Spontaneous magnetization in time-reversal symmetry-breaking unitary superconductors
SSpontaneous magnetization in time-reversal-breaking unitary superconductors
Lun-Hui Hu ∗ and Xuepeng Wang † Department of Physics, Pennsylvania State University, University Park, State College, PA 16802, USA National Laboratory of Solid-State Microstructures,School of Physics, Nanjing University, Nanjing 210093, China
In this work, we study the spontaneous magnetization (spin-polarization) in the time-reversalsymmetry (TRS) breaking superconductors with unitary pairing potentials. Generally, the spin-single and spin-triplet pairings coexist in superconductors without inversion symmetry. And theTRS can be spontaneously broken once a relative phase of ± π/ Introduction–.
In condensed matter physics, supercon-ductivity and magnetism are generally antagonistic toeach other [1–3] and the interplay between them bringsus intriguing phenomena. One of them is the famousFulde-Ferrell-Larkin–Ovchinnikov (FFLO) state [4, 5] inwhich the superconducting Cooper pairs carry a finitemomentum induced by an external Zeeman field. Re-cent theoretical efforts have been made to realize the chi-ral Majorana modes in the topological FF phase [6, 7]and the Majorana mode chain in the topological LOphase [8]. Another fascinating phenomena is the sponta-neous spin-polarization (SP) in a time-reversal symmetry(TRS) breaking superconductor (SC) [9–11], which con-tinuously promotes extensive experimental and theoret-ical research [12, 13]. The TRS breaking candidate SCsinclude Sr RuO [14–16], Re Zr [17, 18], UPt [19–21],PrOs Sb Si [23, 24], SrPtAs [25], Ru B [26, 27], LaNiC [28], LaNiGa [29, 30], Bi/Ni bilayers[31], CaPtAs [32], and others summarized in a recent pa-per [13]. More recently, iron-based superconductors alsoexhibit TRS breaking signatures [33, 34]. These excit-ing experimental discoveries arouse considerable atten-tions, and a great deal of theoretical progress has recentlybeen made on the TRS breaking SCs with mixed pairingstates [35–45], non-unitary pairing states [46–57], Bogli-ubov Fermi surface [58–61].There are mainly two direct ways to probe the TRSbreaking pairing states, including the zero-field Muon-spin relaxation or rotation ( µ SR) [62–64], and the polarKerr effect (PKE) [65, 66]. Firstly, the µ SR is especiallyvery sensitive to a small change of the internal fields,and thus can be used to measure the fields of 10 µ T.And the observation of an enhancement of the zero-fieldmuon spin depolarization rate provides direct evidence tothe stabilized TRS breaking pairing states. Besides, thePKE measures the optical phase difference between twoopposite circular-polarized lights reflected on a samplesurface, thus it gives information about the TRS of thesystem. And a finite PKE unambiguously points to theTRS breaking states. Practically, the non-unitary spin-triplet pairing potential could spontaneously induce the spin-polarization (magnetization) in a homogeneous su-perconductor, naturally explaining the experimental ob-servations by the µ SR and the PKE. However, one maystill wonder if there is any mechanism other than the for-mer theoretically proposed non-unitary spin-triplet pair-ing states to induce the SP of TRS breaking noncen-trosymmetric SCs.The answer is YES. In this work, we report the dis-covery of the spontaneous spin-polarization (SP) in theTRS breaking unitary superconductors. The spin-singletpairing (∆ s ) coexists with the spin-triplet pairing (∆ t )in both noncentrosymmetric superconductors and two-dimensional superconductors that are grown on insulat-ing substrates. Once a relative phase of ± π/ s ± i ∆ t ), the TRS is spontaneously broken.By combing the symmetry analysis and the Ginzburg-Landau theory, we find that the interplay between thespin-orbit coupling (SOC) and the ∆ s + i ∆ t or ∆ s − i ∆ t unitary pairing potential could give rise to the SP of a ho-mogeneous superconductor. The direction of the inducedSP is perpendicular to both the SOC (cid:126)g -vector and thespin-triplet (cid:126)d -vector, while both (cid:126)g and (cid:126)d are real vectors.It leads to a new scenario to TRS breaking superconduc-tors of which TRS breaking signatures are observed bythe µ SR and the polar Kerr effect. Furthermore, we alsodiscuss the effect of the spontaneous spin-polarizationon first-order (second-order) topological superconductorswhich support topological edge (corner) states.
Superconductors with TRS breaking unitary pairing–.
