Magnetic Properties in Non-centrosymmetric Superconductors with and without Antiferromagnetic Order
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Magnetic Properties in Non-centrosymmetric Superconductorswith and without Antiferromagnetic Order
Youichi
Yanase , ∗ and Manfred Sigrist , (Received Today 2007) The paramagnetic properties in non-centrosymmetric superconductors with and without an-tiferromagnetic (AFM) order are investigated with focus on the heavy Fermion superconduc-tors, CePt Si, CeRhSi and CeIrSi . First, we investigate the spin susceptibility in the linearresponse regime and elucidate the role of AFM order. The spin susceptibility at T = 0 is in-dependent of the pairing symmetry and increases in the AFM state. Second, the non-linearresponse to the magnetic field are investigated on the basis of an effective model for CePt Siwhich may be also applicable to CeRhSi and CeIrSi . The role of antisymmetric spin-orbitcoupling (ASOC), helical superconductivity, anisotropic Fermi surfaces and AFM order areexamined in the dominantly s -, p - and d -wave states. We emphasize the qualitatively impor-tant role of the mixing of superconducting (SC) order parameters in the p -wave state whichenhances the spin susceptibility and suppresses paramagnetic depairing effect in a significantway. Therefore, the dominantly p -wave superconductivity admixed with the s -wave order pa-rameter is consistent with the paramagnetic properties of CePt Si at ambient pressure. Wepropose some experiments which can elucidate the novel pairing states in CePt Si as well asCeRhSi and CeIrSi . KEYWORDS: Superconductivity without inversion center; antiferromagnetic superconductor; Pauli para-magnetic effect; spin susceptibility
1. Introduction
Since the discovery of superconductivity in the non-centrosymmetric heavy Fermion compound CePt Si,
1, 2 superconductivity in materials without inversion cen-ter is attracting growing interest. Many new non-centrosymmetric superconductors (NCSC) with un-usual properties have been identified among heavyfermion systems such as UIr, CeRhSi ,
4, 5
CeIrSi ,
6, 7
CeCoGe and others like Li Pd x Pt − x B, Y C , Rh Ga , Ir Ga ,
11, 12 Mg Ir B , Re W and someorganic materials. The aspects of missing inversionsymmetry are also of great interest for other materials.For example, the spin Hall effect in the semiconductor and the helical magnetism in MnSi are very active re-search fields.NCSC adds several unusual aspects to the properties ofsuperconductivity. One immediate consequence of non-centrosymmetricity is the necessity for an extended clas-sification scheme of Cooper pairing states, as parity is notavailable as a distinguishing symmetry. Using the tradi-tional scheme the SC states here may be represented asa mixture of pairing states of even and odd parity, or,equivalently, their spin configuration is a superpositionof a singlet and a triplet component. This is a conse-quence of the presence of antisymmetric spin-orbit cou-pling (ASOC) in non-centrosymmetric materials. Re-cent theoretical studies led to the discussion of various in-triguing properties which could appear in NCSC, such asthe magneto-electric effect, the unusual anisotropicspin susceptibility,
18, 20, 22–28 the occurrence of an anoma-lous coherence effect in NMR 1 /T T ,
21, 29 the unusual origin of nodes in the SC gap,
27, 29–32 the realization ofthe helical SC phase, the possible appearance ofFulde-Ferrel-Larkin-Ovchinnikov (FFLO) state at zeromagnetic field, de Haas-van Alphen effect, variousnovel impurity effects, and vortex core states andunconventional features in quasiparticle tunneling andJosephson effect. The non-centrosymmetric heavy fermion superconduc-tors, e.g. CePt Si, UIr, CeRhSi , CeIrSi and CeCoGe are of particular interests because the Cooper pairingis most likely unconventional (non- s -wave) due to thestrong electron correlation. Although many studies havebeen devoted to this topics, there is no consensus onthe symmetry of pairing in these compounds so far. Thesymmetry of Cooper pairs may be determined by theparamagnetic properties such as the spin susceptibilitybelow T c .In centrosymmetric superconductor, the spin suscep-tibility is a distinguishing feature for the spin config-uration of the pairing state, as it decreases below T c for the spin singlet superconductor and remains con-stant in the case of spin triplet pairing, if the mag-netic field is perpendicular to the d -vector (parallel tothe equal-spin direction).
53, 54
The measurements of theKnight shift which is proportional to the spin suscep-tibility have played an important role for the identifi-cation of SC state in various compounds. For super-conductors with very high H c2 probing effects of para-magnetic limiting can give also insight into the pair-ing symmetry and has been applied in connection withNCSC. However, the response to the magnetic field isnot so straightforward in non-centrosymmetric systems. Full Paper
Youichi
Yanase and Manfred
Sigrist
As mentioned above, spin singlet and triplet compo-nents are mixed in the pairing state. Furthermore theband splitting induced by the ASOC affects the mag-netic properties. Therefore, it is necessary to clarifythe magnetic properties very carefully before drawingstrong conclusions. In this context also the influenceof AFM order on the (magnetic) properties of the SCphase is an important point to investigate, since allpresently known non-centrosymmetric heavy Fermion su-perconductors, i.e. CePt Si, UIr, CeRhSi , CeIrSi andCeCoGe , coexist with the magnetism. In CePt Si atambient pressure, superconductivity ( T c = 0 . T N = 2 .
1, 56, 57
The AFMorder can be suppressed by pressure and vanishing atthe critical value of P ∼ . P > . CeRhSi ,
4, 5
CeIrSi
36, 7 and CeCoGe are AFM at ambient pressure and super-conductivity appears only under substantial pressure. Al-though most of the theoretical studies except for Refs. 27,30 and 61 neglected the AFM order so far, it turnsout that the magnetism affects the electronic state pro-foundly. It has been shown that a gap line-node behaviorcould be induced by the AFM order for the pairing statewith dominantly p -wave component,
27, 30, 62 which mayexplain the experimental results in CePt Si at ambientpressure.
In this paper we investigate the linear as well as thenon-linear response regime of the NCSC in a magneticfield with the aim to provide guidelines to identify thepairing symmetry based on magnetic properties. Beforegoing into details we briefly summarize the main conclu-sions of our study. It is known that in the linear responseregime the paramagnetic properties are universal i.e. thespin susceptibility is independent of the pairing symme-try. In the presence of Rashba-type ASOC, the spin sus-ceptibility along the c -axis is constant through T c while itdecreases along the ab -plane to half of the normal statevalue at T = 0, in absence of AFM order.
18, 22–28
Theinfluence of helicity (Cooper pairs possess a finite mo-mentum) in NCSC on the behavior of the susceptibilityturns out to be negligible. On the other hand, the fold-ing of Brillouin zone due to the AFM order significantlyaffects the spin susceptibility in the SC phase. The spinsusceptibility for the magnetic field perpendicular (par-allel) to the staggered moment is increased (decreased)by the AFM order. The non-linear response to the magnetic field is im-portant when the magnetic field is comparable to orhigher than the standard paramagnetic limiting field H P ∼ . k B T c /µ B . It should be noted that most of theexperimental studies, such as the Knight shift and criti-cal magnetic field H c2 , have been carried out in the non-linear response region.
1, 2, 4–7, 58, 66–70
The pairing state inNCSC can be identified by the measurements in the non-linear response regime because the paramagnetic prop-erties depend on the pairing symmetry in contrast to thesituation in the linear response regime.We show that the critical magnetic field H c2 along the ab -plane is significantly enhanced in the non-linear re-sponse regime by the formation of helical SC state. This enhancement coincides with the non-linear increase ofthe helicity of the SC order parameter. H c2 furthermorerises for the dominantly p -wave state owing to the mix-ing of SC order parameters. These effects, namely (i) theformation of the helical SC state and (ii) the mixing ofSC order parameters, are quantitatively important foranisotropic Fermi surfaces. AFM order significantly en-hances the effect (ii) and also boosts H c2 . In this case,the spin susceptibility remains nearly constant through T c . On the other hand, these effects are negligible in thedominantly spin singlet pairing state. Since the influenceof AFM order is quantitatively important, the paramag-netic properties of the SC phase in the AFM state pro-vide a means to distinguish between pairing states withdominant spin triplet and singlet component.Among the non-centrosymmetric heavy fermion super-conductors, CePt Si has been investigated in most detailbecause the superconductivity exists at ambient pressurewhile others require substantial pressure to become su-perconducting. Therefore, we pay particular attention tothe situation in CePt Si, and discuss the pairing sym-metry by comparing the experiments
1, 2, 7, 58, 66, 67 withour theoretical results. The paramagnetic properties ofCePt Si look puzzling at first sight because the experi-mental results are incompatible with the theoretical re-sults within the linear response theory and without tak-ing into account the AFM order.
18, 22–26, 28
In our presentstudy we show that the experimental results are consis-tent with the theoretical results for the dominantly p -wave state by taking into account the AFM order as wellas the non-linear response to the magnetic field.Moreover we propose further test experiments whichcould strengthen our conclusions. First, the influence ofAFM order can be examined by the pressure which sup-presses the AFM order. Second, the 2-fold anisotropyin the ab -plane arises from the AFM order and theanisotropy is qualitatively different between the domi-nantly p -wave, inter-plane d -wave and intra-plane s - or d -wave states. Future experimental studies of these kindcould help identify the pairing symmetry in CePt Si,CeRhSi , CeIrSi and CeCoGe .The paper is organized as follows. In § § Si which could be also applied to CeRhSi ,CeIrSi and CeCoGe . The paramagnetic properties inthe magnetic field along the ab -plane are investigated in § §
5. The non-linear response to the magnetic fieldfor the dominantly s -wave state is investigated in §
4. In §
5, which is the main part of this paper, we show themagnetic properties in the dominantly p -wave state. Theinfluences of the helical superconductivity, anisotropicFermi surface and AFM order are elucidated. The pairingsymmetry of CePt Si is discussed in § Si, CeRhSi andCeIrSi in §
7. In §
8, nature of the helical SC state isinvestigated in details. We show the crossover from thehelical SC state with long wave length to that with shortwave length. These results are summarized and some dis-cussions are given in § . Phys. Soc. Jpn. Full Paper
Youichi
Yanase and Manfred
Sigrist
2. Linear Response Theory
In this section we investigate the linear responseregime of the NCSC in a magnetic field, and study themagnetic properties in the paramagnetic (PM) and in theAFM phase. The latter we consider both for the case ofa centrosymmetric and a non-centrosymmetric system.
