Polarization of spontaneous magnetic field and magnetic fluctuations in $s+is$ anisotropic multiband superconductors

We show that multiband superconductors with broken time-reversal symmetry can produce spontaneous currents and magnetic fields in response to the local variations of pairing constants. Considering the iron pnictide superconductor Ba$_{1-x}$K$_x$Fe$_2$As$_2$ as an example we demonstrate that both the point-group symmetric $s+is$ state and the C$_4$-symmetry breaking $s+id$ states produce in general the same magnitudes of spontaneous magnetic fields. In the $s+is$ state these fields are polarized mainly in ab crystal plane, while in the $s+id$ state their ab-plane and c-axis components are of the same order. The same is true for the random magnetic fields which are produced by the order parameter fluctuations near the critical point of the time-reversal symmetry breaking phase transition. Our findings can be used as a direct test of the $s+is/s+id$ dichotomy and the additional discrete symmetry breaking phase transitions with the help of muon spin relaxation experiments.

Introduction. Superconducting states with spontaneously broken time-reversal symmetry (BTRS) have been recently in the focus of interest. First such states have been studied in connection with the chiral p-wave order parameter in the superfluid 3 He A phase [1] and Sr 2 RuO 4 superconducting compound [2]. More recently, s + id and s + is states has been suggested as the candidate order parameters in multiband iron pnictide compounds [3][4][5][6][7][8][9]. Recent experiment [10] supports this hypothesis demonstrating the presence of spontaneous currents in the ion irradiated samples of Ba 1−x K x Fe 2 As 2 in the certain doping level interval.
Spontaneous currents were predicted to exist near impurities in s + id superconducting states which spontaneously break the C 4 crystalline symmetry of the parent compound [3]. As for the s+is states, initially it has been claimed that magnetic field can appear only in samples subjected to strain [11]. However, this conclusion was made based on the specific circularly-symmetric model of the impurity.
More general consideration has shown [12,13] that magnetic fields in the s + is state can be generated without strain in the presence of the general-form inhomogeneities of the order parameter. They can be induced e.g. by the domain wall between s + is and s − is states [14], attached to the sample edge or by any external controllable perturbation such as the local heating. Later the particular case of two-dimensional defects elongated along the crystal c-axis and forming square shapes in the ab plane have been studied [15]. In such system the spontaneous magnetic field generated in s + is state is several order of magnitude smaller than in the s + id one. The purpose of the present paper is threefold. First, Figure 1. (Color online) The schematic picture of spontaneous magnetic fields generated in the s + is/s + id superconductors by the cylindrical inhomogeneities with the axis directed either perpendicular (ca-defect) or along the crystal anisotropy axis z(c) (ab-defect). In the general case when both types of defects are present the s + is state has dominant component of B in ab-plane, while in the s + id all components are of the same amplitude.
we show that the spontaneous magnetic field is generated both in the s + is and s + id state due to the generalform inhomogeneities of the pairing interactions. Such form of disorder can exist in the sample even without the externally generated defects just due to the spatiallyinhomogeneous doping level. Second, we demonstrate that in the general case, when the system is inhomogeneous both in the ab-plane and in the c-direction s + is and s + id states yield the same magnitudes of spontaneous fields. However, as shown schematically in Fig.(1) this regime is characterized by the qualitatively different polarizations of the spontaneous field in s + id and arXiv:1804.10281v1 [cond-mat.supr-con] 26 Apr 2018 s + is states. This qualitative prediction can be used for resolving the s + id/s + is dichotomy in real materials. Third, we demonstrate that the order parameter fluctuations near the BTRS phase transition generate random magnetic fields with the critical correlation radius. Thus, the discrete symmetry breaking phase transition can be revealed through the magnetic field fluctuations.