In the absence of the external magnetic field or compet-ing magnetic orders, the non-vanishing spin-polarization(SP) of the superconducting states generally attributesto the breaking of the time-reversal symmetry (TRS). Itcan be generated by the non-unitary pairing states witha complex spin-triplet (cid:126)d -vector, hence the SP directionis parallel to (cid:126)d × (cid:126)d ∗ . However, in this Letter, we explorethe spontaneous SP induced by the TRS breaking unitarypairing states in superconductors (SCs) without inversionsymmetry. For this propose, we start with a single-band a r X i v : . [ c ond - m a t . s up r- c on ] F e b model Hamiltonian without the inversion symmetry [67], H = ξ k σ + α(cid:126)g · (cid:126)σ, (1)where ξ k = k / m − µ is the electron band energy mea-sured from the Fermi energy µ , (cid:126)σ denotes the Pauli ma-trices for spin, and α is the strength of spin-orbit coupling(SOC). The inversion symmetry is broken by (cid:126)g ( k ) = − (cid:126)g ( − k ). In this work, we mainly discuss the effect ofthe Rashba-type SOC which is given by (cid:126)g = ( − k y , k x ),allowed by the C v point group. There are two Fermisurfaces with opposite chirality, and the two bands are (cid:15) k, ± = ξ k ± α | (cid:126)g | , and the Fermi momentum is k F = √ mµ for α → H int = (cid:88) k , k (cid:48) (cid:88) s ,s V k , k (cid:48) c † k ,s c †− k ,s c − k (cid:48) ,s c k (cid:48) ,s , (2)here s , are spin indexes. Applying the mean-field de-compositions, we define the gap functions as ∆ s ,s ( k ) = (cid:80) k (cid:48) V k , k (cid:48) (cid:104) c − k (cid:48) ,s c k (cid:48) ,s (cid:105) . Here (cid:104)· · · (cid:105) represents averag-ing over the thermal equilibrium states. After ignoringthe fluctuations, the mean-field pairing Hamiltonian be-comes, H ∆ = (cid:88) k (cid:88) s ,s ∆ s ,s ( k ) c † k ,s c †− k ,s + h.c. . (3)Due to the breaking of inversion symmetry, the even-parity pairing coexists with the odd-parity pairing [67].In particular, the gap function matrix is given by ˆ∆( k ) =(∆ s ψ ( k ) + ∆ t (cid:126)d ( k ) · (cid:126)σ ) iσ y , with ψ ( k ) = ψ ( − k ) and thereal spin-triplet (cid:126)d -vector (cid:126)d ( k ) = − (cid:126)d ( − k ) required by theFermi statistic. The SC gaps ∆ s ( t ) = | ∆ s ( t ) | exp( iθ s ( t ) )are generally complex for the spin-singlet(triplet) pairingstates. To break the TRS T = iσ y K with K being com-plex conjugate, the relative phase θ ts = θ t − θ s shouldbe nonzero. Typically, θ ts could be pinned to ± π/ s ± i ∆ t ,because of ˆ∆( k ) † ˆ∆( k ) = | ∆ s ψ ( k ) | + | ∆ t | (cid:126)d ( k ) · (cid:126)d ( k ).At the mean-field level, the non-interacting Hamilto-nian H is not diagonalized due to the off-diagonal SOC,which may reveal the possibility to the spontaneous SPof the superconducting states. The angular form fac-tors of the spin-singlet pairing could be either s-wave( ψ s ( k ) ∼ { , cos k x + cos k y , cos k x cos k y } ) or d-wave( ψ d ( k ) ∼ cos k x − cos k y ). As for the p-wave pairing sym-metries, we focus on the in-plane (cid:126)d vectors for simplicity,and there are four possibilities, (cid:126)d A ( k ) = ( − sin k y , sin k x ) , (cid:126)d A ( k ) = (sin k x , sin k y ) ,(cid:126)d B ( k ) = (sin k y , sin k x ) , (cid:126)d B ( k ) = (sin k x , − sin k y ) . (4)They belong to different 1D irreducible representations FIG. 1. Illustrations of the p-wave pairings and the bound-ary spin-polarization. (a) The four p-wave pairing symmetrieswith in-plane (cid:126)d vector, labeled by the black arrows. (b) Thereal space viewpoint of the boundary spin-polarization andthe red arrows show the direction of the SP on each bound-ary of a two-dimensional rectangular sample. (c) The mo-mentum space viewpoint of the spin-polarization. By sym-metry constraints, the (cid:126)M A -type SP could be induced by theTRS breaking unitary pairings ∆ s + i ∆ t , including the s-wavepairing and (cid:126)d A ( k ) (or d-wave and (cid:126)d B ( k )). And the inducedspin-polarization is parallel with (cid:126)g × (cid:126)d . of the C v point group, shown in Fig. 1(a). Note that allthe spin (cid:126)d -vectors in Eq. (4) have odd parity, and all the (cid:126)d -vectors are real so that | (cid:126)d ∗ × (cid:126)d | vanishes. In this work,we consider a relatively small SOC αk F < k B T c so thatthe spin-triplet (cid:126)d -vectors that are not parallel with theSOC (cid:126)g -vector could be stabilized [68, 69]. Spontaneous spin-polarization–.
