Hall conductivity in the normal and superconducting phases of the Rashba system with Zeeman field
HHall conductivity in the normal and superconducting phases of the Rashba systemwith Zeeman field
Suk Bum Chung
1, 2, 3 and Rahul Roy Center for Correlated Electron Systems, Institute for Basic Science (IBS), Seoul 151-747, Korea Department of Physics and Astronomy, Seoul National University, Seoul 151-747, Korea Department of Physics and Astronomy, University of California Los Angeles, Los Angeles, California 90095-1547, USA (Dated: September 24, 2018)We study the intrinsic Hall conductivity of the ordinary and topological superconducting phasesof a Rashba metal in a perpendicular Zeeman field. In this system the normal metal breaks timereversal symmetry while the superconducting order parameter does not, in contrast to the chiralp-wave superconducting state predicted in the monolayer strontium ruthenate (Sr RuO ) whoseHall conductivity has been studied extensively. We study the effects of intra-band and inter-bandpairing and find there is qualitatively larger change in the intrinsic Hall conductivity when there isinter-band pairing, with the change in magnitude linear in the pairing gap. We argue that inter-band pairing leads in general to higher energy costs for the topological phase compared to thetopologically trivial phase and thus that the qualitative behavior of the intrinsic Hall conductivitywith superconductivity in these systems could provide important clues about the nature of pairingin the superconducting phase and even some hints of whether it is topological or not. I. INTRODUCTION
Although the recent interest in intrinsic Hall conduc-tivity largely focuses on its remarkable quantization ininsulators , a nonzero Hall conductivity is possiblefor any systems, including metal and superconductor,that breaks time-reversal symmetry. While there is noquantization of the Hall conductivity in metals, therenonetheless exists the identical geometric picture of itthrough the Karplus-Luttinger formula , which statesthat the intrinsic Hall conductivity is proportional to thenet Berry curvature in the Brillouin zone. The (non-)quantization of the Hall conductivity in insulators (met-als) can be explained by this formula together with thefact that the total Berry curvature for each band in thefirst Brillouin zone is quantized. On the other hand,less has been known about what determines the mag-nitude of intrinsic Hall conductivity of superconductors,in spite of recent detection of time-reversal symmetrybreaking in various unconventional superconductors in-cluding not only Sr RuO but also more recently UPt and URu Si .The possibility of exotic physics in topological super-conductors such as non-abelian statistics of vortex defectshas led to a great deal of theoretical and experimental in-terest in possible candidate materials. Sr RuO whichis thought to be a chiral p x + ip y superconductor has per-haps attracted the most interest. Evidences for brokentime-reversal symmetry in Sr RuO include muon spinrelaxation measurements in addition to the Kerr rota-tion.The Kerr rotation angle is a measure of the intrin-sic Hall conductivity and a non-zero value of this angleindicates time-reversal symmetry breaking. Somewhatsurprisingly however, theoretical calculations show thatin the long wavelength limit, the intrinsic, i.e. impurity-independent, Hall conductance of a pure p + ip supercon- ductor is zero . A non-zero intrinsic Hall conductiv-ity has only been obtained so far in chiral p -wave modelsthat allow for interband pairing . Unlike in insula-tors, where the Hall conductance is quantized, the Hallconductance of superconductorsA different type of time-reversal symmetry breakingsuperconductor that is attracting widespread interest inrecent years is the the one that occurs in 2D metalwith strong Rashba spin-orbit coupling under a perpen-dicular Zeeman field. In this system, a topologicallynon-trivial superconducting phase analogous to the spin-polarized chiral p -wave superconductor can arise for theright range of the chemical potential and the Zeemanfield . There are two key differences between thisRashba superconductor and the chiral p -wave supercon-ductor that affect the Kerr rotation angle. The first isthat the time-reversal symmetry breaking in the Rashbasuperconductor does not originate from the Cooper pair-ing and is already present in the normal phase. The otheris that the Rashba system is inherently multi-band dueto the spin-orbit coupling splitting of the Fermi surface,naturally raising the question whether the Cooper pairingis purely intraband or has a nonzero interband compo-nent. The possibility of the inter-band pairing has beendiscussed in the recent literature and it been notedthat its presence or absence will not affect the possibilityof the topological quantum phase transition. However,the physical consequence of the inter-band pairing hasremained an under-investigated aspect of the field.In this paper, we calculate the Hall conductivity ofthe Rashba superconductor in which an interesting in-terplay of time-reversal symmetry breaking in the nor-mal phase and a time-reversal invariant order parameteroccurs. We separately consider the effects of the intra-band and inter-band pairing and find that the effect of su-perconductivity on the Hall conductivity is qualitativelystronger when there is nonzero inter-band pairing. Amore precise statement of our result is that the change a r X i v : . [ c ond - m a t . s up r- c on ] J u l in the Hall conductivity due to superconductivity is lin-ear in the pairing gap with a nonzero inter-band pairingbut quadratic in pairing gap with a purely intra-bandpairing. The effect of inter-band pairing is consistentwith the recent calculations of the Hall conductivity inmulti-band chiral p -wave superconductor models.This paper is organized as follows. In the section II,we calculate the Hall conductivity of the Rashba metalunder a Zeeman field using the linear response and showthat the result agrees with the Karplus-Luttinger for-mula. In the sections III-IV, we calculate the effect ofCooper pairing on the Hall conductivity of this systemwith purely intra-band pairing and with both intra- andinter-band pairing, respectively. In the section V, we dis-cuss how the two cases would correspond to the topologyof the superconducting phase followed by a discussion inthe conclusion. II. THE 2D RASHBA METAL AND ITS HALLCONDUCTIVITY
We use the term “2d Rashba metal” to describe a twodimensional system with a strong Rasha spin orbit cou-pling governed by an effective Hamiltonian of the form:ˆ H = p m ∗ − µ − α ( p y σ + p x σ ) (1)where m ∗ is the effective mass, α a parameter which char-acterizes the strength of the spin-orbit coupling, and σ , are the Pauli spin matrices; for convenience, we will set (cid:126) = 1.The two bands of the 2d Rashba metal have energies, p / m ∗ − µ ± αp and a Dirac like crossing at p = 0. Thesystem is time-reversal invariant which implies that theHall conductance is zero. This in turn means that the netor integrated Berry curvature over all p of all negativeenergy eigenstates for is zero.In the presence of an effective Zeeman term (as couldpossibly be induced from a tunneling from a magneticinsulator), this picture changes. Time-reversal symmetryis no longer preserved, raising the possibility of a non-zeroHall conductance. We shall now confirm this possibilitywith a detailed calculation. The Hamiltonian with aneffective Zeeman term hσ can be written asˆ H = p m ∗ − µ − d p · σ , (2)where d = ( − αp y , αp x , h ). The current operator for thisHamiltonian is ˆ v x = p x m ∗ − σ α, ˆ v y = p y m ∗ + σ α. (3) and the finite temperature Green function isˆ G ( k , iω n ) = [ iω n − ( ξ k − d k · σ )] − = ˆ P + ( k ) iω n − ( ξ k − d k ) + ˆ P − ( k ) iω n − ( ξ k + d k ) , (4)where ˆ P ± ( k ) = (1 ± d k · σ ) / ω n is the Matsubara frequency, and ξ k ≡ k / m ∗ − µ . We can then compute the optical Hall con-ductivity at T = 0 using the Kubo formula: σ xy ( ω ) = ie ω (cid:90) d kdν (2 π ) tr[ˆ v x ˆ G ( k , iν + ω )ˆ v y ˆ G ( k , iν )] − ( x ↔ y )= − e α h ω (cid:90) d k (2 π ) (cid:88) s = ± d k (2 d k + sω ) × (cid:90) dν π (cid:20) iν − ( ξ k + sd k ) − iν + ω − ( ξ k − sd k ) (cid:21) , (5)where we have used (cid:90) d kdν (2 π ) tr[ˆ v x ˆ G ( k , iν + ω )ˆ v y ˆ G ( k , iν )] − ( x ↔ y )= (cid:90) d kdν (2 π ) (cid:88) s = ± ˆ v x ˆ P s ˆ v y ˆ P − s [ iν + ω − ( ξ k − sd k )][ iν − ( ξ k + sd k )] − ( i ↔ j ) (6)and k F ± are the momenta where ξ k ∓ d k = 0. Thisformula comes out simplest for ω = 0, giving us the Hallconductivity: σ xy ( ω = 0) = e α h (cid:90) d k (2 π ) (cid:88) s = ± s d k sgn( ξ k + sd k )= e α h π (cid:90) k F + k F − kdk ( h + α k ) / = e π h (cid:113) h + α k F − − (cid:113) h + α k F + . (7)For the case where we only have a single Fermi surface,we can just set k F − = 0. Note that this requires µ ≥ | h | .This result could also have been obtained from theKarpus-Luttinger formula for the Hall conductivity ofa metal. The Karpus Luttinger formula states that σ xy = e π (cid:90) d k (2 π ) (cid:88) n F xyn n n ( k ) (8)where n is the band index, F xyn is the Berry curvatureof the n -th band, and n n ( k ) is the occupation number ofthe n -th band at momentum k .It is straightforward to use this formula if one notesthat the Berry curvature is entirely determined by thespin-dependent terms, hence is equal to the skyrmiondensity of the spin, ˆd · ( ∂ k x ˆd × ∂ k y ˆd ). Thus, there is can-cellation between the contribution from the larger Fermipocket which covers the sphere starting from the northpole down to the ‘altitude’ β + ≡ tan − αk F + /h andthat from the smaller Fermi pocket which starts fromthe south pole up to β − ≡ tan − αk F − /h , giving us σ xy = e π π (cid:90) dφ (cid:32)(cid:90) β + − (cid:90) cos β − − (cid:33) d cos β = e π h (cid:113) h + α k F − − h (cid:113) h + α k F + . (9)in agreement with the result obtained from the Kuboformula. III. PAIRING IN THE RASHBASUPERCONDUCTOR WITH ZEEMAN FIELD
Much of interest in the Rashba metal with an effec-tive Zeeman arises from the existence of the topologi-cally non-trivial superconducting phase. A simple formof pairing that has been studied extensively inthis context is H pair = | ∆ | (cid:88) k c † k ↑ c †− k ↓ + h . c . (10)The full first-quantized Hamiltonian including pairing isthen ˆ H BdG = τ ( ξ k − σ · d (cid:107) ) − σ d z + τ | ∆ | (11)in the Nambu basis ( ψ k , ↑ , ψ k , ↓ , − ψ †− k , ↓ , ψ †− k , ↑ ) T , with τ being the Pauli matrices in the electron-hole space. Thiscan be transformed to the band basis (with σ diagonalwith respect to the bands) as ˆ U † ˆ H BdG ˆ U = τ ( ξ k − σ d k ) − | ∆ | ( − τ ˆ k y + τ σ ˆ k x ) (cid:112) − d z + | ∆ | τ σ ˆ d z , (12)where ˆ U gives us the basis transformation be-tween the band basis and the original spin basis( ψ k , ↑ , ψ k , ↓ , − ψ †− k , ↓ , ψ †− k , ↑ ) T . The pairing has both inter-band and intra-band components. We would like to dis-entangle the contributions to the Hall conductivity fromthe two components and ask if there are qualitative dif-ferences between them. Interband pairing is of coursemore likely to arise in systems where the superconduc-tivity is intrinsic, i.e. not induced through the proximityeffect.In the following two sections, we will calculate theHall conductivity of the Rashba system in the super-conducting state. We will show that the change in theHall conductivity from its normal state value will dependqualitatively on absence or presence of significant inter-band pairing. The effect of superconductivity is weak in the absence of the interband pairing, in the sense thatthe Hall conductivity still retains some similarity to theKarplus-Luttinger formula of Eq.(8). The purely intra-band Cooper pairing in the Rashba system is expectedto break time-reversal symmetry, yet, strikingly, it doesnot significantly impact the Hall conductivity. This is insome sense consistent with the theoretical results for thesuperconducting phase of Sr RuO , where the intrinsicHall conductivity in absence of the interband pairing iszero , which is to say, the intrinsic Hall conductivityretains the normal state value in absence of the inter-band pairing. On the other hand, as we shall see later,there is significant impact from time-reversal symmetrypreserving interband pairing . A. The case of purely intra-band pairing
We consider the purely intra-band pairing model thathas an anti-chiral (chiral) pairing gap for the larger(smaller) Fermi surface,ˆ∆ ± = | ∆ ± | ( − τ ˆ k y ± τ ˆ k x ) exp( ∓ iτ φ/
2) (13)( φ being the phase difference between the two gaps), inthe band basis, as this can be regarded as the purely in-traband pairing that is closest to the s -wave pairing ofEq.(10) . In fact, this pairing with | ∆ + | = | ∆ − | and φ = 0 is exactly equal to Eq.(12) minus the inter-band pairing term | ∆ | τ σ ˆ d z , as can be shown from thefull first quantized BdG Hamiltonian with the intrabandpairing of Eq.(13) in the band basis:ˆ U † ˆ H (cid:48) BdG ˆ U = τ ( ξ k − σ d k ) + ˆ P (cid:48) + ˆ∆ + + ˆ P (cid:48)− ˆ∆ − , (14)where ˆ P (cid:48)± = (1 ± σ ) / P (cid:48)± = (1 ± σ ) / P ± = (1 ± σ · d ) / U † [(1 ± σ · ˆ d (cid:107) ) + τ σ ˆ d z ] U/ P (cid:48)± . Topologically, the purely intraband pairing gap ofthis subsection is equivalent to the pure s -wave pair-ing, i.e. it gives us the same topological phases withthe same Read-Green class of the topological quantumphase transition . Since the inner and outer Fermi sur-faces form two independent superconductors in the intra-band pairing model, when there are two Fermi surfacesEq.