PHYSICAL REVIEW B 88, 214507 (2013)

Classification of ground states and normal modes for phase-frustratedmulticomponent superconductors

Daniel Weston1 and Egor Babaev1,2

1Department of Theoretical Physics, KTH Royal Institute of Technology, SE-106 91 Stockholm, Sweden2Department of Physics, University of Massachusetts, Amherst, Massachusetts 01003, USA

(Received 13 June 2013; revised manuscript received 26 September 2013; published 10 December 2013)

We classify ground states and normal modes for n-component superconductors with frustrated intercomponentJosephson couplings, focusing on n = 4. The results should be relevant not only to multiband superconductors,but also to Josephson-coupled multilayers and Josephson-junction arrays. It was recently discussed that three-component superconductors can break time-reversal symmetry as a consequence of phase frustration. We discusshow to classify frustrated superconductors with an arbitrary number of components. Although already for thefour-component case there are a large number of different combinations of phase-locking and phase-antilockingJosephson couplings, we establish that there are a much smaller number of equivalence classes where propertiesof frustrated multicomponent superconductors can be mapped to each other. This classification is related tothe graph-theoretical concept of Seidel switching. Numerically, we calculate ground states, normal modes, andcharacteristic length scales for the four-component case. We report conditions of appearance of new accidentalcontinuous ground-state degeneracies.

DOI: 10.1103/PhysRevB.88.214507 PACS number(s): 74.70.Xa, 74.20.Mn, 74.20.Rp

I. INTRODUCTION

The generalization of BCS theory to the multiband casewith two coupled gaps was predicted in 1959,1,2 but was longwidely considered a theoretical speculation. It attracted wideinterest only after the 2001 discovery3 of superconductivity inMgB2: a clear-cut example of a two-band superconductor.4 Inparallel, theoretical works explored to what extent two-bandsuperconductivity differs from that in ordinary single-bandBCS theory. Since the condensates in two-band superconduc-tors are not independently conserved, they share the samebroken U (1) symmetry as their single-band counterparts. [Forbrevity, and without loss of generality, we do not distinguishbetween global and local U (1) symmetry.] That is, although asimple two-band superconductor can under certain conditionsbe described by a pair of complex fields ψi = |ψi |eiφi (i =1,2), there is a term of the form −η|ψ1||ψ2| cos(φ1 − φ2), sothat two-band systems attain their free-energy minimum eitherwhen the phases are locked (phase difference equal to zero) orwhen the phases are antilocked (phase difference equal to π ).

Nevertheless, as first discussed by Leggett, the individualphases are important degrees of freedom in two-componentsystems. First, there exist collective excitations associated withfluctuations of the phase difference around its ground-statevalue.5,6 Second, the existence of several phases can undercertain conditions give rise to fractional vortices.7

Another qualitatively new feature which can exist in two-component superconductors is that of two coherence lengths,ξ1 and ξ2. (For the definition of coherence lengths in thepresence of intercomponent coupling in Ginzburg-Landaumodels, see Ref. 8; for microscopic models, see Refs. 9and 10.) This feature can give rise to what was recently termedtype-1.5 superconductivity. That is, since there exist twocoherence lengths, a vortex in a two-band superconductor willtypically have two cores of different sizes. In the case whereξ1 < λ < ξ2, where λ is the penetration depth, there can existthermodynamically stable vortices with long-range attractiveand short-range repulsive interaction, giving rise to type-1.5

superconductivity.8–17 This regime possesses an additionalphase, in which there is macroscopic phase separation intodomains of Meissner state and vortex state. For a recent reviewof type-1.5 superconductivity, see Ref. 18.

More than a half a century after two-band superconductivitywas introduced, the problem of three-band superconductivitybecame highly relevant following the discovery of iron-basedsuperconductors,19 which is a subject of rapidly growinginterest.20,21 This raised the question if qualitatively newphysics can result from the addition of a third superconductingband. It was realized that systems with three components maydisplay phase frustration, i.e., it is not necessarily the casethat the Josephson-coupling terms −ηij |ψi ||ψj | cos(φi − φj )can each simultaneously be minimized. Such frustrationmay lead to time-reversal-symmetry breaking (TRSB).22,23

States with TRSB break U (1) × Z2 symmetry.13 This phasefrustration and new broken symmetry leads to interesting newphysics. Recently, the scenario of such U (1) × Z2 symmetrybreaking in some iron-based superconductors was put onmore solid ground;24 related states in other superconductorshave also been discussed.25 The growing number of recentlydiscovered multiband systems has resulted in the growingopinion that multiband superconductivity is the rule rather thanthe exception. This has prompted investigation of multibandgeneralizations of various states, including the Fulde-Ferrell-Larkin-Ovchinnikov state.26

In Ref. 13 it was discussed that phase-frustration-inducedTRSB leads to new collective mixed phase-density modes,different from the phase-only Leggett mode. Normal modeswere also discussed for some multiband cases in Ref. 27.Phase frustration can have a dramatic effect on the magneticresponse of three-band superconductors.13 This is because,due to frustration, the system can have large characteristiclength scales associated with field variations even in thecase of strong Josephson coupling. At the TRSB transitionpoint, the length scale of one of the phase-difference modesdiverges (as has also been discussed earlier in a Londonphase-only model28); this is a necessary consequence of the

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DANIEL WESTON AND EGOR BABAEV PHYSICAL REVIEW B 88, 214507 (2013)

continuous transition to a ground state with higher discretedegeneracy. This divergence places systems with at leastthree components in contrast to two-component systems, inwhich increasing Josephson coupling typically diminishesdisparities of the density variations.8,12 Also, this divergenceimplies that even if the system is type 2 in the TRSBstate (λ > ξi), as one approaches the Z2 phase transition thedivergence of one coherence length makes the system type1.5 with ξi < λ < ξj (provided that the phase transition iscontinuous29). For dynamical aspects of the aforementionedmixed phase-density modes, see Refs. 24 and 30. Other newphysics which have been discussed in connection with three-component systems are possible (meta)stable flux-carryingexcitations characterized by a CP 2 topological invariant,27,31

and fractional fluxons in long Josephson junctions32 withflux quantization similar to that in fractional vortices in[U (1)]3 superconductors.33 Aside from that, the system canhave an anomalous normal state where fluctuations destroysuperconductivity, and yet the resulting normal state retainsbroken time-reversal symmetry.34

The above reviewed appearance of new physics in thethree-component case raises the question of whether super-conductors with four or more components are analogous tothose with three components, or whether further new physicsappears due to the additional components. Experimentally,four or more Josephson-coupled components can be realizedusing proximity effects in layered superconducting systems,in a similar way as in the earlier proposal to realize athree-component TRSB state in a bilayer of s± and s-wavesuperconductors.22

In this paper, we present a classification of superconductingstates in n-component systems (n ∈ N) with various possiblefrustration-inducing combinations of signs of Josephson cou-plings. This classification is related to the graph-theoreticalconcept of Seidel switching. Furthermore, we calculate groundstates, normal modes, and characteristic length scales forfour-component systems. In doing this, we find that the caseof four and larger numbers of components is substantiallyricher, and allows a number of qualitatively new phenomena,as compared to the two- and three-component cases.

II. EQUIVALENT SIGNATURES

We consider multicomponent superconductors that aremodeled by the London free-energy density

f = 1

2(∇ × A)2 +

∑i

1

2|Dψi |2 −

∑j>i

ηij cos φij . (1)

Here, A is the vector potential, D = ∇ + ieA, the ψi =|ψi |eiφi are complex fields representing the superconductingcomponents, and φij = φi − φj . Although we here considera London model (in which |ψi | = const), the results in thissection are valid also for Ginzburg-Landau models, as wellas under the inclusion of arbitrary non-phase-dependent termsin (1).

We initially focus on the four-component case, although wewill find results pertaining to the n-component case. We definethe signature of the Josephson couplings to be the tuple

(sgn η12, sgn η13, sgn η14, sgn η23, sgn η24, sgn η34),

TABLE I. Representatives of the classes of strongly equivalentsignatures for four components. For unfrustrated signatures, theground-state phase configuration is given. For singly frustratedsignatures, the discriminatory coupling is given.

# η12 η13 η14 η23 η24 η34 Weak-equivalence class

1 + + + + + + φ1 = φ2 = φ3 = φ4

2 + + + + + − Singly frustrated (η34)3 + + + + − − Singly frustrated (η14)4 − + + + + − Multiply frustrated5 + + + − − − Multiply frustrated6 + + − + − − φ1 = φ2 = φ3 = φ4 + π

7 + − + + − − Singly frustrated (η12)8 + + − − − − Singly frustrated (η23)9 + − − − − + φ1 = φ2 = φ3 + π = φ4 + π

10 + − − − − − Singly frustrated (η34)11 − − − − − − Multiply frustrated

where sgn denotes the sign function, i.e.,

sgn x =⎧⎨⎩

0, x = 0+, x > 0−, x < 0.

The signature is of interest since each Josephson coupling setsa preferential value for a certain phase difference: if ηij > 0,then φi and φj want to lock (φij = 0), whereas if ηij < 0,then φi and φj want to antilock (φij = π ). We now proceed todiscuss similarities between signatures.

Under the assumption that the Josephson-coupling coeffi-cients ηij are all nonzero, there are 26 = 64 distinct signatures.If two signatures can be mapped to each other via relabelingof the components, then they are obviously equivalent. In thiscase we say that the signatures are strongly equivalent. It iseasily seen that there are 11 classes of strongly equivalentsignatures; representatives of these classes are given in Table I.However, it is not necessary to study each of these equivalenceclasses. Instead, the signatures can be divided into threeequivalence classes in such a way that it is sufficient to studya single representative of each class. We now establish this.

We define the following operators, which act on the phasesand coupling coefficients, respectively:

Pi : φi �→ φi + π and Qi : η(ij ) �→ −η(ij ) (∀ j �= i).

Here, by (ij ) we mean the tuple obtained by sorting i andj in ascending order [e.g., (12) = (21) = 12]. In words, Pi

inverts the ith phase, and Qi changes the sign of all Josephson-coupling coefficients that involve the ith component. The freeenergy is clearly invariant under the simultaneous applicationof Pi and Qi .

Motivated by this observation, we now define anotherequivalence relation on the set of signatures. Two signaturesare considered weakly equivalent if one of the signatures canbe obtained from the other by relabeling of the componentsand application of Qi’s. Clearly, signatures which are stronglyequivalent are also weakly equivalent. If two signatures areweakly equivalent, it is sufficient to study one of them. Thereason for this is that the phase behavior of the secondsignature can be obtained from the phase behavior of thefirst signature via application of the appropriate Pi’s, and all

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characteristics apart from the phase behavior are identical forthe two signatures.

As a very simple example of the above, consider the cases ofattractive and repulsive coupling for a two-component system(in which the phases are, respectively, locked and antilockedin the ground state). These cases are weakly equivalent sincethey can be mapped to each other via either of the two Qi’s.

A. Graph-theoretical approach for classification of frustratedn-component superconductors

Our discussion so far has in no significant way been specificto the four-component case. Before moving on to the specificsof this case, we consider the general n-component case. Indoing this, it is convenient to take a graph-theoretical approach.We let the n components be represented by the (unlabeled)vertices in a graph of order n. The Josephson couplings arerepresented by edges in this graph. If a particular couplingcoefficient is negative, we let the corresponding edge be blue;if a coupling coefficient is positive, we let the correspondingedge be red; if a coupling coefficient is zero, there is nocorresponding edge. As long as we consider the case in whichthe coupling coefficients are all nonzero, we could avoidcoloring the edges by letting the presence of an edge indicateone sign and the absence of an edge indicate the other sign.That the vertices are unlabeled means precisely that if twosignatures are strongly equivalent, then they are representedby the same graph.

For n components, there are clearly n(n − 1)/2 possibleJosephson couplings. Thus, under the assumption that thecoupling coefficients are all nonzero, there are 2n(n−1)/2

signatures (removing this assumption, we get 3n(n−1)/2 sig-natures). The questions of how many strong-equivalence andweak-equivalence classes there are, are much more difficultto answer. However, using our graph-theoretical approach, wewill make progress in this regard. Presently, we continue toassume that all of the couplings are nonzero; subsequently,we briefly consider the case in which some couplings vanish(which is relevant for multilayer or Josephson-junction-arrayrealizations of frustrated systems).

Each strong-equivalence class corresponds to a uniquecomplete graph on n (unlabeled) vertices with edges coloredred and blue. As suggested above, by removing the edgesof one particular color we obtain a bijection from the set ofgraphs of the aforementioned type to the set of graphs onn vertices (without colored edges). Thus, the number Ns(n)of strong-equivalence classes for n components is equal tothe number of graphs on n (unlabeled) vertices. Althoughthere is no known closed-form formula for this number, thecorresponding enumeration problem has been solved usingPolyas enumeration theorem.35 The values of Ns(n) for 2 �n � 6 are given in Table II. The sequence Ns(n) is Sloane’sA000088.

Each weak-equivalence class corresponds to an equivalenceclass of complete graphs on n (unlabeled) vertices withedges colored red and blue. The operation on such a graphcorresponding to Qi is switching of the colors of all edgesconnected to a particular vertex. This is known as Seidelswitching, and the equivalence classes of graphs that can betransformed into each other via Seidel switching are known

TABLE II. Number NJ of Josephson couplings, Nsgn of signatures,Ns of strong-equivalence classes (of which Nsu unfrustrated andNsf frustrated), and Nw of weak-equivalence classes (of which Nwu

unfrustrated and Nwf frustrated) for n components. We assume thatthe couplings are all nonzero.

n NJ Nsgn Ns Nsu Nsf Nw Nwu Nwf

2 1 2 2 2 0 1 1 03 3 8 4 2 2 2 1 14 6 64 11 3 8 3 1 25 10 1024 34 3 31 7 1 66 15 32768 156 4 152 16 1 15

as switching classes (Fig. 1). Thus, the number Nw(n) ofweak-equivalence classes for n components is equal to thenumber of switching classes of complete graphs on n verticeswith edges colored red and blue. As before, there is noclosed-form formula for this number, but the correspondingenumeration problem has been solved.36 The values of Nw(n)for 2 � n � 6 are given in Table II. The sequence Nw(n) isSloane’s A002854.

We now illustrate the physical interpretation of the aboveby considering the question of how many unfrustrated strong-equivalence classes there are for n components. We denotethis number Nsu(n). Without loss of generality, we assumethat φ1 = 0. Clearly, each phase must have a value of 0or π in order for there to be no phase frustration. Thiscreates a partition of the phases into two sets, within whichthere may only be attractive couplings, and between whichthere may only be repulsive couplings. In terms of ourgraph-theoretical approach, this means that the correspondingstrong-equivalence classes are represented by graphs such thatthe subgraph corresponding to the blue edges is a completebipartite graph (note that this includes edgeless graphs; cf.Fig. 1). Thus, we see that Nsu(n) = floor(n/2 + 1).

Finally, we remark that there is a single unfrustrated weak-equivalence class, regardless of the number of components.This is clear since any unfrustrated phase configuration can bemapped via Pi’s to the configuration in which all phases arelocked.

FIG. 1. (Color online) Switching classes of complete graphs onfour vertices with edges colored red (dashed lines, attractive coupling)and blue (solid lines, repulsive coupling). The uppermost switchingclass corresponds to the unfrustrated signatures, the central class to thesingly frustrated signatures, and the lowermost class to the multiplyfrustrated signatures.

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B. Vanishing couplings

We now briefly discuss the topic of systems in which someJosephson couplings vanish. We do this in the most generalsetting allowed by our model. The most natural goal in thiscontext is to enumerate the weak-equivalence classes for anarbitrary number of components, without the assumption ofall couplings being nonzero. In terms of our graph-theoreticalapproach, this means enumerating the switching classes of (notnecessarily complete) graphs on n vertices with edges coloredred and blue. Naturally, the switching operation correspondingto a particular vertex switches the colors of edges connectedto this vertex, but does not affect the presence or absence ofedges. As far as we are aware, this problem has not been solved.Furthermore, we suspect that it is a very difficult problem.However, for any given moderate number of components, it isstraightforward to identify the weak-equivalence classes usingbrute force.

C. Four components

We now apply the above to the four-component case, whichwe shall consider in greater detail. As mentioned above, thereare three weak-equivalence classes of four-component signa-tures. We refer to the signatures in these classes as unfrustrated,singly frustrated, and multiply frustrated, respectively. Thereasons for choosing these terms will become clear. Table Ishows to which weak-equivalence class the signatures in eachstrong-equivalence class belong. Figure 1 illustrates the threeswitching classes of graphs on four vertices which correspondto the three weak-equivalence classes of four-componentsignatures.

In the next section, we calculate ground states, normalmodes, and characteristic length scales for n-componentsystems. Thereafter, we apply these calculations to the twoweak-equivalence classes of frustrated signatures in the four-component case.

III. GROUND STATES AND NORMAL MODES

We now proceed to consider the full Ginzburg-Landau free-energy density corresponding to (1), i.e.,

f = 1

2(∇ × A)2 +

∑i

1

2|Dψi |2 + αi |ψi |2 + 1

2βi |ψi |4

−∑j>i

ηij |ψi ||ψj | cos φij . (2)

It is straightforward to generalize the discussion to includeterms which depend on products of |ψi | with higher powersusing the methods of Ref. 12. In the following, we briefly com-ment also on the case of higher Josephson harmonics. First, letus generalize some results from the case of three components13

to the case of an arbitrary number of components.Our goal in this section is twofold. First, we seek to

determine the ground-state values of the ψi , i.e., the ground-state values of the densities and phases. Second, we wishto determine the normal modes of fluctuations around theseground states, as well as over what characteristic length scalessuch fluctuations decay. Both of these goals will be attained by

expanding the relevant fields around their ground-state values,as follows:

ψi = [ui + εi(r)] exp{i[φi + ϕi(r)]}, (3)

A = a(r)

r(− sin θ, cos θ,0) = a(r)

rθ . (4)

Here, ui and φi are ground-state amplitudes and phases,respectively. Also, r and θ are radial and azimuthal cylindricalcoordinates, respectively. We now proceed to determine theground states of the system in question.

A. Ground states

Inserting the field expansions (3) and (4) into the free-energy density (2) and retaining only those terms which arefirst order in the fluctuations, we obtain∑

i

2uiεi

(αi + βiu

2i

) −∑j>i

ηij (uiεj + ujεi) cos φij

− ηijuiuj (ϕi − ϕj ) sin φij , (5)

where φij = φi − φj . A necessary condition for the values ofui and φi to be ground-state values is that the free-energydensity (2) is stationary with respect to fluctuations aroundthese values. This means precisely that the prefactor of eachεi and ϕi in (5) should be zero. Requiring this, we obtain

0 = αiui + βiu3i − 1

2

∑j �=i

η(ij )uj cos φ(ij ), (6)

0 =∑j �=i

(−1)(i<j )η(ij )uiuj sin φ(ij ) (7)

for 1 � i � 4. When we write a statement in brackets, as in(i < j ), we understand this to be an expression that equals oneif the statement is true, and zero if the statement is false.

Unfortunately, we are unable to solve (6) and (7) analyti-cally. Therefore, we determine the ground-state values ui andφi numerically.

Finally, we note that it is convenient to set one of thephases to zero; this is allowed since an overall phase rotationis a pure gauge transformation. We do this in our numericalminimization, and thus the minimization is actually performedon a space with seven degrees of freedom (not eight degreesof freedom). However, in the following, we continue to workwith eight degrees of freedom for reasons that will becomeclear.

B. Length scales and normal modes

Having considered the terms in the free-energy density (2)which are first order in the fluctuations, we now proceedto consider the second-order terms (which is equivalent tolinearizing the Ginzburg-Landau equations). In doing this, weswitch to a slightly different basis; more precisely, we replaceϕi by πi ≡ uiϕi . The reason for this is that the so-calledmass matrix, which we determine in this section, becomessymmetric in this new basis. We also introduce the notation

v = (ε1, . . . ,εn,π1, . . . ,πn)T,

i.e., we collect the fluctuations of the matter fields in thevector v.

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Inserting the field expansions (3) and (4) into the free-energy density (2) and retaining only terms that are secondorder in the fluctuations or the gradients thereof, we obtain

1

2(∇v)2 + 1

2vTM2v + 1

2r2(∇a)2 + e2

2r2

∑i

u2i a

2. (8)

We note that here the fluctuations in A decouple from thefluctuations in the matter fields. The matrixM2 is the (squared)mass matrix. Writing the corresponding terms in the freeenergy explicitly, we find that

1

2vTM2v =

∑i

ε2i

(αi + 3βiu

2i

) −∑j>i

ηij εiεj cos φij

− ηij

[(uiεj + ujεi)

(πi

ui

− πj

uj

)sin φij

+ uiuj

2

(πi

ui

− πj

uj

)2

cos φij

].

From this we can determine M2. For brevity, we introduce thenotation ηij = (ηij /2) cos φij and ηij = (ηij /2) sin φij . Also,we divide M2 into four submatrices of equal size, and extracta factor of 2, so that

M2 = 2

(Mεε Mεπ

Mπε Mππ

).

We are now ready to write general expressions for the abovesubmatrices. These are

Mεε =(

0 −ηij

−ηj i 0

)+ diag

(αi + 3βiu

2i

),

Mππ =(

0 −ηij

−ηj i 0

)+ diag

⎛⎝ 1

ui

∑k �=i

ukη(ik)

⎞⎠ ,

and

Mεπ = MTπε =

(0 −ηij

ηj i 0

)

+ diag

⎛⎝ 1

ui

∑k �=i

(−1)(k<i)ukη(ik)

⎞⎠ ,

where i and j are row and column indices, respectively.Having written the (squared) mass matrix M2, let us

consider its physical interpretation. First, the eigenvectorsof M2 are the normal modes of the system. If such aneigenvector has more than one nonzero element, we say thatthis normal mode is mixed. We are especially interested incases where there is both a nonzero density element and anonzero phase element, corresponding to mixed phase-densitymodes. Second, the eigenvalues of M2 are the squared massesof the corresponding normal modes, i.e., the inverse squaredcharacteristic length scales for the decay of small excitationsof these modes.

We have seen that by diagonalizing M2 one can determinethe normal modes and characteristic length scales of the sys-tem. Unfortunately, in general this can not be done analytically.However, since we have included all eight (for the case of fourcomponents) degrees of freedom, we can immediately identifya normal mode, namely, the gauge rotation, as well as the mass

of this mode, which is zero. (Note that in an electrically chargedsystem, this mode can acquire a mass via the Anderson-Higgsmechanism.37) Thus, we could have limited ourselves to theseven physically relevant degrees of freedom. However, wechoose not to do this since the aforementioned knowledgeabout the eigenvectors and eigenvalues of M2 provides auseful way to check our numerical results.

C. Massless modes

It is a rather general feature of frustrated multicomponentsuperconductors that they may undergo continuous transitionswhereby discrete ground-state degeneracy arises. Examplesof this are the aforementioned TRSB transitions in three-component superconductors, as well as other such transitionswhich are studied below. Also, such transitions may appearin phase-only models.28 These transitions are quite generallyaccompanied by the presence of at least one massless normalmode, i.e., by the divergence of a characteristic length scalefor the decay of such a mode.

Indeed, consider a potential U (x,α), which depends onthe (generalized) coordinate vector x ∈ Rn and the parameterα ∈ R. In our case, U is the Ginzburg-Landau potential in (2),and x is a vector of densities and phases (or some otherparametrization of the state space). The parameter α can inour case have several meanings; for example, α could be aJosephson-coupling coefficient. We choose the above notationin order to emphasize the generality of the material in thissection.

Assume that U (x,α) ∈ C2(Rn+1). Assume further that, asα is varied, the system undergoes a continuous transitionwhereby a ground state splits into two degenerate groundstates. This situation is illustrated in Fig. 2. (The argumentapplies more generally to any situation where a local minimumis continuously transformed into several local minima, but thisis the main case of interest to us.) We choose our coordinatesystem so that the Hessian of U (x) is diagonal at the transitionpoint. This is possible since the Hessian is symmetric, andthus diagonalizable (by an orthogonal transformation). Also,we observe that this choice of coordinates is such that eachcoordinate corresponds to a normal mode at the critical pointα = αc at which the transition takes place.

Choose a coordinate xi in which there is discrete degeneracyfor α > αc, and consider the curve of ground states in (xi,α)space which is illustrated schematically in Fig. 2. Obviously,

α

xi∂iU = 0 ∂iU = 0

∂2i U ≥ 0 ∂2

i U ≤ 0

FIG. 2. (Color online) The ground-state value of the (generalized)coordinate xi as a (potentially multivalued) function of the parameterα. By assumed continuity, we have that ∂2

i U = 0 at the critical point.Thus, there exists a massless mode at this point.

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DANIEL WESTON AND EGOR BABAEV PHYSICAL REVIEW B 88, 214507 (2013)

each ground-state point (blue curve) is such that ∂iU = 0 and∂2i U � 0 (∂i denotes differentiation with respect to xi). Now,

fix a value α > αc, and consider how ∂iU varies as xi is varied:in other words, as one moves along a vertical line in the right-hand side of Fig. 2. Immediately above the lower ground-statecurve ∂iU > 0, since ∂iU = 0 and ∂2

i U � 0 on the curve and∂iU �= 0 immediately above the curve (lest points immediatelyabove the curve also be ground states). Similarly, ∂iU < 0immediately below the upper curve. Hence, there is a pointbetween the curves such that ∂iU = 0 and ∂2

i U � 0. Sincethis holds arbitrarily close to the critical point, there is somecurve (dashed line in Fig. 2) that emanates from the criticalpoint and along which ∂iU = 0 and ∂2

i U � 0. By the assumedcontinuity of ∂2

i U , we have that ∂2i U = 0 at the critical point.

This implies the existence of a massless mode at this point.Finally, we note that since each of the coordinates we usecorresponds to a normal mode at the critical point, there willbe a massless mode for each coordinate in which discretedegeneracy arises at this point.

IV. SINGLY FRUSTRATED SIGNATURES

In this section, and the next, we apply the results of theprevious section to the four-component case. Recall that wenamed the two frustrated weak-equivalence classes of four-component signatures singly frustrated and multiply frustrated.The singly frustrated signatures are the frustrated signaturesfor which there exists a phase configuration in which onlyone Josephson coupling is frustrated. We call such a phaseconfiguration a singly frustrated phase configuration, and wecall other frustrated phase configurations multiply frustratedphase configurations. For each singly frustrated signaturethere is a unique singly frustrated phase configuration [up tothe overall U (1) symmetry]. Thus, there is, for each singlyfrustrated signature, a unique coupling which is frustratedin the singly frustrated phase configuration. We call thesecouplings the discriminatory couplings (Table I). We nowconsider the effects of varying a discriminatory coupling.

A. Discriminatory couplings

If, for a given singly frustrated signature, the discriminatorycoupling is sufficiently weak, then the phases will assume thesingly frustrated configuration (at least, the singly frustratedconfiguration will be the ground-state configuration). Con-versely, if the discriminatory coupling is sufficiently strong,then the phases will assume a multiply frustrated configuration.At the transition between singly frustrated and multiplyfrustrated phase configurations, there is a massless mode (apartfrom the mode corresponding to the gauge symmetry). This isan example of the general fact that continuous phase transitionsare accompanied by massless modes. At the transition,time-reversal symmetry is broken, and the spontaneouslybroken symmetry changes from U (1) to U (1) × Z2. Thetransformation corresponding to the Z2 symmetry is complexconjugation of the ψi . Furthermore, for strong discriminatorycouplings the corresponding phases may lock (for attractivecouplings) or antilock (for repulsive couplings) leading to asecond transition, this time from U (1) × Z2 back to U (1).In cases of TRSB, the normal modes are generally mixed.

Signature 2 Signature 3 Signature 7

Signature 8 Signature 10

FIG. 3. (Color online) Examples of ground-state phase configu-rations for singly frustrated signatures with discriminatory couplingssomewhat stronger than the critical values. There is Z2 degeneracycorresponding to complex conjugation of the ψi . The correspondingsingly frustrated phase configurations are obtained by reducing themarked angles to zero.

Evidently, the case of singly frustrated signatures is largelyanalogous to the case of frustrated three-component signatures.This is true despite the fact that for three components there isno discriminatory coupling.

We now consider a specific example of a singly frustratedsignature. Arbitrarily, and without loss of generality, wechoose signature 7. We use the free-energy parameters in (2)given by αi = −1, βi = 1, and |ηij | = 1 except that we varythe coefficient of the discriminatory coupling. The singlyfrustrated phase configuration for signature 7 is

φ1 = φ2 + π = φ3 + π = φ4.

This is the ground-state phase configuration for values ofthe discriminatory coupling coefficient η12 smaller than thecritical value ηc

12 = 1.21. For η12 > ηc12, the phases φ1 and

φ2 approach each other by breaking their locking with φ4

and φ3, respectively, by equal and opposite amounts (Fig. 3).The ground states, normal modes, and characteristic lengthscales for the present parameters with 0 � η12 � 3 are shownin Fig. 4. As we observed in the previous section, the normalmodes are given by the eigenvectors of the (squared) massmatrix, and the characteristic length scales are given by thecorresponding eigenvalues. We note that here, in contrast to thecase considered in Ref. 13, there is a phase-only mode (mode3) also in the TRSB regime. Thus, such modes are possible,even though in the case of TRSB the modes are typicallymixed phase-density modes. Finally, the corresponding plotsfor other singly frustrated signatures are identical (up torelabeling of components), except that the ground-state phaseconfigurations are different.

B. Higher ground-state degeneracy

An interesting question is whether there exist ground stateswith higher than twofold degeneracy [we here ignore theoverall U (1) symmetry, which is always present]. We now gosome way towards answering this question for singly frustratedsignatures. One way in which higher discrete degeneracy mayarise is through equivalence of components: If two (or more)components are equivalent, and if the ground-state values ofthe corresponding fields are not equal, then exchanging thevalues of these fields will map a given ground state to a

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FIG. 4. (Color online) Ground states (cf. Fig. 3), (inverse) length scales, and normal modes for signature 7 with αi = −1, βi = 1, and|ηij | = 1 for ij �= 12. η12 is plotted on the x axes. Note that there is a second massless mode at the critical point η12 = ηc

12 = 1.21. Also, severalof the modes are mixed phase-density modes when η12 > ηc

12, whereas none of the modes are mixed phase-density modes when η12 < ηc12.

distinct but equivalent state. We begin by noting that for singlyfrustrated signatures, no more than two components can beequivalent. To see this, note that neither of the two phasescoupled by the discriminatory coupling (for signature 2: φ3

and φ4 in Fig. 3) can be equivalent to either of the other twophases.

Furthermore, apart from equivalence of components, onecould imagine that higher degeneracy could arise through whatmay reasonably be called ground-state equivalence of phases.By this we mean the following: Consider a given ground state,and in particular components i and j . If η(ik)|ψi | = η(jk)|ψj |(i �= k �= j ) for the ground-state values of |ψi | and |ψj |, thenexchanging the values of φi and φj will have no effect onthe potential energy. Thus, if it is also the case that φi �= φj

in a ground state, then there is corresponding ground-statedegeneracy. As far as we are aware, each component alwayshas a unique ground-state value of the density. Assuming this,we have that equivalence of components is a special case ofground-state equivalence of phases.

Let φi and φj be equivalent in a ground state. Notethat upon application of Pi and Qi , φi and φj need nolonger be equivalent in the above sense. Nonetheless, anyground-state degeneracy is unaffected by this transformation.We understand that φi and φj are in fact still equivalent insome weaker sense. For simplicity, we consider signature 2,for which this question does not arise.

Now, if φ1 and φ2 are equivalent in a ground state, this canonly give rise to higher degeneracy in the aforementioned wayif φ1 �= φ2. Somewhat less obvious is the fact that if φ3 andφ4 are equivalent in a ground state, then it is again necessaryto have φ1 �= φ2 in order for this to yield higher degeneracy.The reason for this is that if φ3 and φ4 are equivalent, andφ1 = φ2 in the ground state, then the degeneracy correspondingto exchange of φ3 and φ4 coincides with the degeneracycorresponding to complex conjugation.

We now show that if two phases are equivalent in a groundstate, then φ1 = φ2 in this state, whence higher degeneracycan not arise in the aforementioned way for singly frustratedsignatures. However, we will find that such degeneracy canoccur for multiply frustrated signatures. We now in turnconsider the cases of φ1 and φ2 being equivalent, and of φ3

and φ4 being equivalent.We parametrize the three relevant degrees of freedom in

the phases as shown in the left half of Fig. 5. For brevity,we introduce the notation ηij = −ηij |ψi ||ψj |. The part of thepotential energy which depends on γ is

Fγ = η13 cos(δ1 − γ ) + η23 cos(δ1 + γ )

+ η24 cos(δ2 − γ ) + η14 cos(δ2 + γ ) + η12 cos 2γ.

By the assumed equivalence of φ1 and φ2, we have that η13 =η23 =: η1 and η14 = η24 =: η2. Thus, using a trigonometric

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FIG. 5. (Color online) Parametrizations of the three relevantdegrees of freedom in the phases.

identity, we can write Fγ as

Fγ = 2(η1 cos δ1 + η2 cos δ2) cos γ + η12 cos 2γ. (9)

If η12 < 0, as we assume, then the last term in (9) is minimizedfor γ = 0 and γ = π , both corresponding to φ1 = φ2. Clearly,one of these values of γ also minimizes the first term in (9),and thus we have that φ1 = φ2 in the ground state.

We now parametrize the three relevant degrees of freedomin the phases as shown in the right half of Fig. 5. The part ofthe potential energy which depends on γ1 and γ2 is

Fγ = η13 cos(δ − γ1) + η23 cos(δ + γ2) + η24 cos(δ − γ2)

+ η14 cos(δ + γ1) + η12 cos(γ1 + γ2).

By the assumed equivalence of φ3 and φ4, we have that η13 =η14 =: η1 and η23 = η24 =: η2. Thus we can write Fγ as

Fγ = 2(η1 cos γ1 + η2 cos γ2) cos δ + η12 cos(γ1 + γ2). (10)

Without loss of generality, we assume that cos δ � 0. Since wealso have that η1 < 0 and η2 < 0, we can conclude that the firstterm in (10) is minimized by γ1 = γ2 = 0. Also, the secondtherm in (10) is clearly minimized precisely if γ1 + γ2 = 0.Thus, we again have that φ1 = φ2 in the ground state.

V. MULTIPLY FRUSTRATED SIGNATURES

The multiply frustrated signatures are the frustrated sig-natures for which more than one of the Josephson couplingsare frustrated, regardless of the phase configuration. For suchsignatures, the ground-state degeneracy can be greater than forsingly frustrated four-component signatures, or for fewer thanfour components. First, for some values of the free-energyparameters, there exists continuous ground-state degeneracycorresponding to rotation of a pair of phases relative to theother two phases. Such rotations can occur when the phasesare pairwise equivalent in the aforementioned sense, and occurdespite all phases being coupled. Second, for certain othervalues of the free-energy parameters, there can exist othertypes of additional continuous ground-state degeneracy. Thiscan occur when two or three phases are equivalent.

A. Energetically free phase rotations

We now say something about what parameter values giverise to energetically free phase rotations of the aforementionedtype. In doing this, and in the remainder of this section, wechoose to consider signature 11 (Table I). For this signature,energetically free phase rotations may exist in cases wherethe phases are pairwise antilocked in the ground state. Assume

Signature 4 Signature 5 Signature 4 and 11

FIG. 6. (Color online) Examples of ground-state phase configura-tions for multiply frustrated signatures with complete intercomponentsymmetry. The phases are pairwise locked or antilocked. The phaserotations corresponding to alteration of the angle γ are energeticallyfree.

that φ1 = φ2 + π and φ3 = φ4 + π , and let γ = φ3 − φ1. Thissituation is illustrated in the rightmost part of Fig. 6. In orderfor this to be a ground-state configuration with 0 �= γ �= π , itis necessary that

η13 = η14 = η23 = η24, (11)

where ηij = −ηij |ψi ||ψj |. To see that these equalities arenecessary, note that if they do not hold, one of the phaseswill be subject to a net force, which will tend to alter thephase configuration. Apart from being necessary ground-stateconditions, the conditions in (11) are sufficient for the γ

rotation to be energetically free: the portion of the potentialenergy which depends on γ is

η13 cos γ + η23 cos(π − γ ) + η24 cos γ + η14 cos(π − γ )

= (η13 + η24 − η23 − η14) cos γ. (12)

Conditions equivalent to those in (11) can of course be givenfor the other multiply frustrated signatures. Note that theconditions in (11) are equivalent to the condition of pairwiseground-state equivalence of phases.

In order for the phase configurations parametrized by γ tocorrespond to ground states, it is necessary that the couplingsbetween the paired phases be sufficiently strong, so thatantilocking is maintained. In fact, we find numerically thatthe condition is

η12η34 � η2, (13)

where η := η13 = η14 = η23 = η24. Hence, we have foundthat the necessary and sufficient conditions for additionalground-state degeneracy in the form of energetically free phaserotations are (11) and (13). Figure 7 classifies the possibleground states under the assumption of (11). (Without lossof generality, we set η = 1.) In cases for which η12 � 1and η34 � 1 (region A), the ground state only breaks U (1)symmetry. The energetically free phase rotations correspondto region C. The remaining possibilities (region B) give rise toground states that break U (1) × Z2 symmetry.

We now consider the case of complete symmetry betweenthe four components, which of course leads to fulfillmentof (11) and (13) in the ground state. For illustrative purposes,we choose the parameter values αi = −1, βi = 1, and ηij =−1. For this completely symmetric case, the ground-statephase configurations are precisely those for which the phases

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Region A

Region B

η34

η12

0 1 2 3

3

2

1

0

A B

BC

Region C

FIG. 7. (Color online) Classification of ground states for signa-ture 11 under the assumption of pairwise ground-state equivalenceof phases. In region A there is no TRSB, in region B there is TRSB,and in region C there are energetically free phase rotations and thusdegeneracy between states with and without TRSB. Representativeground states are shown for the three regions. We set η13 = η14 =η23 = η24 = 1. Along the curve η12η34 = 1, the situation is morecomplicated than this classification suggests (Fig. 11).

are pairwise antilocked. Thus, there is continuous ground-state degeneracy. Figure 6 illustrates the ground-state phaseconfigurations for this signature, as well as the correspondingphase configurations for the other two multiply frustratedsignatures. As we expect, the ground states for the differentsignatures can be mapped to each other via relabeling of thecomponents and inversion of the phases (application of Pi’s).

We note that although rotation by the angle γ (γ rotation,Fig. 6) does not alter the potential energy, such rotation doesalter the normal modes and the corresponding length scales.This is clear from Fig. 8, which displays the ground states,normal modes, and characteristic length scales for the systemin question. Also noteworthy is the fact that γ rotation doesnot in itself lead to exploration of the entire family of groundstates since the pairwise antilocking of phases can occur inthree ways, and rotation by γ does not change the antilocking.This is illustrated schematically in Fig. 9.

Now, choose an angle γ such that 0 < γ < π/2, andconsider the corresponding phase configuration in Fig. 6. FromFig. 8 it is clear that no two values of γ in the aforementionedrange give rise to equivalent normal modes. Due to thecomplete intercomponent symmetry, any permutation of φ2,φ3, and φ4 will give an equivalent state (Fig. 9). Furthermore,complex conjugation of each ψi also yields an equivalentstate. Thus, for a given ground state, there are typically 12ground states that are equivalent to this state. However, there

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FIG. 8. (Color online) Ground states, (inverse) length scales, and normal modes for signature 11 with αi = −1, βi = 1, and ηij = −1. Therotational angle γ (Fig. 6) is plotted on the x axes. Note that there is a third massless mode at the points γ = 0 and π . Also, these points arethe only points for which there are no mixed phase-density modes.

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Special case 2Special case 1Ground-state phase configurations

FIG. 9. (Color online) Illustration of ground-state phase configurations for signature 11 with complete intercomponent symmetry. Weassume that φ1 = 0. There are three possible pairwise antilockings, corresponding to the three connected circles. In each circle, the displayedphase configuration corresponds to the black dot; other equivalent possibilities are given by the other dots. Points marked with squares coincide.Permutation of the three unfixed phases corresponds either to moving within the set of black and gray (dark) dots, or to moving within the set ofwhite dots. Complex conjugation corresponds to moving between the set of dark dots and the set of white dots. There are two types of specialcases in which the number of equivalent states is smaller than in the typical case.

are special cases for which the number of equivalent states isthree (special case 1 in Fig. 9) or six (special case 2 in Fig. 9).

Consider the ground states for which antilocked phase pairsare parallel. As can be seen from Fig. 8, there is a thirdmassless mode in these states (the first two massless modesbeing gauge rotation and γ rotation). Naturally, this third modecorresponds to the other possible way of maintaining pairwiseantilocking. The occurrence of this mode is another example ofthe general situation discussed in Sec. III C. One might objectthat in this case there is no parameter actually modifying thepotential. Nevertheless, we can simply replace α by γ withoutinvalidating the argument.

In the above, we have considered a multiply frustratedsignature with maximal intercomponent symmetry. We nowconsider a case with less intercomponent symmetry. We choosethe parameters to be as before, except that we change η12 fromη12 = −1 to η12 = −2. This has the effect of limiting the setof ground-state phase configurations. Whereas previously anyconfiguration with pairwise antilocking was a ground-stateconfiguration, the ground-state configurations are now thosefor which φ1 = φ2 + π and φ3 = φ4 + π (central circle in theleft part of Fig. 9). Since φ1 and φ2 are now more stronglycoupled, they are necessarily antilocked in the ground state.Consequently there is no longer any third massless mode, ascan be seen from Fig. 10, which displays ground states, normalmodes, and characteristic length scales for the case we nowconsider.

We have found that in the cases considered above, differentground states may be inequivalent in the sense of having quitedifferent normal modes and characteristic length scales. Thissuggests that these new degeneracies do not correspond tobroken symmetries. Also, the number of equivalent states isnot the same for all ground states. Furthermore, note that in thecase of complete intercomponent symmetry, the set of groundstates does not form a manifold. (Too see this, consider thespecial points at which there exists a third massless mode.)Thus, there can be no corresponding Lie group.

We note that for multiply frustrated signatures it is ofno significance whether a particular coupling is attractive orrepulsive; only the strengths of the couplings matter. Thisfollows immediately from the fact that any multiply frustratedsignature can be mapped to the signature for which allcouplings are repulsive. This observation is also germane to

the below, in which we consider frustrated three-componentsystems.

B. Phase rotations: Other possibilities

We are interested to know as generally as possible whenenergetically free phase rotations of the aforementioned typecan occur. We begin by noting that phase frustration is required,and thus at least three components are required. The frustratedthree-component signatures are all weakly equivalent; wechoose the signature for which all Josephson couplings arerepulsive. For this signature, one could imagine that two ofthe phases antilock due to strong repulsive coupling. See thecentral image in Fig. 6 for an illustration (imagine that thetwo locked phases are one and the same). If the third phase isequally coupled to the two antilocked phases, then the thirdphase could rotate relative to the antilocked phases at no energycost. We now show that this is not possible.

Requiring that the potential energy be stationary withrespect to variations in the phase φ1, we obtain

−η12|ψ1||ψ2| sin(φ1 − φ2) − η13|ψ1||ψ3| sin(φ1 − φ3) = 0.

Assuming that φ1 and φ2 are antilocked, so that φ1 − φ2 = π ,we find that the first term above is equal to zero. Thus, thesecond term must also be equal to zero, whence φ1 and φ3 areeither locked or antilocked. Thus, there can be no energeticallyfree phase rotations with fewer than four components.

The physical reason for the impossibility of energeticallyfree phase rotations with only three components is clear:First, antilocking is required since without it the third phasewill prefer certain values over others (recall that we assumethat all couplings are repulsive; locking is an equivalentpossibility for frustrated three-component signatures withattractive couplings). Assume, therefore, that φ1 and φ2 areantilocked, and envisage the insertion of φ3 so that φ3 �= φi fori ∈ {1,2}. The couplings involving φ3 will cause φ1 and φ2 tobe subjected to a net force, and thus antilocking will be broken.By the same argument, energetically free phase rotations arenot possible for any singly frustrated signature.

C. Conditions for other continuous degeneracies

We have studied multiply frustrated four-component sys-tems with pairwise equivalence of phases in the ground states,

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FIG. 10. (Color online) Ground states, (inverse) length scales, and normal modes with parameters as in Fig. 8 except that η12 = −2.The rotational angle γ (Fig. 6) is plotted on the x axes. Note that there is no third massless mode at the points γ = 0,π . Also, the modescorresponding to pure density excitations (modes 5 and 7) do not have the same length scales.

and found that such systems can possess the aforementionedenergetically free phase rotations. We now consider four-component systems with ground-state equivalence of twoor three phases. Such systems can possess other types ofcontinuous degeneracies.

We begin by considering the case of ground-state equiv-alence of two phases. (The case of ground-state equivalenceof three phases is of course a special case of the case wenow consider, as are the two cases previously considered.)We seek to investigate whether in the present case degeneracycan arise in the way described in Sec. IV B, and if so underwhat conditions. To this end, we again parametrize the threerelevant degrees of freedom in the phases as in the lefthalf of Fig. 5. The phases assumed to be equivalent are φ1

and φ2.The part of the potential energy which depends on the

phases is

F = η12 cos 2γ + η34 cos(δ1 + δ2)

+ η13 cos(δ1 − γ ) + η23 cos(δ1 + γ )

+ η24 cos(δ2 − γ ) + η14 cos(δ2 + γ ), (14)

where we assume that 0 � γ � π/2 (the other possibility−π/2 � γ � 0 is obtained by complex conjugation). In thefollowing, δi and γ are ground-state values unless the opposite

is stated. By our assumption of ground-state equivalence of φ1

and φ2, we have that η13 = η23 =: η1 and η14 = η24 =: η2.Thus, we can rewrite (14) as

F = 2(η1 cos δ1 + η2 cos δ2) cos γ

+ η12 cos 2γ + η34 cos(δ1 + δ2).

Requiring that this energy be stationary with respect tovariations of δi and γ , we obtain

∂F

∂δi

= −2ηi sin δi cos γ − η34 sin(δ1 + δ2) = 0, (15)

∂F

∂γ= −2(η1 cos δ1 + η2 cos δ2) sin γ − 2η12 sin 2γ = 0.

(16)

As discussed in Sec. IV B, it is only if γ �= 0 that higherdegeneracy can arise in the way described in that section.Therefore, we assume that γ �= 0; this happens precisely if2η12 > |η1 cos δ1 + η2 cos δ2|. Furthermore, it is easily seenthat if γ = π/2 in a ground state, then there are energeticallyfree phase rotations of the type described above. Therefore,we also assume that γ �= π/2; this happens precisely ifη1 cos δ1 + η2 cos δ2 �= 0. In summary, we assume that 0 <

γ < π/2, whence 0 < |η1 cos δ1 + η2 cos δ2| < 2η12.

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Under the aforementioned assumptions, we have that (15)and (16) are equivalent to the following set of equations:

0 = η1 sin δ1 − η2 sin δ2, (17)

0 = 2η1 sin δ1 cos γ + η34 sin(δ1 + δ2), (18)

0 = 2η12 cos γ + η1 cos δ1 + η2 cos δ2. (19)

We note that the first equation above implies that δ1 = 0mod π precisely if δ2 = 0 mod π . This is of interest since ifboth δ1 = 0 mod π and δ2 = 0 mod π , then the degeneracycorresponding to exchange of φ1 and φ2 will coincide with thedegeneracy corresponding to complex conjugation. We thusassume that δ1 �= 0 mod π , whence δ2 �= 0 mod π .

We proceed by using (19) to substitute for cos γ in (14). (Indoing so, we use a trigonometric identity to substitute cos 2γ ,so that the only remaining variables are the δi .) The energyexpression we thus obtain can, upon multiplying by a constantfactor and disregarding an additive constant, be written as

G = − 12 (η1 cos δ1 + η2 cos δ2)2 + η12η34 cos(δ1 + δ2).

Considering G amounts to restricting attention to a certainsurface in phase space, which intersects all minima of interest.The variables δ1 and δ2 parametrize this surface. If, under ourassumptions, δ1 and δ2 minimize F , then these same valueswill of course minimize G. We thus seek minima of G byrequiring stationarity with respect to variations of the δi :

∂G

∂δi

= (η1 cos δ1 + η2 cos δ2)ηi sin δi

− η12η34 sin(δ1 + δ2) = 0. (20)

Using the assumption that δi �= 0 mod π , as well as (17), wecan rewrite (20) as

(η1η2 − η12η34)(η1 cos δ1 + η2 cos δ2) = 0.

We see that if η1η2 − η12η34 �= 0 then η1 cos δ1 + η2 cos δ2 =0, contradicting one of our assumptions. Thus, it is only incases for which

η1η2 − η12η34 = 0 (21)

that higher degeneracy can arise in the way described inSec. IV B, without there being energetically free phaserotations of the type investigated above. In fact, in some casesfor which (21) holds there is additional continuous degeneracy.One such case is that of complete intercomponent symmetryconsidered above; we consider other such cases below.

The fact that continuous degeneracy can arise for parametervalues that fulfill (21) is made plausible by the followingconsiderations. First, if (21) holds then (under our assump-tions) one can reduce (17)–(19) to two equations. With onlytwo equations for three variables, it is not surprising thatone can have continuously connected degenerate minimaof the potential energy. Second, consider the physical roleof the (effective) coupling coefficients η1, η2, η12, and η34.The couplings corresponding to η1 and η2 together favor aphase configuration in which φ1 = φ2 = φ3 + π = φ4 + π .In contrast, the couplings corresponding to η12 and η34 favor aphase configuration in which φ1 = φ2 + π and φ3 = φ4 + π .In the case of (21), there is a balance between these two

η34

η12

φ12

0 1 2 3

3

2

1

0η34

η12

φ13 , φ23

0 1 2 3

3

2

1

0

η34

η12

φ14 , φ24

0 1 2 3

3

2

1

0η34

η12

φ34

0 1 2 3

3

2

1

0 0

1

2

3

4

5

6

FIG. 11. (Color online) Sizes of the ranges over which ground-state phase differences can be varied for signature 11 under theassumption that φ1 and φ2 are equivalent in the ground states (η13 =η23 =: η1 and η14 = η24 =: η2), and that η1η2 − η12η34 = 0. We setη2 = 1. The curve η12η34 = 1 coincides with the corresponding curvein Fig. 7.

tendencies which one could imagine gives rise to continuousground-state degeneracy.

D. Properties of other continuous degeneracies

We now proceed to investigate the continuous ground-statedegeneracies that can occur if (21) is fulfilled [note that theassumption of ground-state equivalence of φ1 and φ2 is implicitin (21)]. Without loss of generality, we set η2 = 1. This leavesus with two degrees of freedom in the ηij . We choose tolet these degrees of freedom be parametrized by η12 and η34;note that variation of these this implies variation of η1. Weconsider the ranges 0 < η12,η34 < 3. For each correspondingpoint in the space of the ηij , we determine the size of therange over which a given phase difference can be variedwithout leaving the set of ground states (Fig. 11). (We avoidthe term ground-state manifold since the ground states do notnecessarily form a manifold.) We see that there are regions withno degeneracy in phase differences, regions with completedegeneracy in certain phase differences, and regions withpartial degeneracy in phase differences.

We now consider the example of three components beingequivalent; this leads to fulfillment of (21), at least underappropriate relabeling. Let the free-energy parameters be suchthat αi = −1, βi = 1, and η23 = η24 = η34 = −1. Considervarious values of the parameters η12 = η13 = η14; we assumethat these are equal in order that ψ2, ψ3, and ψ4 be equivalent.Also, we define η to be the common value of η12, η13, andη14. From Fig. 12, we can see that the ground-state valuesof the phases φ2, φ3, and φ4 are distinct for a range ofvalues of η, and thus there is higher than twofold ground-statedegeneracy. Note that for η = −1 the situation illustratedhere is equivalent to that for γ = π/2 in Fig. 8. For thisparticular value of η, the ground-state degeneracy corresponds

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0

0.5

1Mode 4

0 1 2−1

−0.5

0

0.5

1Mode 5

0 1 2−1

−0.5

0

0.5

1Mode 6

0 1 2−1

−0.5

0

0.5

1Mode 7

0 1 2−1

−0.5

0

0.5

1Mode 8

FIG. 12. (Color online) Ground states, (inverse) length scales, and normal modes for signature 11 with αi = −1, βi = 1, and η23 = η24 =η34 = −1. The magnitude of the parameter η := η12 = η13 = η14 is plotted on the x axes. There are continuous ground-state degeneracies fora range of values of η. Note that for η = −1 the situation illustrated here is equivalent to that for γ = π/2 in Fig. 8; for this value of η, thereare energetically free phase rotations. For some η �= −1 there are other types of continuous degeneracies; η = −0.5 corresponds to Fig. 13 andη = −1.5 corresponds to Fig. 14.

to energetically free phase rotations. For other values ofη (investigated below), there are other kinds of continuousdegeneracies. These degeneracies correspond to the secondmassless mode that is present for an entire range of values ofη (mode 2 in Fig. 12). This mode is a phase-only mode, whichfor η = −1 corresponds to energetically free phase rotations.

Consider the lines given by η12 = 1 in Fig. 11. It is easilyseen that along these lines φ1, φ2, and φ4 are equivalent; weexpress this by saying that φ3 is the special phase. Due to thisequivalence of phases, we expect the lines corresponding toφ12 and φ14 to be equivalent; the same can be said of the linescorresponding to φ13 and φ34. Looking at Fig. 11, we see thatthis appears to be the case. In particular, we see that for η34 � 1there is complete degeneracy in the phase differences φ13 andφ34, whereas for large values of η34 there is no degeneracy inthese phase differences. This is relevant for the cases studiedin Fig. 12, in which φ1 is the special phase. Furthermore, itis easily seen that the same information can be obtained byconsidering the lines given by η34 = 1. In this case, it is φ4

that is the special phase. Also, the scale along these lines isinverted, so that taking the limit η12 → 0 along these linescorresponds to taking the limit η34 → ∞ along lines givenby η12 = 1, and vice versa. Inspection of Fig. 11 appears toconfirm this.

We close this section by considering two more values ofthe parameter η that is varied in Fig. 12 (in addition to thevalue η = −1 already considered). We begin by consideringthe value η = −0.5 (Fig. 13). On the basis of the discussionin the previous paragraph, we expect this value to give riseto complete ground-state degeneracy in the phase differencesφ12, φ13, and φ14. This is indeed what we find. Thus, it mayseem reasonable to say that there are energetically free phaserotations. However, note that here it is not the case that one setof phases can be rotated rigidly relative to another at no energycost. This is evident both from the plot of ground-state phasesin Fig. 13, and from the plot of the second massless mode inthe same figure. The topology of the ground-state manifoldis [U (1)]2 × Z2; this is the product of the broken symmetryU (1) × Z2 and an additional factor of U (1) stemming fromaccidental degeneracy.

We now consider the value η = −1.5 (Fig. 14). Byminimizing the potential energy while keeping φ12 fixed, wefind that φ12 can be varied in the range 1.62 � φ12 � 4.67without leaving the ground-state manifold. By the equivalenceof ψ2, ψ3, and ψ4, this is also true of φ13 and φ14. Note that theground-state degeneracy found here is roughly what one wouldexpect on the basis of the results presented in Figs. 11 and 12.Finally, note that whereas the broken symmetry is U (1) × Z2,

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DANIEL WESTON AND EGOR BABAEV PHYSICAL REVIEW B 88, 214507 (2013)

0 1 2 30

0.5

1

1.5

2Ground−state amplitudes

0 1 2 3

−2

0

2

4

6Ground−state phases

0 1 2 3

0

1

2

3

Masses |ψ1|,m1

|ψ2|,m2

|ψ3|,m3

|ψ4|,m4

φ1,m5

φ2,m6

φ3,m7

φ4,m8

0 1 2 3−1

−0.5

0

0.5

1Mode 1

0 1 2 3−1

−0.5

0

0.5

1Mode 2

0 1 2 3−1

−0.5

0

0.5

1Mode 3

0 1 2 3−1

−0.5

0

0.5

1Mode 4

0 1 2 3−1

−0.5

0

0.5

1Mode 5

0 1 2 3−1

−0.5

0

0.5

1Mode 6

0 1 2 3−1

−0.5

0

0.5

1Mode 7

0 1 2 3−1

−0.5

0

0.5

1Mode 8

FIG. 13. (Color online) Ground states, (inverse) length scales, and normal modes with αi = −1, βi = 1, η23 = η24 = η34 = −1, andη := η12 = η13 = η14 = −0.5 (cf. Fig. 12). The phase difference −φ12 is plotted on the x axes. This phase difference can change by 2π withoutcost in potential energy. This can be seen from Fig. 11 via either of the relabelings 1 ↔ 3 or 1 ↔ 4. Note that the state with φ1 = φ2 isequivalent to that with φ1 = φ4; thus, we could have narrowed the displayed range of φ12 without loss of information.

the topology of the ground-state manifold is in fact [U (1)]2. Inother words, ground states that are related by the Z2 symmetryare in fact connected by accidental continuous degeneracy.

VI. HIGHER HARMONICS

In the above, we have considered only first-harmonicJosephson couplings. In principle, there can also be higherharmonics of the form

|ψi |n|ψj |n cos n(φi − φj ) (n = 2,3, . . .). (22)

In general, the presence of such higher harmonics considerablycomplicates the situation. In particular, higher harmonics cangive rise to metastability. In this section, we make someobservations pertaining to the case of second harmonics.

Consider chiral p-wave superconductors, which in certaincases can be modeled by a two-component model withbiquadratic phase-coupling terms (ψ1ψ

∗2 + c.c.)2, correspond-

ing to n = 2 in (22) (see, e.g., Ref. 38). When expandingthe potential to first order in fluctuations of the densitiesand phases around the ground state, the factor of 2 inthe second-harmonic Josephson coupling simply rescales thestrength of the coupling, as compared to a first harmonic.Thus, the normal modes of two-band superconductors witheither first-harmonic (s or s±) or second-harmonic interband

Josephson coupling are the same. This is true regardless ofthe signs of the couplings, and thus there are four equivalentpossibilities.

From the above we conclude that mixing of phase and den-sity modes is not a generic feature of systems with nontrivial(not 0 or π ) ground-state phase differences (in particular, itis not a consequence of TRSB). Rather, such mixing occurswhen the cosine function in a Josephson coupling term isnot stationary in the ground state, so that perturbation ofthe densities causes perturbation of the ground-state valuesof the phases, and vice versa. We now proceed to give whatis perhaps the simplest example of this, in a system with onlytwo components.

Consider the following Ginzburg-Landau free-energydensity

f = 1

2(∇ × A)2 +

2∑i=1

1

2|Dψi |2 + αi |ψi |2 + 1

2βi |ψi |4

− η12|ψ1||ψ2| cos φ12 − κ12|ψ1|2|ψ2|2 cos 2φ12. (23)

But for the presence of the second-harmonic Josephsoncoupling, this is the same as (2) with n = 2. The inclusion of asecond harmonic can lead to the ground state value of φ12 beingnontrivial. We set η12 = 1 and vary κ12 over the range −0.5 <

κ12 < 0. For definiteness, we also set αi = −1 and βi = 1. By

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CLASSIFICATION OF GROUND STATES AND NORMAL . . . PHYSICAL REVIEW B 88, 214507 (2013)

2 3 40

0.5

1

1.5

2Ground−state amplitudes

2 3 4

−2

0

2

4

6Ground−state phases

2 3 4

0

1

2

3

4Masses |ψ1|,m1

|ψ2|,m2

|ψ3|,m3

|ψ4|,m4

φ1,m5

φ2,m6

φ3,m7

φ4,m8

2 3 4−1

−0.5

0

0.5

1Mode 1

2 3 4−1

−0.5

0

0.5

1Mode 2

2 3 4−1

−0.5

0

0.5

1Mode 3

2 3 4−1

−0.5

0

0.5

1Mode 4

2 3 4−1

−0.5

0

0.5

1Mode 5

2 3 4−1

−0.5

0

0.5

1Mode 6

2 3 4−1

−0.5

0

0.5

1Mode 7

2 3 4−1

−0.5

0

0.5

1Mode 8

FIG. 14. (Color online) Ground states, (inverse) length scales, and normal modes with αi = −1, βi = 1, η23 = η24 = η34 = −1, andη := η12 = η13 = η14 = −1.5 (cf. Fig. 12). The phase difference −φ12 is plotted on the x axes. This phase difference can vary from 1.62 to4.67 (the displayed range) without cost in potential energy. This can be seen (approximately) from Fig. 11 via either of the relabelings 1 ↔ 3 or1 ↔ 4. Note that, due to the equivalence of ψ2, ψ3, and ψ4, the above graphs consist of three equivalent segments (e.g., the state with φ2 = φ3

is equivalent to that with φ2 = φ4).

0 0.2 0.40

0.5

1

1.5

2Ground−state amplitudes

0 0.2 0.4−0.5

0

0.5

1

1.5

2Ground−state phases

0 0.2 0.4

0

1

2

3

4Masses

|ψ1|,m1

|ψ2|,m2

φ1,m3

φ2,m4

0 0.2 0.4−1

−0.5

0

0.5

1Mode 1

0 0.2 0.4−1

−0.5

0

0.5

1Mode 2

0 0.2 0.4−1

−0.5

0

0.5

1Mode 3

0 0.2 0.4−1

−0.5

0

0.5

1Mode 4

FIG. 15. (Color online) Ground states, (inverse) length scales, and normal modes with αi = −1, βi = 1, and η12 = 1 in (23). We plot −κ12

on the x axes. Note that for κ12 < −0.2 there is phase-density mode mixing.

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DANIEL WESTON AND EGOR BABAEV PHYSICAL REVIEW B 88, 214507 (2013)

straightforward extension of the calculations of Sec. III, onecan determine the normal modes and corresponding lengthscales for this system. The results are displayed in Fig. 15.We note that there is indeed phase-density mode mixing whenJosephson coupling terms are not stationary in the groundstate.

VII. CONCLUSION

The abundance of recently discovered multiband supercon-ductors with more than two bands (e.g., iron-based supercon-ductors), and especially the possibility of creating Josephson-junction arrays using these materials, raises the need of alsounderstanding more general multiband superconducting states.Here, we considered phenomenological Ginzburg-Landau andLondon models with an arbitrary number of superconductingcomponents, with particular focus on four-component systems.It should be emphasized that realization of such systems doesnot require an intrinsically multiband system; rather, this canbe achieved using real-space Josephson-coupled multilayersor Josephson-junction arrays.

We discussed the fact that the free energies (1) and (2) areinvariant under the transformation which (i) inverts a givenphase (i.e., adds π to this phase), and (ii) changes the signof all coupling coefficients involving this phase. Thus, apartfrom the trivial equivalence of signatures (corresponding torelabeling of components) which we call strong equivalence,there exists another equivalence of signatures (involvingthe aforementioned transformation), which we call weakequivalence.

We considered a graph-theoretical approach that allowed usto establish equivalence classes of multicomponent systemswith frustrated intercomponent Josephson couplings. In thisapproach, we used a mapping where each component corre-sponds to a vertex and each Josephson coupling correspondsto an edge. We thus found the following for an n-componentsystem: The number of strong-equivalence classes is equalto the number of graphs on n (unlabeled) vertices, and thenumber of weak-equivalence classes is equal to the numberof switching classes of such graphs, i.e., the number ofequivalence classes under Seidel switching.

We calculated ground states, normal modes, and charac-teristic length scales for frustrated multicomponent supercon-ductors modeled by the free-energy density (2). We emphasizethat the reported mixed phase-density modes are decoupledfrom the U(1) sector. In the systems considered here, there canbe fluctuation-driven phase transitions to anomalous normalstates with broken discrete symmetry. The mixed modes cansurvive in these states due to the decoupling. For the caseof four-component superconductors, we considered the twoweak-equivalence classes of frustrated signatures that we call

singly frustrated and multiply frustrated. We found that theproperties of singly frustrated four-component systems arelargely similar to those of phase-frustrated three-componentsystems.

In contrast, we found that multiply frustrated four-component signatures allow for qualitatively new featuresnot present in the cases of two and three components.These are associated with accidental continuous ground-statedegeneracies, e.g., in the form of energetically free phaserotations. These degeneracies exist despite Josephson couplingbetween all phase pairs. More precisely, the degeneracies wehave found can arise when at least two phases are equivalentin a ground state (in the weaker sense mentioned in Sec. IV B).The existence of massless modes for some points in parameterspace could lead to a number of interesting states for a range ofparameters. For example, near such points in parameter spaceone of the coherence lengths can be anomalously large, leadingto type-1.5 superconductivity.13

Note that the aforementioned continuous ground-statedegeneracies, which do not correspond to spontaneouslybroken symmetries of the free energy, are such that theground states do not all have the same length scales andnormal modes. Furthermore, in the frustrated case of completeintercomponent symmetry, we found that the ground states donot form a manifold, whence there can be no correspondingLie group.

Finally, we briefly considered systems with higher harmon-ics in the Josephson couplings. In doing so, we found that suchsystems typically display phase-density mode mixing, evenin the simplest case of only two components. However, thisis not the case for chiral p-wave superconductors;38 rather,these have the same normal modes as two-component s-wavesuperconductors.

In this paper, we used an entirely phenomenologicalapproach. Our results suggest that the four-component caseis substantially richer than the better-investigated frustratedthree-component case. This calls for further microscopicinvestigation of these new states based on approaches likethose in, e.g., Refs. 24 and 39.

ACKNOWLEDGMENTS

We thank J. Carlstrom, J. Garaud, and M. Speight fordiscussions. This work was supported by the Knut and AliceWallenberg Foundation through a Royal Swedish Academyof Sciences Fellowship, by the Swedish Research Council,and by the National Science Foundation CAREER Award No.DMR-0955902. We thank the Swedish National Infrastructurefor Computing (SNIC) at the National Supercomputer Centerat Linkoping, Sweden, for computational resources.

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