In a first step we derive a general expression for thespin susceptibility in the SC state on the basis of theextended BCS Hamiltonian, given by H = H b + H SO + H AF + H ∆ , (1) H b = X ~k,s ε ( ~k ) c † ~k,s c ~k,s , (2) H SO = α X ~k ~g ( ~k ) · ~S ( ~k ) , (3) H AF = − X ~k ~h Q · ~S Q ( ~k ) , (4) H ∆ = − X s,s ′ ,~k [∆ ,s,s ′ ( ~k ) c †− ~k − ,s ′ c † ~k + ,s +∆ ,s,s ′ ( ~k ) c †− ~k − + ~Q,s ′ c † ~k + ,s + h.c. ] , (5)where ~k ± = ~k ± ~q H / ~S ( ~k ) = P ss ′ ~σ ss ′ c † ~k,s c ~k,s ′ and ~S Q ( ~k ) = P ss ′ ~σ ss ′ c † ~k + ~Q,s c ~k,s ′ . Here ~q H is the total mo-mentum of Cooper pairs. Note that ~q H is zero in theusual BCS state while that is finite in the helical SCstate. In NCSC the helical SC state can be realizedunder magnetic field above H c1 . We consider a tetrago-nal crystal lattice and assign the x -, y - and z -axis to a -, b - and c -axis, respectively.The first term in eq. (1) describes the dispersion rela-tion without ASOC and AFM order. In this subsectionwe do not identify the specific dispersion of the electronsand assume ε ( ~k ) as general.The second term H SO describes the ASOC due tothe lack of inversion symmetry. This term preservestime reversal symmetry, if the g -vector is odd in ~k , i.e. ~g ( − ~k ) = − ~g ( ~k ). We consider a Rashba-type spin-orbitcoupling as is realized in CePt Si, CeRhSi , CeIrSi and CeCoGe . Because the detailed momentum de-pendence of ~g ( ~k ) is unknown, we express it in terms ofvelocities ~v ( ~k ) = ∂ε ( ~k ) /∂~k : ~g ( ~k ) = ( − v y ( ~k ) , v x ( ~k ) , / ¯ v .This choice at least preserves the correct periodicity in ~k -space. The detailed form of the g -vector is anyway unim-portant in the following. We normalize the g -vector ~g ( ~k )by the average velocity ¯ v [¯ v = N P k v x ( ~k ) + v y ( ~k ) ]so that the coupling constant α has the dimension ofenergy. We assume the relation | ∆ i,s,s ′ ( ~k ) | ≪ | α | ≪ ε F throughout this paper ( ε F is the Fermi energy). This re-lation is valid for the most of NCSC such as CePt Si,UIr, CeRhSi , CeIrSi and CeCoGe .The third term H AF is taken into account to inves-tigate the role of AFM order which enters through thestaggered field ~h Q . We focus on A-type AFM order, i.e.ferromagnetic sheets in the ab -plane are staggered along the c -axis, giving rise to ~Q = (0 , , π ). This spin structureis realized in CePt Si as well as the centrosymmetricsuperconductor UPd Al where the magnetic mo-ments are aligned in the ab -plane. A different AFM statehas been reported for CeRhSi and the magnetic struc-ture is not clearly identified for CeIrSi so far. However,the qualitative role of AFM order can be captured by ~Q = (0 , , π ) in the simple cases.The last term H ∆ describes the mean field term of theSC order. The order parameter is given by ∆ ,s,s ′ ( ~k ) and∆ ,s,s ′ ( ~k ). The second component ∆ ,s,s ′ ( ~k ) only appearsin the case of superconductivity coexisting with AFMorder (∆ ,s,s ′ ( ~k ) = 0 for ~h Q = 0). The order parameterhas both the spin singlet and triplet components owingto the ASOC.It is more transparent for the following discussion toconsider the order parameter in the band basis becausethe superconductivity is mainly induced by the intra-band Cooper pairing when | ∆ | ≪ | α | . Ignoring the orderparameters describing the inter-band pairing, we obtainthe simplified Hamiltonian as, H band = X γ =1 X ~k ′ e γ ( ~k ) a † γ,~k a γ,~k − [∆ γ ( ~k ) a † γ, − ~k − a † γ,~k + + h.c. ] , (6)where P ~k ′ is restricted to the summation within | k z | <π/
2. The dispersion relation e γ ( ~k ) takes into account theASOC and AFM order and is obtained by the unitarytransformation as,ˆ U † ( ~k ) ˆ H ( ~k ) ˆ U ( ~k ) = ( e i ( ~k ) δ ij ) , (7)where the 4 × H ( ~k ) is expressed as,ˆ H ( ~k ) = ˆ e ( ~k ) − ~h Q ~σ − ~h Q ~σ ˆ e ( ~k + ~Q ) ! . (8)We define ˆ e ( ~k ) = ε ( ~k )ˆ σ (0) + α~g ( ~k ) ~σ and ~σ representthe three Pauli matrices and ˆ σ (0) is the 2 × e γ ( ~k ) are non-degenerate except forthe special momentum, if α = 0. Moreover, the relation e γ ( − ~k ) = e γ ( ~k ) is hold owing to the time-reversal sym-metry.The order parameter is expressed in the band basis as,ˆ∆ band ( ~k ) = ˆ U † ( ~k + ) ˆ∆ spin ( ~k ) ˆ U ∗ ( − ~k − ) , (9)whereˆ∆ spin ( ~k ) = ∆ ,s,s ′ ( ~k ) ∆ ,s,s ′ ( ~k )∆ ,s,s ′ ( ~k + ~Q ) ∆ ,s,s ′ ( ~k + ~Q ) ! . (10)Although the off-diagonal matrix element of ˆ∆ band ( ~k )is finite in general, the low-energy properties below T c are hardly affected by the off-diagonal components when | ∆ | ≪ | α | . Therefore, we simply drop the off-diagonalcomponents and obtain the Hamiltonian eq. (6). The SCorder parameter for each band is expressed by the diag-onal components as, ∆ γ ( ~k ) = ( ˆ∆ band ( ~k )) γγ . J. Phys. Soc. Jpn.
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Youichi
Yanase and Manfred
Sigrist
The normal and anomalous Green functions are ex-pressed in the band basis as, G γ ( ~k + , i ω n ) = (i ω n + e γ ( − ~k − )) /A γ ( ~k, i ω n ) , (11) F γ ( ~k, i ω n ) = − ∆ γ ( ~k ) /A γ ( ~k, i ω n ) , (12)with A γ ( ~k, i ω n ) = (i ω n − e γ ( ~k + ))(i ω n + e γ ( − ~k − )) − | ∆ γ ( ~k ) | , (13)where ω n = (2 n + 1) πT is the Matsubara frequency and T is the temperature.We decompose the uniform spin susceptibility into thePauli part and Van-Vleck part, χ µν = χ P µν + χ V µν . (14)The Pauli susceptibility χ P µν arises from the intra-bandscattering while the inter-band scattering gives rise tothe Van-Vleck susceptibility (VVS) χ V µν . In the followingwe assume the staggered moments along the principalaxis, namely ~h Q k ˆ x, ˆ y or ˆ z . Following the Appendix A,the Pauli susceptibility and VVS are expressed as, χ P µµ = − lim ~q → lim Ω n → X γ,~k ′ A µµγγ ( ~k ) × [ G γ ( k + q ) G γ ( k ) ± F γ ( k + q ) F † γ ( k )] , (15)and χ V µν = X γ = δ X ~k ′ A µνγδ ( ~k ) f ( e γ ( ~k )) − f ( e δ ( ~k )) e δ ( ~k ) − e γ ( ~k ) , (16)respectively. The sign in eq.(15) is + for µ = x, y and − for µ = z . We define A µνγδ ( ~k ) = S µγδ ( ~k, ~k ) S νδγ ( ~k, ~k ) where S µγδ ( ~k + ~q, ~k ) is the spin operator in the band basis,ˆ S µ ( ~k + ~q, ~k ) = ˆ U † ( ~k + ~q ) ˆ S µ ˆ U ( ~k ) , (17)with ˆ S µ = (cid:18) ˆ σ ( µ )
00 ˆ σ ( µ ) (cid:19) . (18)The expression of Pauli susceptibility eq. (15) is equiv-alent to the spin susceptibility in a multi-band system.The sign + and − in eq. (15) correspond to the cen-trosymmetric superconductor with spin singlet pairingand that with spin triplet pairing for ~d ⊥ ~H , respec-tively. Thus, the Pauli susceptibility of NCSC decreasesin the ab -plane below T c while that is constant for themagnetic field along the c -axis.It should be noted that the VVS has a temperaturedependence above T c which is similar to that of Paulisusceptibility, because the ASOC is much smaller thanthe Fermi energy ( | α | ≪ ε F ). Therefore, the VVS ineq. (14) should be included in the spin part of mag-netic susceptibility which is extracted by the K - χ plot. Inthis sense, the VVS, arising from the band splitting dueto the ASOC, is quite different from the better knownVVS coming from the orbital degrees of freedom. Notethat both VVS are not affected by the superconductivitywhen T c ≪ | α | . If the order parameter is spatially uniform, namely ~q H = 0, the Pauli susceptibility is described by the mo-mentum dependent Yosida function as, χ P µµ = X γ Z d ~k F A µµγγ ( ~k F ) Y (∆ γ ( ~k F ) , T ) /v γ ( ~k F ) , (19)for µ = x , y, and χ Pzz = X γ Z d ~k F A zz γγ ( ~k F ) /v γ ( ~k F ) , (20)where R d ~k F is the integral on the Fermi surface, and v γ ( ~k F ) is the Fermi velocity of γ -th band. The Yosidafunction is defined as, Y (∆ , T ) = − Z d εf ′ ( p ε + ∆ ) , (21)where f ′ ( E ) = df /dE is the derivative of the Fermi dis-tribution function. Since Y (∆ ,
0) = 0 and Y (0 , T ) = 1,we obtain χ µµ ( T = 0) = χ V µµ for µ = x, y and χ zz ( T =0) = χ Vzz + χ Pzz ( T = T c ) = χ zz ( T = T c ). Thus, the resid-ual spin susceptibility along ab -plane is given by the VVSalone, while for fields parallel to the c -axis both the Pauliand Van-Vleck susceptibility contribute. It should be no-ticed that the spin susceptibility at T = 0 is independentof the pairing symmetry. In this sense the spin suscepti-bility is universal in the linear response regime when thesystem lacks the inversion symmetry. We concentrate now on the uniform state ( ~q H = 0) toinvestigate the residual spin susceptibility χ µµ at T = 0for µ = x, y . The helical SC state with ~q H = 0 will bediscussed later in §
8. In the PM state we set ~h Q = 0and assign the four bands as e , ( ~k ) = ε ( ~k ) ± α | ~g ( ~k ) | , e , ( ~k ) = ε ( ~k + ~Q ) ± α | ~g ( ~k + ~Q ) | so that we can expressthe unitary matrix as,ˆ U ( ~k ) = ˆ U ( ~k ) 00 ˆ U ( ~k + ~Q ) ! , (22)whereˆ U ( ~k ) = 1 √ (cid:18) g x ( ~k ) + i˜ g y ( ~k ) − ˜ g x ( ~k ) − i˜ g y ( ~k ) (cid:19) , (23)with ˜ g µ ( ~k ) = g µ ( ~k ) / | ~g ( ~k ) | . The matrix element of spinoperator is obtained as, S µ ( ~k, ~k ) = − S µ ( ~k, ~k ) = ˜ g µ ( ~k )and S µ ( ~k, ~k ) = − S µ ( ~k, ~k ) = ˜ g µ ( ~k + ~Q ) for µ = x, y while S z γγ ( ~k, ~k ) = 0. We then find χ Pxx = χ Pyy = ρ/ T = T c where ρ is the DOS in the normal state. Since χ xx = χ Pxx + χ Vxx = ρ + O ( α /ε ) in the normal state, theresidual spin susceptibility at T = 0 is obtained as, χ xx ( T = 0) = χ Vxx = χ xx ( T = T c ) / O ( α /ε ) . (24)Thus, the spin susceptibility in the ab -plane at T = 0is half of the normal state value in the limit | α | ≪ ε F . Qualitatively the same result has been obtained inRefs. 18,20,22-28. Fujimoto has shown that the VVS in-creases when the DOS has strong asymmetry and | α | ismoderate. However, the β -band of CePt Si which wewill investigate later does not satisfy this condition. . Phys. Soc. Jpn. Full Paper
Youichi
Yanase and Manfred
Sigrist Since the spin susceptibility decreases below T c for themagnetic field along the ab -plane, the paramagnetic de-pairing effect of H c2 should be observed in NCSC withRashba-type spin-orbit coupling. This is consistent withthe recent observation of the paramagnetic depairing ef-fect in CeRhSi
35, 70 and CeIrSi
37, 68, 78 under high pres-sure where the AFM order is suppressed. However, this isnot the case in CePt Si at ambient pressure (within theAFM phase).