(13) gives us a topologically trivial superconductiv-ity, as we have two weak pairing superconductors whosetopological invariants cancels out to zero due to their op-posing chirality. When the inner Fermi surface vanishesto k = 0 at µ = | h | , ˆ∆ − vanishes as there is no de-generacy at k = 0, and the topological phase transitionsbetween the topologically trivial and non-trivial phasesoccur; note that the phase transition point for the purely s -wave pairing is slightly shifted to µ = h − | ∆ | .Given that the nonzero interband pairing was required forthe purely s -wave pairing, it is natural that the purely intraband pairing of Eq.(13) in general gives us a mixtureof the s -wave and the p -wave pairing in the original spinbasis :ˆ U ( ˆ P (cid:48) + ˆ∆ + + ˆ P (cid:48)− ˆ∆ − ) ˆ U † = − ¯ | ∆ | (cid:20) ( τ (cid:113) − ˆ d z + τ σ · ˆk ˆ d z ) cos φ τ ( σ × ˆk ) · ˆz sin φ (cid:21) + δ | ∆ | (cid:20) τ ( σ × ˆk ) · ˆz cos φ − τ σ · ˆk ˆ d z + τ (cid:113) − ˆ d z ) sin φ (cid:21) (15)where ¯ | ∆ | ≡ ( | ∆ + | + | ∆ − | ) / , δ | ∆ | ≡ ( | ∆ + | − | ∆ − | ) / U ( − τ ˆ k y + τ σ ˆ k x ) ˆ U † = − τ (cid:113) − ˆ d z − τ σ · ˆk ˆ d z , ˆ U ( − τ ˆ k y + τ σ ˆ k x ) σ ˆ U † = τ ( σ × ˆk ) · ˆz . (16)Note that in Eq.(15), the pairing gap breaks time-reversalsymmetry due to the perpendicular Zeeman field, i.e. ˆ d z = h/ √ h + α k (cid:54) = 0 and the phase difference φ be-tween the gaps of the two Fermi surfaces.In the case of the purely intraband pairing, the Hallconductivity calculation for the superconducting phaseis no more complicated than the same calculation for thenormal state. The Kubo formula provides the simplestgauge invariant method for calculating the optical Hallconductivity for a superconductor phase . Whenthe pairing is purely intraband, the Green function, themost important ingredient of the Kubo formula, remainsblock-diagonal in the band basis for the superconductingphase:ˆ U † ˆ G BdG ˆ U ≡ ˆ U † [ iω n − ˆ H (cid:48) BdG ] − ˆ U = − ˆ P (cid:48) + [ iω n + τ ( ξ k − d k ) + ˆ∆ + ] ω n + E − ˆ P (cid:48)− [ iω n + τ ( ξ k + d k ) + ˆ∆ − ] ω n + E − , (17)where E ± = ( ξ k ∓ d k ) + | ∆ ± | is the quasi-particleeigenenergy. The Kubo formula for the superconduct-ing phase can be obtained by inserting the BdG Greenfunction of Eq.(17) into Eq.(5) with an overall factor of1/2 to cancel out the BdG doubling: σ xy ( ω ) = ie ω (cid:90) d kdν (2 π ) tr[ˆ v x ˆ G BdG ( k , iν + ω )ˆ v y ˆ G BdG ( k , iν )] − ( x ↔ y )at T = 0. Its important to note that the current opera-tors have the same expression as in the normal state, i.e. ,they do not get any contribution from the pairing termsof the Hamiltonian.We find that the effect of the Cooper pairing onthe Hall conductivity can be attributed solely to the change in the quasiparticle spectrum in the sense thatthere is contribution only from the normal part of theBdG Green function ˆ g ( k , iω ) ≡ − (cid:80) s = ± ˆ U [ iω n + τ ( ξ k − sd k )] ˆ U † / ( ω n + E s ) but none from the anomalous partˆ f ( k , iω ) ≡ − (cid:80) s = ± ˆ U ˆ P (cid:48) s ˆ∆ s ˆ U † / ( ω n + E s ). In other words,the Hall conductivity still originates from the same pro-cess as in the normal state, the normal propagation of anelectron and a hole from different bands as represented by(a) of Fig.1. Hence the time-reversal symmetry breakingof the intraband pairing playing no role, which includeslack of any dependence on the phase difference φ betweenthe gaps of the two Fermi surfaces. The vanishing ofthe contribution from the anomalous part of the Greenfunction is becausetr (cid:20)(cid:18) k x m ∗ − ασ (cid:19) ˆ U ˆ P (cid:48)± ˆ∆ ± ˆ U † (cid:18) k y m ∗ + ασ (cid:19) ˆ U ˆ P (cid:48)∓ ˆ∆ ∓ ˆ U † (cid:21) − ( x ↔ y )= ± iα [ˆ k x ˆ k y (1 − ˆ d z ) + (ˆ k x ˆ d z + ˆ k y )(ˆ k x + ˆ k y ˆ d z )] × tr[ τ ˆ∆ ± ˆ∆ ∓ ] , (18)where we first traced out band indices usingˆ U † σ ˆ U = − τ σ (ˆ k x + ˆ k y ˆ d z ) − σ ˆ k x ˆ k y (1 − ˆ d z ) − σ ˆ k y (cid:113) − ˆ d z , ˆ U † σ ˆ U = − τ σ ˆ k x ˆ k y (1 − ˆ d z ) − σ (ˆ k x ˆ d z + ˆ k y )+ σ ˆ k x (cid:113) − ˆ d z , (19)vanishes upon angular integration astr[ τ ˆ∆ ± ˆ∆ ∓ ] = ± i | ∆ + || ∆ − | [2ˆ k x ˆ k y cos φ − (ˆ k x − ˆ k y ) sin φ ](20)is odd with respect to the π/ σ xy = e α h π (cid:90) kdk d k E − + E + ) (cid:18) ξ k + d k E − − ξ k − d k E + (cid:19) . (21) Comparison between the integrand of Eq.(7) and Eq.(21)in (a) of Fig.2 shows the change being limited to theelimination of the singularity at the Fermi surface similarto what we see for the occupation number.The change in the Hall conductivity due to infinitesimal pairing gaps has contributions from both the change inthe occupation number n ± ( k ) and the factor that was previously (in the Rashba metal) the Berry curvature σ SCxy − σ Nxy ≈ e π (cid:90) d k (2 π ) (cid:88) n [ { ∆ F xyn } n n ( k ) + F xyn { ∆ n n ( k ) } ] (22)both terms in the order of | ∆ | log | ∆ | for a small pairing gap at the Fermi surface; the first derivative with respectto the pairing gap vanishes. In the limit of small Rashba effect α (cid:28) (cid:112) µ/m ∗ and Zeeman field | h | (cid:28) µ , the changein Hall conductivity comes mostly from ∆ F xyn , giving us σ SCxy − σ Nxy ≈ − e π hµ m ∗ α (cid:88) s | ∆ s | log 4 √ α √ m ∗ µ | ∆ s | . (23)While the factor of log | ∆ − | would not be present in the | ∆ − | term for the topologically non-trivial supercon-ducting phase, we do not expect this to have any sub-stantial effect as | ∆ − | (cid:28) | ∆ + | is physically expectedin the topologically non-trivial phase. Hence, we con-clude that for the case of the purely intraband pairingthere is no qualitative difference between the effect ofsuperconductivity on the Hall conductivity between thetopologically trivial and non-trivial intraband supercon-ductivity. This is consistent with the observation thatthe existence of the chiral Majorana edge state, the keyfeature of the topologically non-trivial superconductivityabsent in the topologically trivial superconductivity, isirrelevant to the response to the electromagnetic field asthe quasiparticle excitation of the Majorana edge stateis charge neutral. B. Effect of interband pairing
In this section, we will show how the effect of supercon-ductivity on the Hall conductivity becomes qualitativelylarger with interband pairing. Following the recent liter-ature on the analysis of the topological superconductivityin the Rashba system , we return to the simplest formof the BCS pairing, H pair = | ∆ | (cid:88) k c † k ↑ c †− k ↓ + h . c ., (24)mentioned at the beginning of this section which gives usthe full first-quantized Hamiltonian ofˆ U † ˆ H ˆ U = τ ( ξ k − σ d k ) − | ∆ | ( − τ ˆ k y + σ τ ˆ k x ) (cid:112) − d z + | ∆ | σ τ ˆ d z (25) x ’ (a) ’ (b) FIG. 1. Contribution to intrinsic Hall conductivity in theband basis, where s, ¯ s, s (cid:48) label bands. Filled and dottedcurves denote the normal and anomalous Green function re-spectively. In both the normal state and the superconductingstate with purely intraband pairing, there is contribution onlyfrom the diagram (a), which originate from the propagationof the electron in the upper band and the hole in the lowerband (or vice versa for the superconducting state). However,the diagram (b) shows that with a nonzero interband pairing,possibility of interband propagation - this particular diagraminvolves the interband normal propagation coming from thecombination of the intraband normal and anomalous propaga-tion with interband pairing - gives rise to additional processescontributing to the Hall conductivity. in the band basis as mentioned in the previous subsection.Note that, in addition to the interband pairings with theopposite chiralities, there is an interband pairing that isnon-chiral . This interband pairing at k = 0 is responsi-ble for shifting the quantum phase transition point from µ = | h | to µ = (cid:112) h − | ∆ | . It is required in order tohave a purely s -wave pairing because with the perpen-dicular Zeeman field, the k and − k states no longer hasthe opposite spins due to partial spin polarization.The interband pairing gives rise to contribution to theHall conductivity that is not present in the normal state, i.e. not representable by (a) of Fig.1, due to the inter-band component of Green function being nonzero. Thiscan be illustrated simply by the case with the infinitesi-mal interband pairing, for which we can setˆ U † ˆ H ˆ U = τ ( ξ k − σ d k ) − | ∆ | ( − τ ˆ k y + σ τ ˆ k x ) , ˆ U † δ ˆ H ˆ U = | δ ∆ | σ τ ˆ d z , (26)where | δ ∆ | (cid:28) | ∆ | . Note that with this model, the Greenfunction has an interband component to the first orderin the interband pairing,ˆ U † δ ˆ G ( k , iω n ) ˆ U = ( iω n − ˆ U † ˆ H ˆ U ) ˆ U † δ ˆ H ˆ U ( iω n − ˆ U † ˆ H ˆ U ) , (27)which has both the normal and anomalous part δ ˆ g = ˆ f δ ˆ H ˆ g + ˆ gδ ˆ H ˆ f ,δ ˆ f = ˆ f δ ˆ H ˆ f + ˆ gδ ˆ H ˆ g, (28)where ˆ g, ˆ f is the normal and anomalous Green functionfor ˆ H . The result is analogous to the multiband chi-ral p -wave model of Sr RuO in having interband pairing turn on a process that contributes to Hall