Next, we use the phe-nomenological Ginzburg-Landau (GL) theory to addressthe spontaneous magnetization induced by ∆ s ± i ∆ t . Inparticular, the symmetry allowed free energy reads, F = F + F + γ (∆ ∗ s ∆ t ) + ( (cid:126)γ · (cid:126)M )∆ ∗ s ∆ t + c.c. , (5)where F = α s ( T ) | ∆ s | + α t ( T ) | ∆ t | and F = β s | ∆ s | + β t | ∆ t | + β st | ∆ s | | ∆ t | . And γ (cid:54) = 0 indicates the pairingbreaks the TRS since the relative phase between the sin-glet and triplet pairings is developed as ± π/ ∗ s ∆ t pines the phase differenceto an arbitrary non-zero value [70] in the low tempera-ture. In this work, we assume ∆ s and ∆ t belong to dif-ferent representations of the lattice symmetry group thusthe bilinear coupling is forbidden. The γ -terms couplethe spin-polarization (cid:126)M and the TRS breaking pairing∆ s ± i ∆ t , which satisfy both the global U (1) symme-try ( θ s,t → θ s,t + δθ ) and the TRS. Once (cid:126)γ (cid:54) = 0, theSP (cid:126)M is spontaneously induced by the ∆ s ± i ∆ t pairingpotentials with the help of the SOC. The bulk magneti-zation vanishes in a purely clean system due to two rea-sons: the translational symmetry and the Meissner effect[54, 71, 72]. As for a disordered system, the SP couldpersist around impurities in the bulk, thus it would beable to be detected by experimental measurements, as dothe µ SR and the PKE.Orders Mirror σ v Mirror σ d R z s-wave + + +d-wave + − − (cid:126)d A + + + (cid:126)d A − − + (cid:126)d B + − − (cid:126)d B − + − M A − − + TABLE I. The character table for all orders (spin-single/triplet pairings and spin-polarization) based on the C v point group. The independent symmetry operators involvethe mirror σ v , σ d and the four-fold rotation R z . Before detailed calculations, we use symmetry analysisto classify the (cid:126)γ -vector. The free energy in Eq. (5) is re-quired to preserve all the crystalline symmetries, whichimposes strict constraints on the direction of the bound-ary magnetization. In this work, we consider the pointgroup C v to identify the ferromagnetic-type SP inducedby the TRS breaking unitary pairing potential. Also,a full classification for the (cid:126)γ -terms by different sym-metry groups is left for future works. The C v pointgroup is generated by there independent symmetry oper-ators ( σ v , σ d and R z ), here σ v are the vertical reflectionplanes along x and y , σ d are the diagonal reflection planesalong the x ± y lines, and R z is the four-fold rotationalong z axis. To break both σ d and σ v , we focus on the (cid:126)γ = (0 , , γ z ) case, where the spontaneous SP on eachboundary are aligned along the z-direction (perpendicu-lar to the sample), as shown in Fig. 1(b). It is marked asthe M A -type SP in Table. I. By symmetry constraints,the SP could be induced by the interplay of the s-wave(d-wave) singlet pairing and the (cid:126)d A ( k ) ( (cid:126)d B ( k ))-tripletpairing, ˆ∆ s + ip = (cid:16) ∆ s Ψ s ( k ) + i ∆ t (cid:126)d A ( k ) · (cid:126)σ (cid:17) iσ y , ˆ∆ d + ip = (cid:16) ∆ s Ψ d ( k ) + i ∆ t (cid:126)d B ( k ) · (cid:126)σ (cid:17) iσ y . (6)Both the s + ip - and d + ip -pairing states are fully gappedin a 2D superconductor, while there may be nodal pointsin 3D superconductors.The underlying microscopic avenue to the SP is be-yond the symmetry arguments. In this work, we analyzethe SP for a single-band Hamiltonian in the lack of theinversion symmetry, and the results here could also begeneralized to the multi-band