1, 2, 7, 58, 66, 67
This observation leads us tostudy the influence of AFM order in the following part.
In order to clarify the influence of AFM order, we firstinvestigate the spin susceptibility in the SC state with inversion symmetry for ~h Q = 0. Owing to the inversionsymmetry, the residual spin susceptibility depends on thepairing symmetry in the usual way. Here we discuss thespin singlet pairing state while the spin susceptibility isconstant through T c in the spin triplet pairing state. Thespin susceptibility consists of the Pauli part and Van-Vleck part as in § T = 0.As a result of the simple calculation, we obtain for thePauli susceptibility above T c , χ P µµ = X ~k [ δ ( e ( ~k )) + δ ( e ( ~k ))] = ρ, (25)for ~H k ~h Q and χ P µµ = X ~k ε − ( ~k ) ε − ( ~k ) + h [ δ ( e ( ~k )) + δ ( e ( ~k ))] , (26)for ~H ⊥ ~h Q . Here, e , ( ~k ) = ε + ( ~k ) ± q ε − ( ~k ) + h with h Q = | ~h Q | and ε ± ( ~k ) = ( ε ( ~k ) ± ε ( ~k + ~Q )) /
2. The Paulipart of spin susceptibility for the magnetic field perpen-dicular to the AFM moment decreases with growing ~h Q ,i.e., χ P µµ ( h Q = 0) > χ P µµ ( h Q = 0) for ~H ⊥ ~h Q . A Van-Vleck part is induced by the AFM order and leads tothe residual spin susceptibility for ~H ⊥ ~h Q at T = 0. Incontrast, the Van-Vleck part and the residual spin sus-ceptibility vanish for the magnetic field parallel to theAFM moment.In Fig. 1 we show the numerical results for the spinsusceptibility in the SC state at T = 0. For this numericalanalysis we assume a tight-binding model approximatingthe so-called β -band of CePt Si, ε ( ~k ) = 2 t (cos k x + cos k y ) + 4 t cos k x cos k y +2 t (cos 2 k x + cos 2 k y ) + [2 t + 4 t (cos k x + cos k y )+4 t (cos 2 k x + cos 2 k y )] cos k z + 2 t cos 2 k z − µ c . (27)The chemical potential µ c is determined so that the elec-tron density per site is n . The Fermi surface of the β -band, which has been obtained in the band structure cal-culations without taking AFM order into account, is reproduced by choosing the parameters as( t , t , t , t , t , t , t , n ) =(1 , − . , − . , − . , − . , − . , − . , . , (28) and α = 0 . t as the unit energy. As shownin Fig. 1, the spin susceptibility for the magnetic fieldperpendicular to the AFM moment m = χ xx ( ~Q, h xQ isincreased in the SC state by the AFM order while thatin the normal state is little affected. m χ b = χ c χ n χ s α =0 Fig. 1. Spin susceptibility along the b - and c -axes at T = 0against the staggered magnetic moment m along the a -axis. TheASOC is zero ( α = 0) and the staggered field ~h Q = h xQ ˆ x is as-sumed. The staggered magnetic moment is defined as m = | < P s,s ′ σ (x) ss ′ c † i,s c i,s ′ > | so that m = 1 is the full moment. Thesolid and dashed lines show the spin susceptibility in the normalstate and in the spin singlet SC state, respectively. The spin sus-ceptibility along the a -axis is zero in the spin singlet SC state at T = 0. At this point we can discuss the role of the band struc-ture. According to eq. (26), the Pauli part of the spinsusceptibility is small for the magnetic field ~H ⊥ ~h Q ,if the quasiparticle dispersion is quasi-two dimensionaland ε − ( ~k ) is small. Although the β -band of CePt Si hasa three dimensional Fermi surface, the band dispersionis weak along the k z -axis according to the result of bandcalculation. This means that the AFM order signifi-cantly would affect the SC state in this band of CePt Si.The presence of quasi-two dimensional Fermi surface isalso expected in CeRhSi . There are cases where AFM order plays indeed animportant role in a centrosymmetric material. For ex-ample, UPd Al is a spin singlet superconductor with T c = 2K which coexists with the AFM order. TheAFM state has a high N´eel temperature of T N = 14Kand a large staggered magnetic moment, m = 0 . µ B .This moment is directed to the ab -plane of tetragonallattice and ~Q = (0 , , π ). This is the same spin struc-ture as CePt Si. NMR measurements show the decreaseof Knight shift below T c with a large residual part. TheVVS arising from the AFM order may induce the largeresidual spin susceptibility, although the multi-orbital ef-fect is another possible origin. This is consistent withthe large H c2 which exceeds the standard paramagneticlimit. The result in § J. Phys. Soc. Jpn.
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Youichi
Yanase and Manfred
Sigrist non-centrosymmetric system for the magnetic field per-pendicular to the AFM moment. We have shown theresults for the spin susceptibility along the a - and b -axes by assuming the dispersion relation eqs. (27), (28), α = 0 . ~h Q k ˆ x to describe the electronic structure ofCePt Si below T N (Fig. 4 of Ref. 27). For fields ~H ⊥ ~h Q the normal state and SC state susceptibility merge for in-creasing staggered moment, suggesting a diminishing ofthe reduction of the spin susceptibility in the SC state.On the other hand, the behavior is opposite for ~H k ~h Q .Thus, a remarkable 2-fold anisotropy is expected in thespin susceptibility below T c even if the anisotropy is weakin the normal state. The condition ~H ⊥ ~h Q is generallyfavored because the magnetization energy is maximallygained for the field direction with largest spin suscep-tibility. However, the meta-stable state ~H k ~h Q can berealized in the weak magnetic field which is smaller thanthe anisotropy energy of AFM moment.The role of AFM order is suppressed by increasing theASOC. We have confirmed that χ yy in the SC state isdecreased by increasing α when ~h Q k ˆ x . The AFM orderplays a quantitatively important role when the ASOC ismuch smaller than the Fermi energy.If the AFM moment is parallel to the c -axis as inCeCoGe , the spin susceptibility along both a - and b -axes is increased in the SC state by the AFM order, whilethat along the c -axis is not affected.
3. Effective Model for CePt Si, CeRhSi andCeIrSi In preparation for the discussion of the non-linear re-sponse to the magnetic field we will introduce here aneffective model for CePt Si, CeRhSi and CeIrSi . Thisis important as we will show that the non-linear spinsusceptibility significantly depends on the symmetry oforder parameter in contrast to the universal spin suscep-tibility in the linear response theory (see § p -wave character.We analyze the following effective model, H = H b + H SO + H AF + H Z + H I , (29) H Z = − X ~k ~h · ~S ( ~k ) , (30) H I = U X i n i, ↑ n i, ↓ + ( V − J/ X n i n j + J X ( ~S i · ~S j − S x i S x j ) (31)= 12 X ~k,~k ′ ,~q,s [ V C ( ~k − ~k ′ ) c †− ~k − ,s c † ~k + ,s c ~k ′ + ,s c − ~k ′− ,s + { U + ( V − J C ( ~k − ~k ′ ) } c †− ~k − , ¯ s c † ~k + ,s c ~k ′ + ,s c − ~k ′− , ¯ s − J C ( ~k − ~k ′ ) c †− ~k − ,s c † ~k + ,s c ~k ′ + , ¯ s c − ~k ′− , ¯ s ] , (32)where n i,s is the electron number at the site i with spin s , n i = n i, ↑ + n i, ↓ , ¯ s = − s and C ( ~k ) = 2(cos k x + cos k y ).The spin operator in the real space basis is defined as ~S i = P s,s ′ ~σ ss ′ c † i,s c i,s ′ . The bracket < i, j > denotes thesummation for the nearest neighbor sites in the ab -plane,namely j = i ± ~a or j = i ± ~b with ~a and ~b the unit vectorsalong the a - and b -axis, respectively.The first three terms in eq. (29) have been defined ear-lier in eqs. (2-4). For the dispersion relation, we adoptthe tight-binding model eq. (27) with using the parame-ter set eq. (28) and α = 0 .
3, reproducing the β -band ofCePt Si.