con-ductivity. However, while the results for Sr RuO obtainHall conductivity due to the anomalous interband Greenfunction, we find that only the normal interband Greenfunction contributes to the Hall conductivity, δσ xy = ie ω (cid:90) d kdν (2 π ) { tr[ˆ v x ˆ g ( k , iν + ω )ˆ v y δ ˆ g ( k , iν )]+ tr[ˆ v x δ ˆ g ( k , iν + ω )ˆ v y ˆ g ( k , iν )] } − ( x ↔ y ) , (29)through the process represented by (b) of Fig.1. Giventhat this process involves anomalous - that is, electronto hole or vice versa - propagation at some point, we ex-pect it to maximize at the Fermi surfaces. Since with thenonzero interband pairing the Hall conductivity receivescontribution from process different from that of the nor-mal state, we can expect that this leads to qualitativelylarger change in the Hall conductivity from its normalstate value. Hence we expect the dependence of the Hallconductivity on the superconducting gap for the purely s -wave pairing to be qualitatively different from that forthe purely intraband pairing.For the Hamiltonian of Eq. 25, we find that the Hall conductivity is σ SCxy = e α h (cid:90) d k (2 π )
1( ˜ E k + + ˜ E k − ) (cid:34) ξ k − d k ) ˜ E k + ˜ E k − (cid:18) − ξ k − d k ˜ E k + ˜ E k − (cid:19) − | ∆ | (cid:40) ξ k − d k )(3 ξ k − d k + 4 α k )˜ E k + ˜ E k − + 6( ξ k − d k + 2 α k )˜ E k + ˜ E k − (cid:41) − | ∆ | (cid:32) ξ k − d k + 4 α k ˜ E k + ˜ E k − + 6˜ E k + ˜ E k − (cid:33) − | ∆ | E k + ˜ E k − (cid:35) (30)where ˜ E k ± = ξ k + d k + | ∆ | ± (cid:113) ξ k d k + | ∆ | h (31)(see Appendix A for derivation); note that in the | ∆ | → ξ k ± d k = 0. From the case of infinitesimalinterband pairing examined above, we have seen that the interband propagation can lead to a large change at theFermi surfaces . Comparison between the integrand of Eq.(30) and that of Eq.(7) shown in (b) of Fig.2 clearlyshows this striking change at the Fermi surface. The two plots of Fig.2 strongly indicate that the change in the Hallconductivity is going to be qualitatively larger with the interband pairing than with the purely intraband pairing.Indeed, from differentiating Eq.(30) with respect to | ∆ | , we find that the Hall conductivity is linear in the pairinggap in the | ∆ | → | ∆ | log | ∆ | dependence we find forthe purely intraband pairing case of the previous section. We obtain σ SCxy − σ Nxy ≈ − e α h (cid:20) k F + v F + αk F + ( h + α k F + ) + k F − v F − αk F − ( h + α k F − ) (cid:21) | ∆ | (32)( v F ± ’s are the velocity on the outer/inner Fermi surfaces) for the case where we have two Fermi surfaces. Given thatthe Landau-Ginzburg theory gives us ∆ ∝ ( T c − T ) / , the above result implies δσ SCxy ∼ ( T c − T ) / below T c .Our results imply that for the purely s -wave pairing, the effect of pairing on the Hall conductivity does not k (cid:144) k F (cid:45) k F F xy (cid:144) R K (cid:72) a (cid:76) k (cid:144) k F (cid:45) k F F xy (cid:144) R K (cid:72) b (cid:76) FIG. 2. Comparison between the integrand of the Hall con-ductivity of the normal state in Eq.(7) with that of (a) thepurely intraband pairing superconductivity in Eq.(21) and (b)the purely s -wave superconductivity in Eq.(30). We havedefined k F ≡ √ µm ∗ , R K = 2 π/e (‘the effective Berrycurvature’ F xy normalized to give the normal state valuefor | ∆ | = 0) and set for both (a) and (b) m ∗ α/k F = 0 . h/αk F = 0 . | ∆ | /αk F = 0 . change across the quantum phase transition at h = µ + | ∆ | . To see this, note that in Eq.(32), ( ∂/∂ | ∆ | ) σ SCxy does not vanish on either side of the transition. Theonly difference for the topologically non-trivial supercon-ducting phase is that the k F − contribution of Eq.(32)vanishes, which leaves unchanged the contribution fromthe larger Fermi surface (the k F + -dependent term) inEq.(32). This is fully in accord with our numerical re-sults in (b) of Fig.2 which show the | ∆ | -linear dependenceto originate at the Fermi surfaces. IV. PHYSICS OF INTERBAND PAIRING