systems. Therefore, wecalculate the coupling coefficient γ for the Hamiltonian H + H ∆ , γ i = 1 β (cid:88) k ,ω n Tr (cid:104) G h ( (cid:126)d · (cid:126)σ ) G e σ i G e σ (cid:105) , (7)with i = { x, y, z } , β = 1 / ( k B T ) is the inverse of tem-perature. In the Nambu basis, the Matsubara Green’sfunction is G e ( k , iω n ) = [ iω n − H ( k )] − with ω n =(2 n + 1) π/β and G h ( k , iω n ) = G ∗ e ( − k , − iω n ). Afterstraightforward calculation, we find that the directionof (cid:126)M is perpendicular to both (cid:126)d vector and spin-orbitcoupling (cid:126)g , namely, (cid:126)γ (cid:107) (cid:88) k (cid:68)(cid:16) (cid:126)g ( k ) × (cid:126)d ( k ) (cid:17) · ψ s ( k ) (cid:69) F S , (8)where (cid:104)· · · (cid:105) F S denotes the average over the entire Fermisurfaces. Here we take the ∆ s + i ∆ t with s-wave pairingand (cid:126)d A ( k ) as an example [see Fig. 1(c)], where (cid:126)g × (cid:126)d A ≈ (cid:126)e z (sin k x + sin k y ). By considering the weak SOC limit( αk F < k B T c ), we find the nonzero z-component of (cid:126)γ tothe leading order of αk F /k B T c as, γ z = − i ζ (3)8 π αk F k B T c , (9)where ζ ( z ) is the Riemann zeta function. Since the SPis induced by the TRS breaking unitary pairing ∆ s + i ∆ t via the Rashba SOC, the coupling coefficient γ z isproportional to the SOC strength α . And the sign of theSP is determined by sign( M z ) =sign( α ∆ t ∆ s ). SOC strength | M | z FIG. 2. The spontaneous spin-polarization induced by the d + ip unitary pairing potential given by Eq. (6). The aver-aged SP on the boundary is calculated as a function of SOCstrength. Parameters used here are: m = ∆ s = ∆ t = 1. To be more explicitly, we perform a numerical calculatefor the averaged spin-polarization based on the solutionof the Bogoliubov–de Gennes (BdG) Hamiltonian. Bycombing the non-interacting Hamiltonian in Eq. (1) withthe pairing Hamiltonian in Eq. (3), we have the BdGHamiltonian as, H BdG = (cid:18) H ( k ) H ∆ H † ∆ −H ∗ ( − k ) (cid:19) , (10)where the Nambu basis (cid:16) c † k , ↑ , c † k , ↓ , c − k , ↑ , c − k , ↓ (cid:17) T is used.And the averaged boundary spin-polarization is definedas, ˜ M z = 1 N l (cid:88) (cid:126)l (cid:88) E n (cid:104) E n ( (cid:126)l ) | ˆ P † e σ z ( (cid:126)l ) ˆ P e | E n ( (cid:126)l ) (cid:105) BdG , (11)where (cid:126)l is the “edge coordinate” on a N x × N y rectangu-lar lattice and (cid:126)l ∈ { (1 , i y ) , ( i x , N y ) , ( N x , i y ) , ( i x , } withtotal sites N l = 2 N x + 2 N y −
4, ˆ P e is the projection op-erator into the particle subspace, and | E n (cid:105) is solution ofthe BdG equation H BdG | E n (cid:105) = E n | E n (cid:105) .We take the d + ip pairing states in Eq. (6) for an ex-ample. And the numerical result is shown in Fig. 2, con-firming that the averaged spontaneous SP ˜ M z = 0 in theabsence of SOC and it increases as we increase the SOCstrength α . So the numerical calculation is consistentwith the analytical analysis for γ z in Eq. (9). Besides,one may estimate the value for ˜ M z ∼ . α = 2 . · nm, k F = 1 nm − , ∆ s = 1 meV, ∆ t = 1meV and T c ≈
10 K. Consequently, we argue that theSP induced by the TRS breaking unitary pairing statesis large enough for experimental measurements. There-fore, our results could provide an alternative explanationto both the µ SR and the PKE experiments where TRSbreaking superconducting states are observed for noncen-trosymmetric superconductors.
The effects of the SP on topological SCs–.