We choose the β -band, because it has sub-stantial Ce 4 f -electron character and the largest DOSat the Fermi energy, namely 70% of the total DOS. Be-sides the sizable jump in specific heat, also the remark-able isotropy of H c2 between the ab -plane and c -axis also indicates that the three-dimensional Fermi surfaceof the β -band is mainly responsible for the superconduc-tivity in CePt Si. In Appendix B we will investigate theother dispersion relation which favors the d x − y -wavesuperconductivity.As for the AFM order, we assume the staggered fieldpointing along the [100]-direction ~h Q = h Q ˆ x with ~Q =(0 , , π ) following the experimental results of CePt Si. For the magnitude we assume h Q ≪ W , the band width.This is consistent with the small observed magnetic mo-ment ∼ . µ B in CePt Si. The AFM moment is ex-pected to be small also in CeRhSi and CeIrSi sincesuperconductivity occurs near the AFM quantum criti-cal point.The fourth term H Z is the Zeeman coupling termdue to the applied magnetic field. We have defined ~h = gµ B ~H where g is the g -factor of quasiparticles and µ B is the Bohr magneton. The paramagnetic depairing effecton the superconductivity is characterized by the dimen-sionless coupling constant h/T c with h = | ~h | .The last term H I describes the effective interactionleading to the SC instability and includes three couplingconstants, U , V and J . We assume the on-site interaction U and the interaction between the nearest neighbor sitesin the ab -plane V . The coupling constant J describes thepart of interaction arising from the AFM order which isanisotropic. According to the random phase approxima-tion (RPA) for the Hubbard model,
27, 62 the SC orderparameter is affected by the AFM order mainly throughthe anisotropy of effective interaction, which can be de-scribed by the J -term in eq. (31).In the following we examine two parameter sets,(A) U > , V = − . U, (33)(B) U < , V = 0 . (34)The amplitude of U is chosen so that T c = 0 .
01 at zeromagnetic field. The ground state is dominantly (A) p -wave and (B) s -wave, respectively. Hereafter we simplycall these states p -wave and s -wave state, respectively.The parameter set (A) is the most important for our pur-pose, because the p -wave symmetry is the most promisingcandidate for the pairing state in CePt Si.
21, 27, 29, 85, 86
Although the spin triplet superconductivity is handi-capped due to the lack of inversion symmetry in non-centrosymmetric systems according to the Anderson’stheorem, the depairing effect arising from the ASOCvanishes (or is at least smallest) in the p -wave state . Phys. Soc. Jpn. Full Paper
Youichi
Yanase and Manfred
Sigrist with ~d ( ~k ) k ~g ( ~k ). This condition is not satisfied inthe realistic model, however the depairing effect due tothe ASOC is almost avoided in the p -wave state with ~d ( ~k ) = − p y ˆ x + p x ˆ y . Another parameter set (B) is inves-tigated as a typical model for the dominantly spin singletpairing state. We will investigate the dominantly d x − y -wave state in Appendix B and obtain qualitatively thesame results as the s -wave state.Before analyzing the effective model in eq. (29), wecomment on the RPA theory applied to the Hubbardmodel for the β -band of CePt Si.
27, 62
This theory leadsto two possible pairing states due to spin fluctuation me-diated interaction: the s + P -wave and the p + D + f -wavestate. The former is dominated by the p -wave componentand can be viewed as an intra-plane pairing state, whilethe latter is described by the inter-plane pairing dom-inated by the d xz - and d yz -wave components. Here wefocus on the s + P -wave state whose order parameteris reproduced by assuming the parameter set (A) and J = 0 . V ( J = 0) for h Q = 0 .
125 ( h Q = 0) in eq. (29).On the other hand, the p + D + f -wave state is not re-alized in eq. (29) because the inter-plane interaction isneglected. It is expected that the paramagnetic proper-ties in the inter-plane d -wave state are qualitatively thesame as those in the intra-plane s - and d -wave states.The other characteristic properties of the p + D + f -wavestate will be discussed in § § i,s, ¯ s ( ~k ) = − T X n,~k ′ { U + ( V − J C ( ~k − ~k ′ ) }× F i,s, ¯ s ( ~k ′ , ω n ) , (35)∆ i,s,s ( ~k ) = − T X n,~k ′ C ( ~k − ~k ′ ) { V F i,s,s ( ~k ′ , ω n ) − J F i, ¯ s, ¯ s ( ~k ′ , ω n ) } . (36)The normal and anomalous Green functions G i,s,s ′ ( ~k ′ , ω n ), F i,s,s ′ ( ~k ′ , ω n ) in the spin basis areobtained by the Dyson-Gorkov equation, ˆ G N ( ~k + , ω n ) − ˆ∆ spin ( ~k )ˆ∆ † spin ( ~k ) − ˆ G TN ( − ~k − , − ω n ) − ! × ˆ G ( ~k + , ω n ) ˆ F ( ~k, ω n )ˆ F † ( ~k, ω n ) − ˆ G T ( − ~k − , − ω n ) ! = ˆ1 . (37)where ˆ X ( ~k ) ( X = G, F, ∆ spin ) is the 4 × X ( ~k ) = X ,s,s ′ ( ~k ) X ,s,s ′ ( ~k ) X ,s,s ′ ( ~k + ~Q ) X ,s,s ′ ( ~k + ~Q ) ! . (38)The normal Green function in the normal state,ˆ G N ( ~k, ω n ) is obtained as ˆ G N ( ~k, ω n ) = (i ω n ˆ1 − ˆ H ( ~k )) − by using eq. (8) with ~h Q = h Q ˆ x and ˆ e ( ~k ) = ε ( ~k )ˆ σ (0) + α~g ( ~k ) · ˆ ~σ − ~h · ˆ ~σ .We here discuss the symmetry of the SC state on thebasis of the following parameterization of order parame- ters:∆ ,s,s ′ ( ~k ) = − d x ( ~k ) + i d y ( ~k ) Φ( ~k ) + d z ( ~k ) − Φ( ~k ) + d z ( ~k ) d x ( ~k ) + i d y ( ~k ) ! , (39)where we use the even parity scalar function Φ( ~k ) andthe odd parity vector ~d ( ~k ). Although a second component∆ ,s,s ′ ( ~k ) appears in the AFM state, the basic propertiesand symmetries are little affected by ∆ ,s,s ′ ( ~k ). P-wave state Even parity part Odd parity partPM at ~h = 0 κ ( δ + c x + c y ) ( − s y , s x , ~h = 0 κ ( δ + η c x + c y ) ( − s y , β s x , ~h = h ˆ y κ ( δ + η c x + c y ) ( − s y , β s x , − i γ s y )Table I. Symmetry of order parameter in the dominantly p -wavestate. We use the abbreviations c x , y = cos k x , y and s x , y =sin k x , y . We assume the PM state at zero magnetic field, theAFM state at zero magnetic field and the AFM state under themagnetic field ~h k ˆ y from the top to the bottom. The parameters β , γ , κ , δ and η are real. In Table I we summarize the order parameters in the p -wave state. The admixture with the even-parity com-ponent due to the ASOC is expressed by the parameter κ which is in the order of α/ε F . We obtain κ ∼ .
15 for α = 0 .
3. The even-parity part Φ( ~k ) is dominated by theextended s -wave component and δ ∼ . s -wave component is suppressed by the strongon-site repulsion U . The d z -component of the odd parityvector ~d ( ~k ) is induced by the magnetic field γ ∝ h/ε F togain the Zeeman energy. The SC state is mainly affectedby the parameter β which is unity in the absence of AFMorder and magnetic field. The magnetic field along the[010]-axis ([100]-axis) decreases (increases) β . The influ-ence of the AFM order depends on the value of J . We find β ∼ . β ∼ .
7) for J = 0 . V ( J = 0) at h Q = 0 . h = 0. The deviation from β = 1 can be viewedas the mixing between ~d ( ~k ) = ( − sin k y , sin k x ,
0) andanother p -wave state ~d ( ~k ) = (sin k y , sin k x , D symmetry, they are mixed due tothe presence of the symmetry reducing AFM moment ormagnetic field.In general, the dominantly s -wave state is admixedto the p -wave state due to the ASOC and belongs tothe same irreducible representation as the dominantly p -wave state realized for the parameter set (A). How-ever, only the conventional s -wave component Φ( ~k ) = 1appears and ~d ( ~k ) = ~ V = J = 0). Generally speaking, the admixture of spinsinglet and triplet order parameters plays no importantrole when the ASOC is much smaller than the Fermi en-ergy, | α | ≪ ε F .The ”helicity” ~q H -vector is perpendicular to the mag-netic field as will be discussed in § ~q H below T c should be determined to maximizethe condensation energy. However, here we determine ~q H at T = T c ( h ) and neglect the temperature dependence J. Phys. Soc. Jpn.
Full Paper
Youichi
Yanase and Manfred
Sigrist below T c for simplicity. The transition temperature T c ( h )is determined by linearizing the mean field equation as, λ ( ~q )∆ i,s, ¯ s ( ~k ) = − T X n,~k ′ { U + ( V − J C ( ~k − ~k ′ ) }× φ i,s, ¯ s ( ~k ′ , ω n ) , (40) λ ( ~q )∆ i,s,s ( ~k ) = − T X n,~k ′ C ( ~k − ~k ′ ) ×{ V φ i,s,s ( ~k ′ , ω n ) − J φ i, ¯ s, ¯ s ( ~k ′ , ω n ) } , (41)where φ i,s,s ′ ( ~k ′ , ω n ) is obtained by linearizing theanomalous Green function F i,s,s ′ ( ~k ′ , ω n ) with respect toˆ∆ spin ( ~k ′ ). We optimize the eigenvalue λ ( ~q ) with respectto the order parameter ˆ∆ spin ( ~k ) and the helicity ~q = ~q H .The transition temperature T c ( h ) is determined by thecriterion λ ( ~q H ) = 1.We have estimated the condensation energy below T c and found that the magnitude of ~q H increases as decreas-ing the temperature. However, we have confirmed thatthe temperature dependence of ~q H can be ignored forthe magnetic properties discussed in the following part. S -wave State For the discussion of non-linear response to the mag-netic field in NCSC we first discuss the simplest case,namely the s -wave state without AFM order. We ad-dress the enhancement of the critical magnetic field h c2 = gµ B H c2 due to the ASOC, assuming the param-eter set (B) U < V = 0. Figure 2 shows the phasediagram, temperature T /T c versus magnetic field h/T c along the [100]- or [010]-direction. The critical field h c2 for both the uniform state ( ~q H = 0) and the helical state( ~q H = 0) are depicted, whereby also the behavior in theabsence of ASOC ( α = 0) is included for a comparison.As we focus here on the paramagnetic limiting effect,we neglect the orbital depairing for simplicity. Note that h c2 in the helical s -wave state has been investigated inRef. 35 for an isotropic Fermi surface.The data in Fig. 2 demonstrate that the h c2 is signifi-cantly enhanced by the ASOC. This is partly due to theresidual spin susceptibility in the SC state induced bythe ASOC. Neglecting the magnetic field dependence ofthe spin susceptibility, we obtain a simple estimation forthe critical magnetic field, h c2 = s E c χ N − χ S . (42)where E c is the condensation energy and χ S and χ N are the spin susceptibility in the SC and normal state,respectively. According to eq. (42), h c2 increases by afactor of √ χ S = χ N at T = 0. In fact, h c2 is enhanced even moredue to the magnetic field dependence of spin suscepti-bility. A further enhancement of h c2 is caused by theformation of a helical SC state, which exceeds the en-hancement in centrosymmetric superconductor owing tothe presence of an FFLO state. T/T c h c / T c α =0.3 uniform α =0.3 helical α =0 uniform α =0 FFLO Fig. 2. (Color online) The H - T phase diagram in the s -wave statefor the magnetic field along the [100]- or [010]-axis. The dia-monds (circles) show the reduced critical magnetic field h c2 /T c = gµ B H c2 /T c in the helical (uniform) SC state against the re-duced temperature T/T c . We assume U < V = 0, J = 0, α = 0 . h Q = 0. The phase diagrams in the absence ofASOC ( α = 0) are shown for a comparison. The dashed andsolid lines show the h c2 /T c in the uniform ( ~q H = 0) and FFLO( ~q H = 0) states, respectively. T/T c h c / T c PM, h//[010]AFM, h//[010]AFM, h//[100]
Fig. 3. (Color online) The H - T phase diagram in the helical s -wave state with AFM order. We assume ~h Q = 0 . x . The otherparameters are the same as Fig. 2. The circles and triangles showthe h c2 for the magnetic field along the [010] and [100]-axis,respectively. The h c2 in the PM state is shown for a comparison(diamonds). We here investigate the influence of AFM order on the s -wave SC state. Figure 3 shows that the h c2 in the s -wave state is increased by the AFM order, however theenhancement is very small. According to the simple es-timation eq. (42) and the universal spin susceptibilityin the linear response theory (see § h c2 alongthe [010]-axis ([100]-axis) is enhanced (suppressed) bythe AFM order through the increase (decrease) of χ S .However, the enhancement (suppression) is much smallerthan expected in this simple estimation. This is mainlybecause of the formation of helical SC phase which in-duces the non-linear spin susceptibility at high fields. In § p -wave state. . Phys. Soc. Jpn. Full Paper
Youichi
Yanase and Manfred
Sigrist
5. The p -wave State We here investigate the dominantly p -wave state whichis the most promising candidate for the pairing state inCePt Si. In this section we assume the parameter set (A)
U > V = − . U . T/T c h c / T c p-waves-waved-wave Fig. 4. (Color online) The H - T phase diagram in the helical p -wave state (circles). We assume U > V = − . U , J = 0, α =0 . h Q = 0. The h c2 in the the s -wave state (diamonds) andin the d x − y -wave state (squares) are shown for a comparison.The parameter set for the d x − y -wave state is shown in theAppendix B. To illuminate the difference with the dominantly spinsinglet pairing state we again turn to the PM state. Wefind that paramagnetic depairing effect is naturally lesseffective in suppressing the onset of superconductivity.Figure 4 shows h c2 for the p -wave state which is muchhigher than for the case of s -wave as well as d -wave pair-ings. This is rather surprising because the h c2 is inde-pendent of the pairing symmetry considering only thesimple estimation in eq. (42). Actually, h c2 of the p -wavestate is enhanced by the modification of SC order param-eters due to the mixing with ~d ( ~k ) = (sin k y , sin k x ,
0) inaddition to the formation of helical SC state. Since the p -wave superconductivity has a multi-component orderparameter with respect to the spin, the order parametercan be modified to optimally cope with the competitionbetween the Zeeman coupling energy and ASOC. This isnot the case in the dominantly spin singlet pairing state.This is the main reason why the paramagnetic depairingeffect in NCSC depends on the symmetry of the leadingorder parameter. We see that the h c2 curves in Fig. 4merge in the low magnetic field region where the linearresponse theory is justified.In order to shed light on the mechanisms stabilizingthe p -wave superconductivity at high magnetic fields,i.e., (i) the formation of helical SC state, and (ii) themodification of SC order parameters, we compare h c2 with the one for the uniform state with ~q H = 0 (tri-angles in Fig. 5) and the one for the SC state with ~d ( ~k ) = ( − sin k y , sin k x , ~k ) = 0 and ~q H = 0 (dia-monds in Fig. 5). Both (i) and (ii) are neglected in the T/T c h c / T c p-wave, helicalp-wave, uniformp-wave, OP fixed Fig. 5. (Color online) The circles (triangles) show the H - T phasediagram in the helical (uniform) p -wave state. We show the phasediagram in the SC state with ~d ( ~k ) = ( − sin k y , sin k x , ~k ) =0 and ~q H = 0 for a comparison. The parameters are the same asin Fig. 4. latter (diamonds) while (i) is neglected in the former (tri-angles). The comparison between the triangles and dia-monds shows the enhancement of h c2 by optimizing theSC order parameter. Actually, the d x - ( d y -)component of d -vector decreases in the magnetic field along the [100]-([010]-)axis to avoid the paramagnetic depairing effect.The h c2 is furthermore enhanced below T = 0 . T c byforming the helical SC state (circles). Thus, the p -wavesuperconductivity can be stabilized in the magnetic fieldwhich is much higher than the standard paramagneticlimit owing to the combination of mechanisms (i) and(ii). T/T c χ a = χ b h=0.1T c h=1T c h=1.5T c h=2T c Fig. 6. (Color online) The spin susceptibility along the [100]- and[010]-directions in the helical p -wave SC state without AFM or-der. The magnetic field is chosen as h = 0 . T c , h = T c , h = 1 . T c and h = 2 T c from the bottom to the top. The other parametersare the same as in Fig. 4. A large critical magnetic field h c2 generally indicatesthat a SC state with a large spin susceptibility is stabi-lized at high magnetic fields. The general spin suscepti-bility defined by χ a = χ b = M x , y /h is obtained from thecalculation of the uniform magnetization, M µ = X k Tr ˆ S µ ˆ G ( k ) , (43) Full Paper
Youichi
Yanase and Manfred
Sigrist where ˆ S µ is the spin operator defined in eq. (18). Thecorresponding spin susceptibility for the p -wave state isshown in Fig. 6. The spin susceptibility at h = 0 . T c drops to half of its normal state value at T = 0. This isconsistent with the linear response theory in § p -wave stateis modified in order to avoid the paramagnetic depairingeffect. Therefore, the spin susceptibility at h = 2 T c is al-most constant through T c , although the critical tempera-ture remains high, ( T c ( h = 2 T c ) = 0 . T c ( h = 0)). Theseresults should be contrasted to the d x − y -wave case dis-cussed in Appendix B. The T c of d x − y -wave state isreduced more strongly ( T c ( h = 2 T c ) = 0 . T c ( h = 0)),but the decrease of spin susceptibility below T c is largerthan that in the p -wave state (Fig. B.1). A further important role in this context is playedby the shape of the Fermi surface. The band structureof the β -band is complicated but has one eye-catchingproperty: the cross sections of the Fermi surface in therange k z > π/
27, 80
We show theschematic figures for the Fermi surface in Fig. 7 wherethe anisotropy is stressed for simplicity. The anisotropyof the Fermi surface affects h c2 in two ways, by facili-tating (i) the formation of the helical SC phase and (ii)the modification of the SC order parameters, as we willdiscuss now. (a) H=0 (b) H// [010] (Uniform) (c) H// [010] (Helical) (d) H// [110] II IIII
Fig. 7. (Color online) The schematic figure for the Fermi surfaceand SC gap in the p -wave state. We show the cross section for afixed k z . Two solid lines show the Fermi surfaces which are splitby the ASOC. The dashed lines show the magnitude of SC gap oneach Fermi surface. (a) The uniform BCS state at ~H = 0. Thedirection of d -vector is shown by the arrows. (b) The uniformBCS state for ~H// [010]. The parts of Fermi surface “I” and “II”are shown. The SC gap on the part “II” is suppressed. (c) Thehelical SC state for ~H// [010]. The SC gap on the part “II” oflarge Fermi surface is increased. (d) The helical SC state for ~H// [110]. First, (i) the helical SC phase is stable for theanisotropic Fermi surface not only in the p -wave statebut also in the s - and d -wave states. This is simply be-cause a set of quasi-particles with ~k = ± ~k + ~q H / ~q H ∼ h/v F ˆ x (Fig. 7(c)) because of the nesting ofFermi surface along the [100]-direction. This leads to thestrong enhancement of h c2 . This is not the case in theisotropic system where the Fermi surface is not nested.Second, the anisotropic Fermi surface enhances (ii) themixing of order parameters and increases in this way h c2 in the p -wave state. Because of the structure of g -vector ~g ( ~k ) ∝ ( − v y ( ~k ) , v x ( ~k ) , d x -componentof spin triplet order parameter while the d y -componentis the main source of the SC gap on the other part(Fermi surface “II” in Fig. 7(b)). Since the d x - and d y -components induce the Cooper pairing on different partsof the Fermi surface, the coupling is weak between thesetwo order parameters. Hence, the splitting of energy be-tween ~d ( ~k ) = ( − sin k y , sin k x ,
0) (most stable state) and ~d ( ~k ) = (sin k y , sin k x ,
0) (second most stable state) dueto the ASOC is small, and they can be easily mixed bythe applied magnetic field.
Isotropic FS Tetragonal FSd(k)=(k x ,k y ,0)(-k x ,k y ,0)(-k y ,k x ,0)(k y ,k x ,0) (-sink y ,sink x ,0)(sink y ,sink x ,0)(-sink x ,sink y ,0)(sink x ,sink y ,0) Fig. 8. (Color online) The schematic figure for the energy lev-els in the dominantly p -wave state. The isotropic and tetrag-onal symmetries are assumed in the left and right figures, re-spectively. The 2-fold degeneracy in the isotropic system be-tween ~d ( ~k ) = ( k y , k x ,
0) and ~d ( ~k ) = ( − k x , k y ,
0) is lifted to ~d ( ~k ) = (sin k y , sin k x ,
0) and ~d ( ~k ) = ( − sin k x , sin k y ,
0) in thetetragonal system. In case of the β -band of CePt Si, ~d ( ~k ) =(sin k y , sin k x ,
0) has lower energy.
