Berry curvature effects on quasiparticle dynamics in superconductors
BBerry curvature effects on quasiparticle dynamics in superconductors
Zhi Wang, ∗ Liang Dong, ∗ Cong Xiao, † and Qian Niu School of Physics, Sun Yat-sen University, Guangzhou 510275, China Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
We construct a theory for the semiclassical dynamics of superconducting quasiparticles by following theirwave-packet motion and reveal rich contents of Berry curvature effects in the phase-space spanned by positionand momentum. These Berry curvatures are traced back to the characteristics of superconductivity, includingthe nontrivial momentum-space geometry of superconducting pairing, the real-space supercurrent, and thecharge dipole of quasiparticles. The Berry-curvature effects strongly influence the spectroscopic and transportproperties of superconductors, such as the local density of states and the thermal Hall conductivity. As a modelillustration, we apply the theory to study the twisted bilayer graphene with a d x + y + id xy superconducting gapfunction, and demonstrate Berry-curvature induced effects. Introduction. – The Chern number of Bogoliubov-de Gennesband structure has commonly been used to characterize thetopology of exotic superconductors [1–5], while much lessattention has been given to the physical effect of the momentumspace Berry curvature which makes up the Chern number[6, 7]. In the presence of inhomogeneity due to external fieldsor a supercurrent, we may also expect to find other componentsof the Berry curvature in the phase space, such as those inthe real space as well as in the cross planes of position andmomentum [8]. Phase space Berry curvatures are known tobe important on the dynamics of Bloch electrons, ubiquitouslyaffecting equilibrium and transport properties of solids [9–16]. It is therefore highly desirable to construct a semiclassicaltheory for quasiparticle dynamics in superconductors, whichsystematically takes into account these Berry curvatures, inorder to provide an intuitive and effective basis for analyzingvarious response properties of superconductors.In this Letter, we introduce the semiclassical quasiparticleas a wave packet in the background of slowly varying gaugepotentials and the superconducting order parameter. Apartfrom Berry curvatures inherited from the parent Bloch states,we identify new contributions due to the non-conserving na-ture of the quasiparticle charge and the phase space structureof the order parameter which is nontrivial in all but the con-ventional s-wave superconductors. The quasiparticle also nat-urally possesses a charge dipole moment, which can coupleto a magnetic field through the Lorentz force and induce fielddependent Berry curvatures.To demonstrate the utility of the semiclassical theory, wediscuss how these Berry curvatures modify the phase-spacedensity of states of the quasiparticles and the impact on elec-tron tunneling spectroscopic measurements. We also presentresults on thermal Hall conductivity due to quasiparticles, andreveal its relationship with topological contribution from thecondensate that has been discussed extensively as a measureof topological superconductors [17, 18]. We assume spin con-servation in this work for simplicity, and illustrate our resultsusing a twisted bilayer graphene model [19] with a d+id su-perconducting gap function for the order parameter.
Quasiparticle wave packet in second quantization formal-ism. —In order for a semiclassical theory of superconduct-ing quasiparticles to be feasible, we assume that all the pos- sible inhomogeneities in the considered system are smoothin the spread of a quasiparticle wave packet, whose cen-ter position is marked as r c . For example, in the mixedstates of type-II superconductors, we focus only on the re-gion far away from the vortex core, where the pairing po-tential can be perceived as slowly varying. A local Hamil-tonian description of the wave packet hence emerges, namely H c = ∫ d r c † σ r ( ˆ h c − µ ) c σ r − ∫ d r d r (cid:48) g ( r c , r − r (cid:48) ) c †↑ r c †↓ r (cid:48) c ↓ r (cid:48) c ↑ r ,where c † σ r is the creation operator for an electron with spin σ ( = ↑ , ↓) at position r , ˆ h c ≡ h ( r , − i ∇ r − e A ( r c , t ) ; { β i ( r c )}) is the spin degenerate single-electron Hamiltonian in the localapproximation (set (cid:126) = µ is the chemical potential, and g isthe effective attractive interaction between electrons. We onlyconsider spin-singlet superconductors with intraband pairingand without spin-orbit coupling. The slowly varying pertur-bation fields { β i } ( i = , , .. ) and the electromagnetic vectorpotential A are represented by their values at r c . ˆ h c possesseslocal eigenfunctions e ie A ( r c , t )· r ψ n σ k ; r c ( r ) , where ψ n σ k ; r c ( r ) are the local Bloch functions of h ( r , − i ∇ r ; { β i ( r c )}) , withlocal Bloch bands ξ n k ; r c . Here n and k are the indices for band(with two-fold spin degeneracy) and wave-vector, respectively,and r c enters in the eigenstates parametrically as a characterof the local description.The interaction term can be treated within a mean-fieldapproach, ending with [20] H c = (cid:213) n σ k E n k ; r c γ † n σ k ; r c γ n σ k ; r c , (1)where the creation/anihilation operators for the local eigen-state are introduced by the Bogoliubov transformation, e.g., γ † n ↑ k ; r c = µ ∗ n k ; r c c † n ