Time reversal symmetry breakdown in normal and superconducting states in frustrated three-band systems
TTime reversal symmetry breakdown in normal and superconducting states infrustrated three-band systems
Troels Arnfred Bojesen, Egor Babaev,
2, 3 and Asle Sudbø Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway Department of Theoretical Physics, The Royal Institute of Technology, 10691 Stockholm, Sweden Physics Department, University of Massachusetts, Amherst, Massachusetts 01003, USA (Dated: October 29, 2018)We discuss the phase diagram and phase transitions in U (1) × Z 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 Z time reversal symmetry. By contrast, for weak frustrationof interband coupling the energy of a domain wall between different Z states is low and thusfluctuations restore broken time reversal symmetry in the superconducting state at low temperatures. In recent years, the discovery of superconductors suchas the Iron Pnictides , has generated much interest formultiband superconducting systems. From a theoreti-cal viewpoint, one of the main reasons for the stronginterest is that in contrast to previously known two-bandmaterials, iron-based superconductors may exhibit dra-matically different physics due to the possibility of frus-trated inter-band Josephson couplings originating withmore than two bands crossing the Fermi-surface . Intwo-band superconductors the Josephson coupling locksthe phase differences between the bands to 0 or π . Bycontrast, if one has three bands and the frustration of in-terband coupling is sufficiently strong, the ground stateinterband phase difference can be different from 0 or π .This leads to a superconducting state which breaks timereversal symmetry (BTRS) . From a symmetry view-point such a ground state breaks U(1) × Z . Recently,such a scenario has received solid theoretical support inconnection with hole-doped Ba − x K x Fe As . The pos-sibility of this new physics arising also in other classesof materials is currently under investigation . For otherscenarios of time reversal symmetry breakdown in iron-based superconductors discussed in the literature, see .Three band superconductors with frustrated inter-band Josephson couplings feature several properties thatare radically different from their two-band counterparts.These include (I) the appearance of a massless so-calledLeggett mode at the Z phase transition ; (II) the exis-tence of new mixed phase-density collective modes in thestate with broken time-reversal symmetry (BTRS) in contrast to the “phase-only” Leggett collective modein two-band materials ; and (III) the existence of(meta-)stable excitations characterized by CP topologi-cal invariants .So far the phase diagram of frustrated three-band su-perconductors has been investigated only at the mean-field level . However, the iron-based materials featurerelatively high T c , as well as being far from the type-I regime. Hence, one may expect fluctuations to be ofimportance.In this paper, we study the phase diagram of a three-band superconductor in two spatial dimensions in theLondon limit, beyond mean-field approximation. Theresults should apply to relatively thin films of iron-based superconductors where, owing to low dimension-ality, fluctuation effects are particularly important. Themain findings of this work are as follows. (I) Whenthe frustration is sufficiently strong, the phase diagramacquires an unusual fluctuation-induced metallic statewhich is a precursor to the BTRS superconducting phase.This metallic state exhibits a broken Z time-reversalsymmetry. A salient feature is that, although the stateis metallic and non-superconducting, it nevertheless fea-tures a persistent interband Josephson current in mo-mentum space which breaks time reversal symmetry. (II)When the frustration is weak (i.e. when phase differencesare only slightly different from 0 or π ) we find that thesystem can undergo a fluctuation driven restoration ofthe Z symmetry at very low temperatures.The London model for a three-band superconductor isgiven by F = (cid:88) α =1 , , | ψ α | ∇ θ α − e A ) + (cid:88) α,α (cid:48) >α η αα (cid:48) | ψ α || ψ α (cid:48) | cos( θ α − θ α (cid:48) ) + 12 ( ∇ × A ) . (1)Here, | ψ α | e i θ α denotes the superconducting condensatecomponents in different bands labeled by α = 1 , ,
3, while the second term represents interband Josephsoncouplings. The field A is the magnetic vector potential a r X i v : . [ c ond - m a t . s up r- c on ] O c t that couples minimally to the charged condensate matter fields. By collecting gradient terms for phase differences,it can also be rewritten as F = 12 (cid:37) (cid:32)(cid:88) α | ψ α | ∇ θ α − e(cid:37) A (cid:33) + 12 ( ∇ × A ) + (cid:88) α,α (cid:48) >α | ψ α | | ψ α (cid:48) | (cid:37) [ ∇ ( θ α − θ α (cid:48) )] + η αα (cid:48) | ψ α || ψ α (cid:48) | cos( θ α − θ α (cid:48) ) , (2)where (cid:37) = (cid:80) α | ψ α | . 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 η αα (cid:48) are positive, eachJosephson term by itself prefers to lock phase differenceto π , i.e. θ α − θ α (cid:48) = 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 ψ α . Thus, by choosing one of thesephase locking patterns the system breaks time reversalsymmetry .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 Z 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 2 π , i.e. ∆ θ ≡ (cid:72) ∇ θ = 2 π, ∆ θ =2 π, ∆ θ = 2 π . 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 Z sector. In addition, the model featuresother topological defects discussed in detail in Refs. 12and 13. These are Z domain walls (several solutionswith different energies), fractional-flux vortices with lin-early divergent energy, as well as CP skyrmions whichare combined vortex-domain wall defects carrying topo-logical charges in both the U(1) and Z sectors of themodel. This spectrum of topological excitations distin-guishes this model from other U(1) × Z systems, like e.g. XY -Ising model . The model is also principally differentfrom [U(1)] superconductors, since in such systems frac-tional vortices have logarithmically divergent energy andthus drive BKT phase transitions . The difference be-tween a Z -ordered and disordered state is illustrated inFig. 2.In two dimensions, the effective magnetic field pen-etration length is inversely proportional to the filmthickness . We thus begin by discussing the limit ofvery large penetration length, in which we may neglectthe coupling to the vector potential. We discuss the phase diagram of the model in the case of a finite penetrationlength in our summary.The partition function of the lattice version of themodel (1) reads Z = (cid:89) α,i (cid:20)(cid:90) π − π d θ α,i π (cid:21) exp( − βH ) , (3)where the Hamiltonian is given by H = − (cid:88) (cid:104) i,j (cid:105) ,α cos ( θ α,i − θ α,j )+ (cid:88) i,α (cid:48) >α g αα (cid:48) cos ( θ α,i − θ α (cid:48) ,i ) . (4) i, j ∈ { , , . . . , N = L × L } denote sites on a lattice ofsize L × L and (cid:104) i, j (cid:105) indicates nearest neighbor latticesites (assuming periodic boundary conditions). β is the(properly rescaled) coupling (“inverse temperature”) and g αα (cid:48) are interband Josephson couplings. Here, we con-sider the case of similar prefactors for the three gradientterms.Algebraically decaying correlations and frustrationeffects typically render two dimensional U(1) × Z -symmetric models difficult to investigate numerically .In this work we emply a non-equilibrium approach,namely that of short time critical dynamics (STCD). Seee.g. the review articles 19 and 20, and references therein.See online supplementary material for details First, we consider the case g = g = g = g , whichis shown in Fig. 3. The phase transitions for the Z and U(1) symmetries are close, but clearly separated forall values of g . This means that beyond the mean-fieldapproximation there appears a new phase. As the tem-perature increases from the low-temperature maximallyordered phase, an unbinding of vortex-antivortex pairs ofcomposite vortices first takes place. In the resulting statethe free composite vortices (1 , ,
1) and ( − , − , −
1) donot further decompose into (1 , , , , , ,
1) frac-tional vortices, because Josephson coupling provides lin-ear confinement of the constituent fractional vortices ).Due to this confinement, the proliferation of (1 , ,
1) and( − , − , −
1) vortex-antivortex pairs disorders only theU(1) sector of the model described by the first term inEq. (2). However, these defects do not restore Z sym-metry. This results in a formation of a new state whichis non-superconducting but breaks broken time-reversal θ θ θ (a) Phases of the fields.(b) +1 (c) − FIG. 1. (Colors online) (b) and (c) are examples of phaseconfigurations for the two Z symmetry classes of the groundstates, shown at a 2 × xy plane. Here g > g > g >