In general, the tetragonal anisotropy of the Fermi sur-face reduces the splitting between the most stable andthe second most stable pairing states. For an isotropicFermi surface, the second most stable pairing statehas 2-fold degeneracy; ~d ( ~k ) = ( k y , k x ,
0) is degener-ate with ~d ( ~k ) = ( − k x , k y , . Phys. Soc. Jpn. Full Paper
Youichi
Yanase and Manfred
Sigrist schematic figure (Fig. 8). This lift of degeneracy de-creases the difference of condensation energy between ~d ( ~k ) = ( − sin k y , sin k x ,
0) and ~d ( ~k ) = (sin k y , sin k x , ~d ( ~k ) = ( − sin k x , sin k y , T/T c h c / T c p-wave, h//[100]p-wave, h//[110]s-wave, h//[100]s-wave, h//[110] Fig. 9. (Color online) The critical magnetic field h c2 along the[110]-axis in the helical p -wave (open circles) and s -wave (opensquares) states. Those along the [100]-axis are shown by theclosed symbols for a comparison. Furthermore, a strong anisotropy of the Fermi surfaceinduces a pronounced 4-fold anisotropy in the paramag-netic properties. Figure 9 shows that the h c2 along the[110]-direction is much smaller than that along the [100]-direction in case of the p -wave state. This is mainly be-cause the state ~d ( ~k ) = ( − sin k y , sin k x ,
0) is admixed bythe magnetic field along the [110]-direction, with ~d ( ~k ) =( − sin k x , sin k y , ~d ( ~k ) = (sin k y , sin k x , β -band of CePt Si. On the other hand, the 4-foldanisotropy is weak for the s -wave state as shown by thesquares in Fig. 9. This indicates that the anisotropicFermi surface enhances the h c2 in the p -wave state mainlythrough the mixing of SC order parameters.Finally, we comment on orbital depairing which wehave neglected so far. The orbital depairing effect isreduced by the mixing of order parameters in the p -wave state. For example, the parameter β in ~d ( ~k ) =( − sin k y , β sin k x , − i γ sin k y ) is decreased by the mag-netic field along the [010]-direction, and reduces the or-bital depairing effect, because the coherence length alongthe [100]-direction shrinks. Thus, the h c2 in the p -wavestate is enhanced by modifying the order parameterthrough the suppression of the orbital depairing effectas well as the paramagnetic depairing effect. In the discussion of the influence of AFM order on the p -wave SC state we focus on staggered moments alongthe [100]-axis with the magnetic field parallel to the [010]-axis, since the AFM moment favors to be perpendicularto the field. The situation of the magnetic field parallelto the moment is described in § p -wave statesignificantly depends on the anisotropic spin-spin inter- action, the J -term in eq. (31). The critical field h c2 de-picted in Fig. 10 with h Q = 0 .
125 in the AFM orderedphase shows a clear trend. While the AFM leads to areduction of h c2 in the absence of the anisotropic spininteraction ( J = 0), a strong enhancement is obtainedfor J = 0 . T/T c h c / T c PMAFM, J=0.3VAFM, J=0
Fig. 10. (Color online) The H - T phase diagram in the p -wavestate for the magnetic field along the [010]-axis in the presenceof AFM moment along the [100]-axis. The squares and trianglesshow the h c2 for J = 0 . V and J = 0, respectively. We fix h Q = 0 .
125 and choose the other parameters as in Fig. 4. The h c2 in the PM state is shown for a comparison (circles). We understand these results by analyzing the param-eter β of ~d ( ~k ) = ( − sin k y , β sin k x ,
0) at zero magneticfield.For β <
1, the superconductivity is dominant on Fermisurface region “I” in Fig. 7(b), while the magnetic fieldalong the [010]-axis suppresses Cooper pairing on theFermi surface “II”. For this reason, this SC state is robustagainst the magnetic field along the [010]-axis. The mag-netic field reduces β even more enhancing the anisotropyof the SC gap. The enhancement of h c2 due to the AFMorder is much more significant than expected in the sim-ple estimation eq. (42). In fact, the suppression of para-magnetic depairing effect in case of β < βT c appears in thiscase and induces the strong non-linearity. This is thereason why the influence of AFM order is much moreimportant in the p -wave state than in the s -wave state.If we assume J = 0, the parameter β is more than unity,which is incompatible with our RPA analysis for the Hub-bard model. On the other hand, we obtain β ∼ . J = 0 . V and h Q = 0 . χ b for J =0 . V ( β <
1) and J = 0 ( β >
1) in Figs. 11 (a) and(b), respectively. For low magnetic fields ( h = 0 . T c ) χ b is enhanced by AFM order in both cases consistentwith the linear response theory ( § χ b is furthermore enhanced for the moderate magnetic field h = T c with β < T c ( h ) is little decreased. According to these Full Paper
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T/T c χ b h=0.1T c h=1T c h=1.5T c h=2T c (a)(b) T/T c χ b h=0.1T c h=T c h=2T c Fig. 11. (Color online) The spin susceptibility along the [010]-direction in the helical p -wave state with AFM order. We assume J = 0 . V leading to β < J = 0 leading to β > theoretical results, the NMR Knight shift measurement by Yogi et al. and the µ SR measurement by Higemoto et al. were carried out in the non-linear response regime.In contrast to β <
1, the moderate magnetic field h = T c little affects the spin susceptibility (Fig. 11(b)). Thenon-linearity of spin susceptibility appears only in thehigh field region close to the critical magnetic field. Thisis a characteristic property of the SC state with strongparamagnetic depairing effect such as the spin singletpairing state in centrosymmetric system. Qualitativelythe same magnetic field dependence is obtained in thedominantly d x − y -wave state (see Fig. B.1 in AppendixB).
6. Pairing symmetry in CePt Si Measurements of H c2 and the Knight shift are consis-tent with p -wave superconductivity in CePt Si at ambi-ent pressure. The temperature dependence of H c21, 2, 7, 58 implies the absence of paramagnetic depairing. NMR and µ SR Knight shift data show no decrease below T c ,
66, 67 al-though T c remains rather high at applied magnetic fields.These findings could be understood based on the p -wavestate with AFM order for which the theoretical resultshave been shown in Figs. 10 and 11(a).We here note that the other possible mechanismsfor the high critical field H c2 are unlikely relevant in CePt Si. For example, a small g -factor has been sug-gested for CeCoIn ( g ∼ . It is expected that the g -factor of CeCoIn is significantly renormalized by thestrong AFM correlation in the ab -plane. However, thisis not the case in CePt Si where the spin correlationin the ab -plane are dominantly ferromagnetic.
27, 57
Thestrong coupling effect which has been ignored in thispaper is another possible cause of high H c2 . But, thejump of the specific heat at T = T c does not indicatestrong coupling effects in CePt Si,
1, 2, 7, 65 in contrast toCeIrSi . The p -wave state is consistent with the coherence peakin NMR 1 /T T
21, 29, 92 and the line node behaviors invarious quantities.
27, 30, 31, 63–65
Moreover the microscopictheory within an RPA theory suggests an in-plane p -wavestate induced by the β -band of CePt Si. For ~H ⊥ ~h Q the experimental magnetic propertiesof CePt Si at ambient pressure are consistent with the p -wave state with β <
1. This indicates the stronganisotropy of the effective spin interaction, which is de-scribed by the J -term in eq. (31) and is compatible withthe RPA analysis. It does however not agree with thenaive second order perturbation theory which leads tothe p -wave state with β > This is because the roleof spin fluctuation is underestimated within the pertur-bation theory. Based on this fact we may state thatthere is some evidence for spin-fluctuation-mediated su-perconductivity in CePt Si.When the magnetic field is parallel to the AFM mo-ment ~H k ~h Q , the paramagnetic depairing effect is en-hanced (suppressed) in the p -wave state with β < β > h c2 for ~H k ~h Q with β > h c2 for ~H ⊥ ~h Q with β < T c would mark theSC transition. Under such circumstances p -wave stateswith both β > β < Si.
1, 2, 7, 58, 66, 67
We here comment on the inter-plane d -wave statewhich we found as another possible pairing state on thebasis of the RPA theory. Although the 2-fold degen-eracy exists in this state ( d xz - and d yz -wave), the orderparameter has no internal degree of freedom with respectto the spin. Therefore, the paramagnetic depairing effectcannot be avoided by modifying the order parameter incontrast to the p -wave state. Hence, the magnetic prop-erties are qualitatively the same as those in the s -wavestate which seem to be incompatible with the experimen-tal results in CePt Si. The inter-plane d -wave state is in-compatible with the coherence peak in the NMR 1 /T T too.
7. Proposals for test experiments
Here we discuss several experiments which could helpto establish the pairing symmetry for CePt Si as well asCeRhSi and CeIrSi .The influence of antiferromagnetism on the magneticproperties can be tested by using the fact that AFM . Phys. Soc. Jpn. Full Paper
Youichi
Yanase and Manfred
Sigrist order can be suppressed by pressure in these materi-als. It follows from our results in § § ~H k ab and the spin susceptibility should decrease be-low T c in the low-magnetic field regime. Actually recentmeasurements of H c2 along ab -plane in CeRhSi
35, 70 andCeIrSi
37, 68, 78 imply a clear paramagnetic depairing ef-fect in the purely SC region, consistent with the theo-retical view. Notably paramagnetic depairing seems lesseffective in the AFM state of CeIrSi . This is compat-ible with p -wave pairing. No studies of this kind havebeen performed so far for CePt Si.A further aspect is the 2-fold in-plane anisotropy inthe AFM state. Since the [100]- and [010]-axes are notequivalent in the AFM state, a 2-fold anisotropy appearsin the ab -plane. Although the AFM moment perpendicu-lar to the uniform magnetic field is generally favored, thesituation ~H k ~h Q can nevertheless be realized for mag-netic fields low enough to leave the orientation of theAFM moment unchanged.We summarize the 2-fold anisotropy expected for eachpairing state in Fig. 12 taking also the orbital depairingeffect into account. We assumed here that the H c2 de-termined by the orbital depairing is much higher thanthe standard paramagnetic limit field in CePt Si,
1, 2, 7, 58
CeRhSi
34, 5 and CeIrSi .
7, 68
The upper critical field dueto orbital depairing is naturally enhanced by the heavymass of quasi-particles in these heavy Fermion com-pounds. Under such conditions paramagnetic depairingcan play a role in the high-field regime.Fig. 12(a) shows the H - T phase diagram in the p -wavestate. For ~H k ~h Q the paramagnetic depairing effect isenhanced (suppressed) with β < β > ~H ⊥ ~h Q (see Fig. 10). There-fore, a significant 2-fold anisotropy of H c2 could appearat high magnetic fields for either β < β >
1, pro-vided the AFM moment remains pinned. Qualitativelythe same anisotropy would occur at low magnetic fields,because the orbital depairing effect is anisotropic owingto the in-plane anisotropy of coherence length, namelythe difference of ξ a and ξ b . On the basis of the RPA the-ory for CePt Si we have estimated the anisotropy as ∂H c2 /∂T | T = T c ( ~H k [100]) : ∂H c2 /∂T | T = T c ( ~H k [010]) = ξ a : ξ b = 0 .
