-con].

[11] A. J. Leggett, Progress of Theoretical Physics 36, 901(1966).

[12] J. Garaud, J. Carlstrom, and E. Babaev, Physical Re-view Letters 107, 197001 (2011), arXiv:1107.0995 [cond-mat.supr-con].

[13] J. Garaud, J. Carlstrom, E. Babaev, and M. Speight,Phys. Rev. B 87, 014507 (2013), arXiv:1211.4342 [cond-mat.supr-con].

[14] E. Granato, J. M. Kosterlitz, J. Lee, and M. P. Nightin-gale, Phys. Rev. Lett. 66, 1090 (1991).

[15] J. Smiseth, E. Smørgrav, E. Babaev, and A. Sudbø,Phys. Rev. B 71, 214509 (2005), arXiv:cond-mat/0411761.

[16] E. Babaev, Nuclear Physics B 686, 397 (2004),arXiv:cond-mat/0201547.

[17] J. Pearl, Applied Physics Letters 5, 65 (1964).[18] M. Hasenbusch, A. Pelissetto, and E. Vicari, Journal

of Statistical Mechanics: Theory and Experiment 2005,P12002 (2005).

[19] Y. Ozeki and N. Ito, Journal of Physics A: Mathematicaland Theoretical 40, R149 (2007).

[20] E. V. Albano, M. A. Bab, G. Baglietto, R. A. Borzi,T. S. Grigera, E. S. Loscar, D. E. Rodriguez, M. L. R.Puzzo, and G. PSaracco, Reports on Progress in Physics74, 026501 (2011).

[21] G. Blumberg, A. Mialitsin, B. S. Dennis, M. V. Klein,N. D. Zhigadlo, and J. Karpinski, Phys. Rev. Lett. 99,227002 (2007).

[22] Y. Ozeki, K. Ogawa, and N. Ito, Phys. Rev. E 67, 026702(2003).

[23] M. Matsumoto and T. Nishimura, ACM Trans. Model.Comput. Simul. 8, 3 (1998).