Three-band model. Here we develop general treatment of spontaneous magnetic fields in BTRS states further considering inhomogeneities created by the spatial variation of pairing constants in the minimal three-band microscopic model [6,16,17] with three distinct superconducting gaps ∆ 1,2,3 residing in different bands. The pairing which leads to the BTRS state is dominated by the competition of two interband repulsion channels η 1,2 > 0 described by the following coupling matrix We assume for simplicity that the density of states ν 0 is the same in all superconducting bands. This model can be used for both the s + is and s + id states. In the former case ∆ 1,2 correspond to the gaps at hole pockets and ∆ 3 is the gap at electron pockets, so that u hh = ν 0 η 1 and u eh = ν 0 η 2 are respectively the hole-hole and electron-hole interactions [6,17]. The same model (1) can be used to describe the s + id states but there, ∆ 1,2 describe gaps in electron pockets and ∆ 3 is the gap at the hole Fermi surface, so that u eh = ν 0 η 1 and u ee = ν 0 η 2 are electron-hole and electron-electron interactions respectively [13,18]. The physical origin of spontaneous magnetic field generation follows from that the total current is the sum of partial currents in each of N bands j = N k=1 j k and therefore can generated by the gradients of relative phases [12]. Since j k =λ −2 k (∇θ k −ẽA)/(4πẽ) the London expression for magnetic field is modified in multiband superconductors as follows where we use the units with = c = 1. Hereλ k are in general the tensor coefficients characterizing contribution of each band to the Meissner screening. In the clean limit they can be expressed as followŝ whereK k = v k v k is the anisotropy tensor, v k is the Fermi velocity in k-th band normalized to the certain band-independent characteristic velocityv F . We normalize the gaps by T c / √ ρ, where ρ = n πT 3 c ω −3 n ≈ 0.1 and T c is the critical temperature, magnetic field by B 0 = T c ν 0 /ρ, which is close to the thermodynamic critical field at zero temperature [19]. Length is normalized by the Cooper pair size ξ 0 =v F /T c , and we introduce the dimensionless Cooper pair charge isẽ = 2eξ 2 0 B 0 . The London penetration depth is given byλ −2 is the tensor coefficient characterising magnetic field generated by the variations of interband phase differences.
In contrast to the usual London electrodynamics, the multicomponent superconducting systems can generate spontaneous magnetic fields due the second term in Eq.(2) acting as a source. It appears in general due to the presence of interband phase difference gradients. They can be generated e.g. by the spatially-dependent pairing interactions (1) according to the two distinctive mechanisms described below.
First, the second term in Eq. (2) is non-zero due to the anisotropy of superconducting state characterized by the tensorsγ ki . This scenario is generic for the s + id state when all three spatial components of these tensors are different. The s + is state is isotropic in ab (xy) plane, but there is anisotropy in c (z) direction, so that γ x ki = γ y ki = γ z ki . In this case in the linear order the abplane inhomogeneities are decoupled from the magnetic field. However, in general the systems is inhomogeneous along the c-axis direction as well, which yields the magnetic response of the same magnitude as the s + id state. Second, the magnetic field can be generated even in the particular case of s + is state with 2D spatial inhomogeneities in the ab-plane. However, this mechanism relies on the non-linear coupling between the current and the gradients if pairing coefficients. Indeed in the linear response regime we can assume thatγ ki = γ kiÎ , wherê I is the unit matrix and γ ki = const when the second term in Eq.(2) vanishes. The non-linear coupling is realized when the relative density gradients ∇γ ki (r) are non-collinear with that of relative-phase gradients ∇θ ki . Below we show that such a coupling generically appear in s+is states described by the model (1) with the spatiallydependent pairing coefficient η 2 = η 2 (r), although the magnitude of the spintaneous magnetic field is significantly smaller than in the s + id state.