672 : 1 at h Q = 0 . H - T phasediagram is highly anisotropic in both high and low mag-netic field region as shown in Fig. 12(a).The strong 2-fold anisotropy in the ab -plane appearsalso in the inter-plane d -wave state due to the anisotropyof the coherence length. The 2-fold degeneracy betweenthe d xz - and d yz -wave states is lifted by the AFM or-der. The staggered moment along the [100]-axis favorsthe d xz -wave state and yields a coherence length whichis longer along the [100]-axis than along the [010]-axis.For this reason H c2 close to T = T c is smaller for themagnetic field along the [010]-axis. This anisotropy issuppressed at high magnetic fields because the paramag-netic depairing effect is nearly isotropic as in the s -wavestate (Fig. 3). These considerations lead to the schematicphase diagram in Fig. 12(b). (a) p-wave ( β <1) H//[100]H//[010] H//[100]H//[010] (b) d xz -wave(c) d xz -, d yz -wave ABC A: d xz -waveB: d xz +id yz -waveC: d yz -wave (d) d x -y -, d xy -, s-wave H//[100]H//[010]
Fig. 12. (Color online) Schematic figure for the 2-fold anisotropyin the H - T phase diagram. We assume the AFM order alongthe [100]-axis. (a) The p -wave state with β <
1. The oppositeanisotropy is expected for β >
1. (b) The inter-plane d -wave( d xz -wave) state. (c) Possible multiple phase transitions in theinter-plane d -wave state for ~H k [010]. (d) The intra-plane d -wave( d x − y -wave or d xy -wave) and s -wave states. It should be noted that the in-plane anisotropy of H c2 in the inter-plane d -wave state does not vanish if thequantum critical point of the AFM order is approached.This is in contrast to the p -wave state where the in-plane anisotropy is suppressed by decreasing the AFMmoment. In the vicinity of AFM quantum critical point,multiple phase transitions can occur for the inter-plane d -wave state as discussed in Ref. 27. These multiple phasesin the H - T plane are shown in Fig. 12(c) for the magneticfield along the [010]-axis. Pure d xz - and d yz -wave statesappear in the high-temperature region and in the high-magnetic field region, respectively. The chiral d xz ± i d yz -wave state is stabilized at low temperatures and fields. Ifthe multiple phase transitions were observed in the H - T plane or in the P - T plane, it would be a strong evidencefor the inter-plane d -wave state. Although some indica-tions for a second SC transition have been reported inCePt Si,
64, 94–96 it remains unclear whether it representsan intrinsic property or is caused by the sample inhomo-geneity.In contrast to the p -wave and inter-plane d -wavestates, the 2-fold anisotropy of H c2 is very weak in theintra-plane d -wave and s -wave states because the para-magnetic depairing effect as well as the orbital depairingeffect are nearly isotropic. Therefore, we obtain a simplephase diagram in Fig. 12(d).Since the 2-fold anisotropy of H c2 is quite differentbetween the dominantly p -wave, inter-plane d -wave andintra-plane spin singlet pairing states, the future exper-iment in the AFM state could identify the pairing sym-metry in CePt Si, CeRhSi and CeIrSi . It should be no-ticed that this experiment can be performed in CePt Siwithout applying the pressure.
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8. Helical Superconductivity
In this section we discuss the nature of the helical SCstate which is a novel SC phase specific to NCSC. The SCphase with a finite total momentum of Cooper pairs ~q H isstabilized in the presence of Rashba-type spin-orbit cou-pling under a magnetic field in the ab -plane. Thisstate bears some similarity with the FFLO state incentrosymmetric superconductors, but has also impor-tant differences. First, the helical SC phase is stabilizedimmediately above H c1 which is much lower than H c2 in extremely type II superconductors. This is in contrastto the FFLO state which appears in a narrow regionnear H c2 only. Second, the phase of SC order parameteris modulated as ∆( ~r ) = ∆ e i ~q H ~r in the helical SC state(which is the same form as in the Fulde-Ferrel (FF) state)while the Larkin-Ovchinnikov (LO) state with the spa-tial modulation of the amplitude, ∆( ~r ) = ∆ cos ~q H ~r =∆( e i ~q H ~r + e − i ~q H ~r ) /
2, is more stable than the FF state. Because the two momenta ~q H and − ~q H are equivalent inthe centrosymmetric system, the order parameter has adouble q structure in the LO state. On the other hand, ~q H is not equivalent to − ~q H in the non-centrosymmetricsystem under a magnetic field. For this reason the helicalSC phase is realized in the NCSC at least just below thecritical temperature. At higher fields and low tempera-ture also a “stripe SC state” can be realized,
35, 36 whichis similar to the LO state.Experimental evidence for the FFLO state has beenobtained for CeCoIn , more than forty years afterthe theoretical proposal.
88, 97
This is partly because theFFLO state is suppressed by a weak disorder. Thestripe SC state, which is resembles the FFLO state, canbe suppressed by a weak disorder too. In contrast to thesestates the helical SC state is realized even in the disor-dered material, if the superconductivity is present. Al-though there is no experimental verification of the helicalSC state in NCSC so far, the existence of the helical SCphase is a mandatory features from a theoretical pointof view.Now we turn to the effect of finite ~q H on the param-agnetic properties. Although the influence of the helicalsuperconductivity has been taken into account in § §
5, the following discussion will be important for a deeperunderstanding.One of the characteristic properties in the helical SCstate is the presence of a finite spin magnetization. In thelow magnetic field region this magnetization is expressedas ~M = ~M + ˆ χ ′ ~H with finite ~M . For simplicity, wehere consider the PM state and assume the SC orderparameter without gap nodes. Then, the magnetizationis obtained as, ~M = 14 X ~k ~ ˜ g ( ~k )( ~B ( ~k ) · ~q H − ~B ( ~k ) · ~q H ) (44) ∼
12 [ Z d ~k F ~ ˜ g ( ~k F )( ~v ( ~k F ) · ~q H ) /v ( ~k F ) − (1 ↔ D (ˆ z × ~q H ) , (46)with D ∝ α . We define ~B γ ( ~k ) = d( e γ ( ~k ) /E γ ( ~k )) / d ~k where γ is a band index. As shown in eq. (46), the mag- netization is oriented along the direction perpendicularto ~q H .The helical superconductivity also affects the differen-tial spin susceptibility χ ′ µµ = d M µ / d H µ when the SCgap has a node. According to eqs. (11-13), the quasi-particles suffer a Doppler shift in the helical SC stateand the single particle excitation energy is expressed as q e γ ( ~k ) + | ∆ γ ( ~k ) | ± ~v γ ( ~k ) · ~q H /
2. Following eq. (15), thePauli part of differential spin susceptibility is obtainedas, χ ′ P µµ = X γ Z d ~k F A µµγγ ( ~k F ) × Y H ( ~v γ ( ~k F ) , | ∆ γ ( ~k F ) | , T ) /v γ ( ~k F ) , (47)for µ = x, y where Y H ( ~v, ∆ , T ) is the generalized Yosidafunction, Y H ( ~v, ∆ , T ) = − Z d ε [ f ′ ( p ε + ∆ + ~v · ~q H / f ′ ( p ε + ∆ − ~v · ~q H / . (48)Since Y H ( ~v, ∆ ,
0) = p − / ( ~v · ~q H ) for | ∆ | < | ~v · ~q H | /
2, the Doppler shift boosts the differential spinsusceptibility in the SC state with a gap node like inCePt Si.
In fact, the uniform BCS state is favored at H = 0 andthe helical SC state is induced by an infinitesimal mag-netic field owing to the linear coupling between the mag-netization and the helicity ~q H (eq. (46)) with ~q H ⊥ ~H .Since the amplitude of ~q H is linear in the magnetic field | ~H | , the formation of helical SC state leads to a correctionto the linear response theory in § | α | ≪ ε F be-cause the amplitude of ~q H is small, | ~q H | ∼ ( α/ε F ) h/v F inlinear order of small parameter α/ε F .The helicity can play a quantitatively more importantrole in the non-linear response regime, because the am-plitude of ~q H increases from | ~q H | ∼ ( α/ε F ) h/v F in the lowfield region to | ~q H | ∼ h/v F in the high field region witha rapid crossover around h ∼ T c . For example, Fig. 13shows the magnetic field dependence of the helicity | ~q H | in the p -wave state, with a sharp increase of the helicityabove h = T c . As a result the critical field h c2 is signif-icantly enhanced at high fields as shown in Figs. 2 and5. The nature of the crossover from | ~q H | ∼ ( α/ε F ) h/v F to | ~q H | ∼ h/v F becomes obvious viewing the momentumdependence of eigenvalues λ ( ~q ) in eqs. (40) and (41). Fig-ures 14(a) and (b) show the numerical results in the PMand AFM states, respectively. In Fig. 14(a), λ ( ~q ) pos-sesses a crossover from a single to a double peak struc-ture, yielding a rapid increase of the helicity. This resultimplies that the nature of the helical SC phase is differ-ent below and above the crossover magnetic field. Actu-ally, the “stripe SC state” can be stabilized above thecrossover field. As shown in Fig. 14(b), the crossoverfrom the single to the double peak structure is suppressedby the AFM order. The eigenvalue λ ( ~q ) has a single peakeven in the magnetic field much higher than the standard . Phys. Soc. Jpn. Full Paper
Youichi
Yanase and Manfred
Sigrist h/T c | q H | h Q =0h Q =0.125 Fig. 13. (Color online) The amplitude of the helicity | ~q H | justbelow T c . The circles and squares show the results in the PM andAFM states, respectively. We assume the magnetic field alongthe [010]-direction which leads to ~q H along the [100]-direction. J = 0 . V is assumed in the AFM state. The other parametersin the PM and AFM states are the same as in Figs. 4 and 10,respectively. paramagnetic limit. This is simply because the AFM or-der suppresses the paramagnetic depairing effect in the p -wave state.We would like to point here that CePt Si is a goodcandidate for an experimental observation of the heli-cal SC phase. Actually, the large critical field H c2 leadsto the helical SC phase with large ~q H ( | ~q H | ∼ h/v F ) ina large part of the H - T phase diagram. It seems to bedifficult to detect the helical SC phase with small helic-ity | ~q H | ∼ ( α/ε F ) h/v F because the wave length is muchlonger than the coherence length. Thus the high fieldphase with | ~q H | ∼ h/v F is more promising for the ex-perimental observation. The high field phase is stable inthe p -wave state above h = T c as shown in Figs. 4, 10and 13. However, this phase shrinks in the SC state withdominantly spin singlet pairing and/or the strong orbitaldepairing effect which leads to small H c2 .