It is alsoworthwhile to study the effect of the boundary SP onfirst-order (second-order) topological superconductors,which host topological Majorana edge (corner) states.From the viewpoint of the symmetry arguments, we learnthat two singlet-triplet mixed gap functions ∆ s + ip and∆ d + ip in Eq. (6) not only break the TRS but also lead tothe boundary SP with the help of the Rashba-type SOC.We firstly discuss the s + ip pairing states. It is wellknown that a purely p-wave superconductor supports apair of helical Majorana edge modes (HMEM) protectedby the TRS [73–75], which is descried by the edge Hamil-tonian H edge ( (cid:126)l ) = A ( (cid:126)l ) k l s z . Here k l is the momentum onedge l and (cid:126)s are the Pauli matrices defined for the HMEMbasis. Both the s-wave pairing potential and the bound-ary SP are the sources for the mass generation to theHMEMs, resulting in the edge Hamiltonian as, H edge ( (cid:126)l ) = A ( (cid:126)l ) k l s z + ( M s ( (cid:126)l ) + M sp ( (cid:126)l )) s x , (12)with M s ( (cid:126)l ) M sp ( (cid:126)l ) >
0, thus the HMEMs are fully gappedon each boundary with the same mass sign as expected.As for the d + ip case, a second-order topological super- conductor (HOTSC) is achieved, [76–93], which supportstopological Majorana corner states (MCS). Since boththe d-wave gap function and the boundary SP M z are oddunder σ d , they serve as staggered mass potentials for theHMEMs in the edge theory. Once the staggering massesare obstructed, the Jackiw-Rebbi zero modes [94] appearon each corner, protected by the combined σ d T symme-try. To show this precisely, we construct the edge Hamil-tonian in the spirit of k · p theory and take the l -edge foran example. Firstly, we solve H BdG ( i∂ x , k y = 0) | χ ± (cid:105) = 0for the HMEMs basis ( | χ (cid:105) , | χ − (cid:105) ) for x ≥
0. Aftersome algebra, we find | χ (cid:105) = ( e iθ | p (cid:105) + e − iθ | h (cid:105) ) e − ξx /N with tan θ = ( α + ∆ t ) / (cid:112) ∆ t − α and N is the normal-ization constant. Here we set | p (cid:105) = (0 , , − i, T / √ | h (cid:105) = ( i, , , T / √
2. The TRS-partner satisfies | χ − (cid:105) = T | χ (cid:105) . And the localization length ξ satisfies( ξ + µ ) − (cid:112) ∆ t − α ξ = 0, which gives rise to the crite-rion for the helical Majorana states to exist, ∆ t − α > H l ( k y ), H l ( k y ) = A ( l ) k y s x + ( M s ( l ) + M sp ( l )) s y , (13)where A ( l ) = − (∆ t − α cos 2 θ ), M s ( l ) = − ∆ d /
2, and M sp ( l ) = M z cos 2 θ . We also find that M s ( l ) M sp ( l ) > M sp = 0 for the α = 0 case,which is consistent with the GL theory. Similarly, wealso can the mass terms for other edges, M s ( l i ) = (cid:26) ∆ d / , − ∆ d / . (14)and M s ( l i ) M sp ( l i ) >
0. It indicates that the boundarySP enlarges the edge gap to protect the MCSs.
Conclusion and discussion–.
In a short conclusion, wefind that the TRS breaking unitary pairing states couldinduce the spontaneous spin-polarization with the helpof the spin-orbit coupling for the noncentrosymmetricsuperconductors. In the weak spin-orbit coupling limit( αk F < k B T c ), we propose that both s + ip and d + ip could give rise to the SP along the z-axis for a two-dimensional superconductor. And the SP is perpendic-ular to both the real spin-triplet (cid:126)d -vector and the SOC (cid:126)g -vector. Moreover, the estimated averaged boundary SPmight be in the same order as the superconducting gaps,thus it is able to be detected in experiments. There al-ready existing some candidate materials, including Re Zr[17, 18], SrPtAs [25], LaNiC [28], Bi/Ni bilayers [31] andCaPtAs [32], where strong evidences for the TRS break-ing signatures are observed by the µ SR and the PKE.In contrast to the previous theoretical works which areall focus on the non-unitary pairing states to explain theexperimental measurements, our theory provides a newunderstanding to these experiments. We also notice arecent theoretical work demonstrating that the pairingsymmetry might be d + ip [95] for Sr RuO , so that ourtheory might apply near an interface where spin-orbit ap-pears to explain the observations by the µ SR [14] and thePKE [15]. We hope our theory leads to a deeper under-standing of the TRS breaking superconductors, as wellas topological superconductors.
Acknowledgments–.
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