0. The spatial contributionto the energy is minimized by making the spatial gradientzero (hence breaking the global U(1) symmetry). Then thereare two classes of phase configurations, one with chirality +1and one with chirality -1, minimizing the energy associatedwith the interband interaction. The chirality is defined as +1if the phases (modulo 2 π ) are cyclically ordered θ < θ <θ , and − −→ , −→ , −→ ) correspond to( θ , θ , θ ), as shown in (a). symmetry. The effective model which describes this newstate is given by the last terms in Eq. (2), F Z = (cid:88) α,α (cid:48) >α (cid:110) | ψ α | | ψ α (cid:48) | (cid:37) [ ∇ ( θ α − θ α (cid:48) )] + η αα (cid:48) | ψ α || ψ α (cid:48) | cos( θ α − θ α (cid:48) ) (cid:111) (5)Secondly, at higher temperatures the Z 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 Z 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 Z -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, see Fig. 2. (a) A Z broken, U(1)symmetric configurationwith +1 chirality. (b) A Z 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. In (a) the Z symmetry is broken (and of +1 chirality) since the phasesof all lattice sites have the same cyclic ordering. There ishowever no spatial ordering of the phases, hence the configu-ration is U(1) symmetric. In (b) neither the Z nor the U(1)symmetry is broken. The arrows ( −→ , −→ , −→ ) correspondto ( θ , θ , θ ), as shown in Fig. 1a.0 5 10 15 20 250 . . . . . . Z symmetry. Ordi-nary U(1) symmetry. Z symmetry bro-ken. Algebraicallydecaying U(1)-phase. Z symmetry bro-ken. OrdinaryU(1) symmetry. g β β Z β U(1)
FIG. 3. (Color online) Phase diagram for the three-bandmodel with g = g = g = g . g ∈ [1 , . . . , β U(1) line lies above the β Z line for the investigated values of g .Error bars are smaller than symbol sizes. Lines are guides tothe eye. 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 Z domain walls can be made arbitrarily small.Thus, one can interchange critical temperatures of U(1)and Z phase transitions. Moreover, inclusion of fluctu-ations can in a certain limit dramatically suppress thecritical temperature of the Z 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 field cou-ples only to the U(1) sector of the model, making the U(1)symmetry local. It also makes the energy of (1 , ,
1) and( − , − , −
1) composite vortices finite . As a result, atany finite temperature, there is a finite probability of ex-citing 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, thisphysics manifests itself as a conversion of a BKT tran-sition to a broad crossover which takes place at lowercharacteristic temperatures than the U(1) phase transi-tion in the global U(1) × Z model. On the other hand,since the Z phase transition is not directly affected bythis coupling, the Z ordered non-superconducting statepersists. Thus, in the thermodynamic limit a supercon-ducting system with finite penetration length featuresU(1) × Z superconductivity at zero temperature, whileat any nonzero temperature it resides in a Z metallicstate, up to the temperature where the Z -symmetry isrestored. In other words, a finite penetration length in-creases the phase space of a metallic state with brokentime-reversal symmetry. The arguments above carry overto three dimensions as well. . . . . . . β β Z β U(1) . . π π (b) θ g θ FIG. 4. (a) The phase diagram for the three-band model withunequal Josephson couplings. We set g = g = 15 andvaried g . Error bars are smaller than symbol sizes. Linesare guides to the eye. (b) The phase difference θ betweenband 2 and 3 in the ground state, as defined in the phasevector inset. θ = 0 for g < . θ = θ = (2 π − θ ) / g . 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-inducednon-superconducting state which also exhibits a sponta-neously broken time reversal symmetry, associated withpersistent interband Josephson currents in k -space. Thisstate is distinct from an ordinary metallic state wherethere is no such broken symmetry. Experimentally, itcan be distinguished from superconducting and ordi-nary normal states by a combination of local (e.g. tun-neling) and transport measurements. Another way ofpossibly detecting this state would be by observing anOnsager anomaly in the specific heat in the normalstate. These predictions could also be used to verify ifBa − x K x Fe As breaks time reversal symmetry at cer-tain doping. A related phase should also exist in othersuperconductors which break time reversal symmetry ,as well as in interacting multicomponent Bose conden-sates. ACKNOWLEDGMENTS
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