9. Summary and Discussions
We have investigated the paramagnetic properties inNCSC. The SC states with leading p -wave, d -wave or s -wave order parameter have been examined in view of theheavy Fermion superconductors, CePt Si, CeRhSi andCeIrSi .First, the linear response to the magnetic field has beeninvestigated with the particular interest on the role ofAFM order. The spin susceptibility is universal in thesense that it is independent of the pairing symmetry at T = 0, if the ASOC is much larger than the SC gap,and results from the band splitting due to the ASOC.The spin susceptibility below T c is increased in the AFMstate due to the folding of unit cell, if the magnetic field isapplied perpendicular to the AFM moment. The resultis opposite for the magnetic field parallel to the AFMmoment.Second, we have shown that the non-linear responseto the magnetic field depends on the symmetry of theleading SC order parameter. In particular, the spin sus-ceptibility and H c2 for the p -wave state are significantlyenhanced by non-linear effects through (i) the formation (a)(b) -0.06 -0.04 -0.02 0 0.02 0.04 0.06 q x λ ( q ) h=T c h=2T c h=3T c -0.06 -0.04 -0.02 0 0.02 0.04 0.06 q x λ ( q ) h=T c h=2T c h=3T c Fig. 14. (Color online) The momentum dependence of the eigen-value λ ( ~q ) in the linearized mean field equation (eqs. (40) and(41)) with ~q = ( q x , , of a helical SC state and (ii) the mixing of SC order pa-rameters. The anisotropy of Fermi surface can increasethese non-linear effects, and strengthen the influence ofAFM order. Taking these aspects into account, the exper-imental results
1, 2, 7, 58, 66, 67 for CePt Si at ambient pres-sure (within the AFM state) are consistent with the p -wave state admixed with a secondary s -wave component.This is the pairing state which has been proposed re-cently by Frigeri et al. and identified by the microscopicRPA theory.
27, 62
It has been shown that this p -wavestate is consistent with the line node behavior
27, 62–65 and also with the coherence peak in NMR 1 /T T .
21, 29, 92
Although the RPA theory has identified the inter-plane d -wave ( d xz - and d yz -wave) state as further candidate,this state seems to be incompatible with the Knight shift, H c2 and NMR 1 /T T measurements at ambient pressure.According to these comparisons between the theory andexperiment, CePt Si is rather likely the first identifiedspin triplet superconductor in Ce-based heavy fermionsystems.We have proposed several experiments which can pro-vide further evidences for the pairing state in CePt Sias well as in CeRhSi and CeIrSi . The first proposalis the pressure dependence in various quantities. If theAFM order is a major cause of the unusual properties inCePt Si, a pronounced pressure dependence is expectedin NMR, specific heat, thermal transport, superfluid den-sity and so on. If CePt Si has a leading p -wave orderparameter, the following behaviors are expected above Full Paper
Youichi
Yanase and Manfred
Sigrist the critical pressure P ∼ . T c for the magnetic field along the ab -plane and below the standard paramagnetic limit. (b)The paramagnetic depairing effect is enhanced for ~H k ab but not for ~H k c . (c) The low-energy excitations due tothe accidental line nodes are decreased. (d) The coher-ence peak in NMR 1 /T T is enhanced by the isotropicSC gap. These pressure dependences are not expectedin the intra-plane d -wave ( d x − y - and d xy -wave) and s -wave states. The pressure dependence (c) is expected alsoin the inter-plane d -wave state and then the additionalphase transition occurs in the P - T and H - T plane. Another proposal for a future experiment is the 2-foldanisotropy arising from the AFM order. The strong 2-fold anisotropy is expected in the dominantly p -wavestate while the anisotropy is negligible in the s -wave andintra-plane d -wave states. In the inter-plane d -wave statethe strong 2-fold anisotropy is expected near T = T c but the anisotropy is suppressed at high magnetic fields.The experimental observation of the 2-fold anisotropy inthe AFM state could provide an important evidence forthe pairing symmetry in CePt Si, CeRhSi and CeIrSi .Thus, the response to the AFM order can be a signatureof the pairing symmetry in non-centrosymmetric super-conductors. Acknowledgments
The authors are grateful to D. F. Agterberg, J. Akim-itsu, J. Flouquet, P.A. Frigeri, S. Fujimoto, J. Goryo,N. Hayashi, K. Izawa, N. Kimura, Y. Kitaoka, Y. Mat-suda, V. P. Mineev, H. Mukuda, E. Ohmichi, Y. ¯Onuki,T. Shibauchi, R. Settai, T. Takeuchi, H. Tanaka, T.Tateiwa, H. Tou and M. Yogi for fruitful discussions.This study has been financially supported by the NishinaMemorial Foundation, Grants-in-Aid for Young Scien-tists (B) from the Ministry of Education, Culture,Sports, Science and Technology (MEXT), the Swiss Na-tionalfonds and the NCCR MaNEP. Numerical compu-tation was carried out at the Yukawa Institute ComputerFacility.
Appendix A: Linear Response Theory
The dynamical spin susceptibility in the linear re-sponse regime is obtained by the Kubo formula as, χ µν ( q ) = − X γ,δ X k ′ [ S µδγ ( ~k + ~q, ~k ) S νγδ ( ~k, ~k + ~q ) G δ ( k + q ) G γ ( k ) − S µγδ ( ~k + + ~q, ~k + ) S νγδ ( − ~k − − ~q, − ~k − ) F δ ( k + q ) F † γ ( k )] . (A · q = ( ~q, iΩ n ), k = ( ~k, i ω n ) and ~q is the momentumalong the ab -plane. The spin operator S µγδ ( ~k + ~q, ~k ) in theband basis has been given in eq. (17).Taking the limit Ω n → ~q →
0, we obtain theuniform spin susceptibility χ µν = lim ~q → lim Ω n → χ µν ( q )which can be decomposed into a Van-Vleck and Paulipart as, χ V µν = lim Ω n → lim ~q → χ µν ( q ) , (A · χ P µν = χ µν − χ V µν . (A ·
3) We obtain the following expressions, χ P µν = − lim ~q → lim Ω n → X γ X k ′ [ S µγγ ( ~k, ~k ) S νγγ ( ~k, ~k ) G γ ( k + q ) G γ ( k ) − S µγγ ( ~k + , ~k + ) S νγγ ( − ~k − , − ~k − ) F γ ( k + q ) F † γ ( k )] , (A · χ V µν = − X γ = δ X k ′ [ S µγδ ( ~k, ~k ) S νδγ ( ~k, ~k ) G δ ( k ) G γ ( k ) − S µγδ ( ~k + , ~k + ) S νγδ ( − ~k − , − ~k − ) F δ ( k ) F † γ ( k )] . (A · | ∆ γ ( ~k ) | , v F | ~q H | / ≪ | α | where v F is theFermi velocity, the Van-Vleck part of spin susceptibilityeq. (A.5) is obtained as in eq. (16).When we restrict to the AFM moment along the prin-cipal axis, namely ~h Q k ˆ x, ˆ y or ˆ z , the relation ˆ U ( − ~k ) = e i θ ˆ I ˆ U ( ~k ) holds with θ an arbitrary phase factor. Here wedenote ˆ I = (cid:18) ˆ I ± ˆ I (cid:19) , (A · I = (cid:18) − (cid:19) . (A · I in ˆ I is + for ~h Q k ˆ x , ˆ y and − for ~h Q k ˆ z . According to eqs. (17) and (18),we obtain S µγγ ( − ~k, − ~k ) = − S µγγ ( ~k, ~k ) for µ = x, y and S z γγ ( − ~k, − ~k ) = S z γγ ( ~k, ~k ). If v F | ~q H | / ≪| α | , the coefficient in eq. (A.4) is approximated as S µγγ ( ~k + , ~k + ) S νγγ ( − ~k − , − ~k − ) ∼ S µγγ ( ~k, ~k ) S νγγ ( − ~k, − ~k ) andthe Pauli part of spin susceptibility is obtained aseq. (15). Appendix B: magnetic Properties in the d -waveState For the discussion for the intra-plane d -wave state weadopt the model eq. (29) but assume the tight bindingparameters in eq. (27) as,( t , t , n ) = (1 , . , . , (B · d x − y -waveSC state for the parameter set (A) U > V = − . U .The order parameters are described as Φ( ~k ) = δ + η cos k x − cos k y with δ = 0 and η = 1 at h Q = h = 0. Ouranalysis confirms | δ | , | − η | ≪
1. In general, the d x − y -wave state is admixed with the f x(x − y ) - and f y(x − y ) -wave order parameters owing to the ASOC. However, the f -wave component does not appear in the mean field so-lution of the effective model eq. (29) because interactionsbeyond the nearest neighbor sites are neglected.We calculate the critical magnetic field h c2 by solv-ing the linearized mean field equation eqs. (40) and (41)and show the result in Fig. 4. The spin susceptibility iscalculated on the basis of eq. (43) by solving the meanfield equation eqs. (35-38). In Fig. B.1 we show the spinsusceptibility below T c for various magnetic fields. Theseresults should be contrasted to those for the p -wave state(Figs. 4, 6, 10 and 11). . Phys. Soc. Jpn. Full Paper
Youichi
Yanase and Manfred
Sigrist T/T c χ a = χ b h=0.1T c h=T c h=1.5T c h=2T c Fig. B ·
1. (Color online) The spin susceptibility along the [100]-and [010]-directions in the d -wave state without AFM order. Weassume U > V = − . U , J = 0, α = 0 . h Q = 0. Themagnetic field is chosen as h = 0 . T c , h = T c , h = 1 . T c and h = 2 T c from the bottom to the top.1) E. Bauer, G. Hilscher, H. Michor, Ch. Paul, E. W. Scheidt, A.Gribanov, Yu. Seropegin, H. Noel, M. Sigrist, P. Rogl: Phys.Rev. Lett (2004) 027003.2) E. Bauer, I. Bonalde and M. Sigrist: J. Low. Temp. Phys. (2005) 748; E. Bauer, H. Kaldarar, A. Prokofiev, E. Royanian,A. Amato, J. Sereni, W. Bramer-Escamilla and I. Bonalde: J.Phys. Soc. Jpn. (2007) 051009.3) T. Akazawa, H. Hidaka, T. Fujiwara, T. C. Kobayashi, E. Ya-mamoto, Y. Haga, R. Settai and Y. ¯Onuki: J. Phys. Soc. Jpn. (2004) 3129.4) N. Kimura, K. Ito, K. Saitoh, Y. Umeda, and H. Aoki, T.Terashima: Phys. Rev. Lett. 95 (2005) 247004.5) N. Kimura, Y. Muro and H. Aoki: J. Phys. Soc. Jpn. (2007)051010.6) I. Sugitani, Y. Okuda, H. Shishido, T. Yamada, A. 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