Our results indicate that the Hall conductivity pro-vides a clear diagnostic for presence of interband pairing.It is therefore imperative to consider the circumstancesunder which the interband pairing could arise.In an intrinsic superconductivity with infinitesimallyweak interaction, the Cooper pairing occurs between elec-trons of the same energy near the Fermi level, as the pair-ing of states at different energies cannot save as muchenergy. Thus intrinsic pairing between different bands isunlikely to be energetically favorable unless the pairingis of the FFLO type with a nonzero center of mass mo-mentum or the pairing interaction is strong compared tothe gap between bands at the Fermi energy.Since we have not considered FFLO pairing, the re-sults of the previous section are relevant only when thepairing interaction is strong or when the superconduc-tivity is induced through proximity effect and producesa pairing gap comparable to the splitting between the twobands 2 d k . From both the analysis of the infinitesimalinterband pairing in Eqs.(28), (29) and the numerical re-sults for the purely s -wave pairing in Fig.2, we see thatthe effect of interband pairing on the Hall conductivityis significant mainly at the Fermi surfaces.We emphasize that the physically relevant question iswhether there can be interband pairing comparable tointraband pairing. It needs to be pointed out here thatphysically even the proximity effect will not induce a purely s -wave pairing. Tunneling between the Rashbasystem and the s -wave superconductor allows for spin-flip processes due to the spin-orbit coupling and may in-duce inverse-proximity effect resulting in a nonzero spin-triplet pairing correlation on the s -wave superconductoror a suppression of superconductivity in the s -wave su-perconductor.We conclude here that interband pairing is less likelyin the topologically non-trivial superconducting phasewhich requires a single Fermi surface in the presence ofthe Zeeman field. We will show that usually the gapbetween the bands at the Fermi surface are larger thanin the case of a single Fermi surface than in the case ofFermi surfaces in both bands. This in turn makes inter-band paring more energetically unfavorable in the caseof topologically non-trivial superconductivity than in thecase of the trivial superconducting phase. To illustratethis point, it is useful to examine certain specific rangesof the parameters.Consider the special point µ = 0 at which we have atopologically non-trivial superconductivity for any h > | ∆ | >
0. At the limit of h → k F + = 2 m ∗ α , and hence the band splitting at theFermi surface would be 2 αk F + = 4 m ∗ α . In the h (cid:28) m ∗ α limit, the band splitting at the Fermi surface wouldbe much larger then h and hence also the pairing gap | ∆ | .Possibility for the interband pairing only exists in theexperimentally challenging regime of | h | (cid:38) | ∆ | (cid:29) m ∗ α .There are less constraints in the topologically triv-ial superconducting phase, since for a fixed h, | ∆ | , thechemical potential µ needs not be fine-tuned. One pos-sible scenario is in the limit of small spin-orbit cou-pling and Zeeman energy, i.e. µ (cid:29) | h | , mα . In thiscase there is no restriction against the band splitting2 d k ≈ (cid:112) h + 2 µm ∗ α becoming comparable to | ∆ | , andthe interband pairing on the both Fermi surfaces will giveus σ SCxy − σ Nxy ≈ − √ e µ / ( m ∗ α ) / h | ∆ | ( h + 2 µm ∗ α ) . (33)Another scenario is for the case where we are in the trivialsuperconducting phase, yet close to the quantum phasetransition, e.g. , 0 < µ − | h | (cid:28) µ . For | ∆ | comparableto | h | , there can be significant interband pairing at thesmaller Fermi surface giving us σ SCxy − σ Nxy ≈ − e αk F − h | ∆ | . (34) V. CONCLUSION AND DISCUSSION
We have studied the intrinsic Hall conductivity in thenormal and superconducting phases of the Rashba sys-tem under perpendicular magnetic field. In this system,the normal state itself has broken time-reversal symme-try and a non-zero intrinsic Hall conductivity, in contrastto Sr RuO , where the normal system is time-reversalinvariant and Cooper pairing breaks time-reversal sym-metry. We have compared two cases for this system, onewhere the Cooper pairing is exclusively intra-band andthe other where we allowed for the inter-band pairingas well; both cases have the same topologically trivialand non-trivial superconducting phases and the identi-cal class of the quantum phase transition between themtuned by the chemical potential µ . For either case, wefind no qualitative difference between the Hall conductiv-ity in the topologically trivial and non-trivial supercon-ducting phases. On the other hand we find that betweenthe case with zero and non-zero inter-band pairing, thedependence of the Hall conductivity on the pairing gap isqualitatively different, with the non-zero inter-band pair-ing case having a Hall conductivity linear in the pairinggap.Experimentally, our result suggests that while the ob-servation of the linear dependence of intrinsic Hall con-ductivity on the pairing gap is more likely to be associ-ated with the topologically trivial