Ginzburg-Landau calculation. To go beyond local approximation we can calculate spontaneous magnetic fields using Ginzburg-Landau theory derived for s+is/s+ id states [13,18] corresponding to the model (1). The general free energy density, normalized to B 2 0 is given by F = F s + B 2 /8π, where the Ginzburg-Landau (GL) free energy describing both the s+is and s+id states is given by whereΠ = ∇ − iẽA. This model is formulated in terms of the two order parameters ψ 1 and ψ 2 which are related to the individual gap functions within separate bands as (∆ 1 , ∆ 2 , ∆ 3 ) = (aψ 2 − ψ 1 , aψ 2 + ψ 1 , ψ 2 ), where a = (η 1 − η 2 1 + 8η 2 2 )/4η 2 . The coefficients that determine gradient terms in Eq.(4) are combined from the anisotropy tensors characterising each superconducting band as followŝ The difference between s+is and s+id symmetries is determined by the structure of mixed-gradient coefficients (7) in ab plane. That is for s+is state k 12,x = k 12,y ≡ k ab 12 and for s + id state k 12,x = −k 12,y ≡ k ab 12 . Despite having quite different properties in the ab plane both states are characterized by the same anisotropy along the c axis determined by the coefficients K z k ≡ K c k = K ab k . The 122 iron pnictide compounds has been shown to feature moderate anisotropy [20] therefore we will use for estimations that K c k /K ab k ≈ 1 − 3. In case of the s + is superconductor it is of the crucial importance for the linear coupling between the magnetic field and pairing constant inhomogeneities that the at least two bands should have different anisotropies, e.g. K c 1 /K ab Otherwise, by using the scale transformation the problem is reduced to the case of the fully isotropic s + is superconductor when only the non-linear coupling is possible yielding much smaller spontaneous currents.  The other coefficients in GL expansion (4) are given by α k = a k , β k = b kk , γ = b 12 and δ = b J , expressed, in terms of the coefficients of the coupling matrix (1) as where G 1 = 1/η 1 and G 2 = η 1 + η 2 1 + 8η 2 2 /4η 2 2 are the positive eigenvalues of the matrix (1)Λ −1 and G 0 = min(G 1 , G 2 ).
Below, we consider the spontaneous magnetic field produced by the superconducting currents generated by the inhomogeneities of the pairing constant η 2 = η 2 (r). At first let us consider the 2D inhomogeneities in the ab plane, so that η 1 = 1 and η 2 (x, y) = 1 + 0.5 sin(x/2) sin(y/2).
This model allows for demonstrating differences between the linear and non-linear mechanisms of the spontaneous current generation, where the former takes place for s+id and the latter is s + is pairings. The magnetic field produced by xy-inhomogeneities has only z-component (Fig.1). The calculated distribution of B z = B z (x, y) is shown in Fig.2. The vector potential in the numerical calculations is normalized by the value of Φ 0 /(2πξ 0 ) where Φ 0 is the superconducting flux quantum, so the the magnetic field is given in the units ofẽB 0 which has an order of the second critical field H c2 . One can see that for one and the same set of parameters s + is state yields the spontaneous magnetic field response about 10 3 times smaller than s + id, which is consistent with the results obtained before [15]. Except of the special case of ab-plane inhomogeneities, in general the s+is and s+id states produce the magnetic fields of comparable amplitudes. To demonstrate this we compare responses produced by the Gaussian pairing constant variation given by η 2 = 1 + 0.5e −(x 2 +y 2 )/2 s + id state (13) The former inhomogeneity (13) corresponds to the abplane defect, while the latter (14) is the ca-plane defect.
To obtain spontaneous fields produced by ca-plane Gaussian defects in s + is case we assume that there is c-axis anisotropy set by the choice of coefficient ratio in different bands K ab 1 = 1, K ab 2 = 1.5, K ab 3 = 0.5, K c 1 = 1.5, K c 2 = 2, K c 3 = 0.75. Such system yields the magnetic field component B y (x, z) shown in Fig.3c. One can compare it with the qualitatively similar distribution of B z component produced by the Gaussian ab-plane inhomogeneity (13) in the s+id state shown in Fig.3f. These two characteristic cases demonstrate that spontaneous magnetic responses of s + is and s + id in general are the   (13,14) and η1 = 1. The upper row corresponds to the ca-plane defect in s + is superconductor with anisotropy parameters for s + is are K ab 1 = 1, K ab 2 = 1.5, K ab 3 = 0.5, K c 1 = 1.5, K c 2 = 2, K c 3 = 0.75. The lower row corresponds to ab-plane defects in s + id state characterized by K ab 1 = 1, K ab 2 = 1.5, K ab 3 = 0.5. GL parameterẽ = 4 for both cases.
same. However, one can conclude that the largest spontaneous magnetic field in the s + is case appears in the direction perpendicular to the anisotropy axis. Therefore on can distinguish between these states by analysing the polarization of spontaneous magnetic fields.