superconductivity, ifnot quite ruling out the topologically non-trivial super-conductivity. This is because the interband pairing inthe topologically non-trivial superconducting phase re-quires either a very strong Zeeman field or a very lowelectron density and therefore would be very difficult torealize, whereas in the case of the topologically trivial su-perconducting phase one merely needs a sufficiently weakspin-orbit coupling.Our Hall conductivity results on the Rashba systemare consistent with those on the chiral p -wave supercon-ductor. While the chiral p -wave superconductor alwaysbreaks time-reversal symmetry, its topology depends onthe number of pockets crossing the Fermi level, being is non-trivial (trivial) for odd (even) number of pock-ets. A non-zero Hall conductivity in an impurity-freechiral p -wave superconductor requires inter-band pairingregardless of whether the superconductivity is topologi-cally trivial or non-trivial. Our results are consistent inboth aspects : on the importance of the inter-band pair-ing on the Hall conductivity and on the absence of anyqualitative dependence on the topology of the supercon-ducting phase.The non-quantization of the Hall conductivity in su-perconductors is another consequence of the sharply dif-ferent electromagnetic response of superconductors andinsulators. The quantization in insulators can be ex-plained by an argument which relies on an adiabatic in-sertion of h/e flux through a ring in the Corbino ge-ometry. Since the inserted flux can be “gauged away”,the system must return to its initial state with a possibletransport of an integer number of electrons from the inneredge to the outer edge . However, in superconductor,flux insertion leads to Meissner screening, and eventu-ally, by the time a h/ e flux is inserted, the phase slip bywhich the superconductor acquires the 2 π phase windingaround the hole occurs in the superconductor to reducethe kinetic energy. Since the 2 π phase winding cannotoccur adiabatically, the h/e flux cannot be adiabaticallyinserted in a superconductor and the quantization argu-ment does not apply.We would like to thank Catherine Kallin, SudipChakravarty and Srinivas Raghu for useful discussionsand suggestions. This work has been supported by theUCLA Startup Funds (RR, SBC) and the Institute forBasic Science in Korea through the Young Scientist grant(SBC). Appendix A: Hall conductivity for the purely s -wave pairing Rashba system The fact that the model is not block-diagonal in the band basis introduces a little complication to the Green’sfunction:ˆ G ( k , iω n ) = [ iω n − ˆ H ( k )] − = ˆ g ( k, iω n ) − α (cid:80) i =1 , σ i [ k x ˆ g ix ( k, iω n ) + k y ˆ g iy ( k, iω n )] − σ h ˆ g ( k, iω n )( ω n + ˜ E k + )( ω n + ˜ E k − ) . (A1)where ˜ E k ± = ξ k + d k + | ∆ | ± (cid:113) ξ k d k + | ∆ | h , ˆ g ( k, iω n ) = − iω n ( ω n + ξ k + d k + | ∆ | ) − τ ξ k ( ω n + ξ k − d k + | ∆ | ) − τ | ∆ | ( ω n + ξ k + α k − h + | ∆ | ) , ˆ g x ( k, iω n ) = − ˆ g y ( k, iω n ) =2 iω n ξ k − τ ( ω n − ξ k + d k + | ∆ | ) + τ ξ k | ∆ | , ˆ g x ( k, iω n ) = ˆ g y ( k, iω n ) = − τ h | ∆ | , ˆ g ( k, iω n ) =( ω n − ξ k + d k − | ∆ | ) − τ iω n ξ k − τ iω n | ∆ | (A2)(note that all ˆ g ’s are independent of spin and ˆk ). However, the above formula does permit writing down in a relativelysimple form the optical Hall conductivity at T = 0 after tracing over the spins: σ xy ( ω ) = e α h (cid:90) d kdν (2 π ) tr[ˆ g ( k, iν )ˆ g ( k, iν + ω ) − ˆ g ( k, iν )ˆ g ( k, iν + ω )] ω ( ν + ˜ E k + )( ν + ˜ E k − )[( ν − iω ) + ˜ E k + ][( ν − iω ) + ˜ E k − ] (A3)(tr here is only over the electron and hole), from which we obtain Eq.(30) in the ω → | ∆ | in the | ∆ | → ∂∂ | ∆ | (cid:12)(cid:12)(cid:12)(cid:12) | ∆ |→ σ SCxy = − e α h (cid:90) d k (2 π ) lim | ∆ |→ ξ k − d k + 2 α k ( ˜ E k + + ˜ E k − ) ˜ E k + ∂∂ | ∆ | | ∆ | ˜ E k − = − e α h lim | ∆ |→ | ∆ | (cid:90) d k (2 π ) ξ k − d k + 2 α k ( | ξ k + d k | + | ξ k − d k | ) ( | ξ k | + d k ) × (cid:20) | ξ k | − d k ) + | ∆ | (1 − h / | ξ k d k | ) − | ∆ | (1 − h / | ξ k d k | ) { ( | ξ k | − d k ) + | ∆ | (1 − h / | ξ k d k | ) } (cid:21) ≈ − e α h π lim | ∆ |→ | ∆ | (cid:90) dk (cid:88) s = ± k F s ( α k F s )( h + α k F s ) / (cid:20) v F s ( k − k F s ) + | ∆ s | − | ∆ s | { v F s ( k − k F s ) + | ∆ s | } (cid:21) = − e α h (cid:20) k F + v F + αk F + ( h + α k F + ) + k F − v F − αk F − ( h + α k F − ) (cid:21) , (A4)where | ˜∆ ± | ≡ E k ± − ( ξ k ∓ d k ) ≈ | ∆ | α k F ± / ( h + α k F ± ). 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