Based on the above analysis one can suggest the polarization-sensitive test of the superconducting state symmetry based. That is, under general conditions, the spontaneous magnetic field in s + is state is directed mostly in the ab-plane, with the typical ratio of components B z /B ab ∼ 10 −3 . On the other hand, s + id state produces spontaneous fields which have in general all components with the same order B z /B ab ∼ 1.
Critical magnetic fluctuations. The spontaneous magnetic field produced by the order parameter inhomogeneities allows for the direct observation of the critical phenomena and fluctuations near the the phase transition to the broken time-reversal symmetry state. To demonstrate that we introduce the order parameter η = −i(ψ 2 ψ * 1 )/|ψ 1 |. Further we assume that ψ 1 = |ψ 1 |e iϕ1 and |ψ 1 | = const. Introducing the gauge-invariant momentum Q = A − ∇ϕ 1 /ẽ, the real and imaginary parts of the complex order parameter η = η r + iη im allows for representing the GL free-energy as follows Hereα r = α 2 + |ψ| 2 (δ − γ) andα im = α 2 + |ψ 1 | 2 (δ + γ). Equationα r (T ) = 0 gives the critical temperature of Z 2 transition T = T Z2 . In the vicinity of this transition only the fluctuations of η r are important as theα im is positive and non-vanishing. Therefore we can describe the timereversal symmetry breaking phase transition in terms of the real-valued order parameter η r : Note that the real order parameter η r is still coupled to the magnetic field, which can be demonstrated as follows.
The above derivation demonstrates that the spatial inhomogenuities of η r with necessity produce the spontaneous magnetic field. That means fluctuations near the Z 2 critical point become magnetic. The variance of magnetic field components, e.g. the one lying in ab-plane B y is given by For simplicity we consider the limiting case when the cross-coupling gradient terms in the functional (16) are rather smallk 12 √ k 11 k 22 when the feedback of magnetic field fluctuations can be neglected. Then, fluctuations of the order parameter η r near the critical temperature T Z2 can be calculated using the conventional expression [21] η 2 r (q) = T /[2V (k 22 q 2 + |α r |)] . Now, we can calculate the correlation function of the spontaneous magnetic field component B y (0)B y (r) = V d 3 q B 2 y (q) e iqr . Considering the long-range part of the fluctuations we obtain B y (0)B y (r) = 8πλ 2 k11 (k 12z − k 12x ) 2 ∂ 4 xxzz G(r), where G(r) = (T /8πk 22 )e −r/rc /r is the order parameter correlation function with the correlation radius r c = k 22 /|α r |. Thus one can see that fluctuation of magnetic field has the same critical radius as the time-reversal symmetry breaking order parameter. These spontaneous fields provide therefore the direct access to the previously hidden critical behaviour near the discrete symmetry-breaking phase transitions.
Conclusion. To summarize, we have shown that in general the s+id and s+is phases in multiband superconductors can produce spontaneous currents and magnetic fields in response to the spatial inhomogeneities caused by either the fluctuations of the pairing constants or the critical fluctuations of the order parameter components. This is in contrast to the previous predictions that s + is state has much weaker magnetic signatures. However, the spontaneous field polarization is found to be drastically different in s+is and s+id states making it possible to distinguish between them experimentally. The random magnetic fields produced by the scalar order parameter fluctuations can reveal the critical behaviour near the BTRS transition and in general any additional discretesymmetry breaking phase transition deep in the superconducting state.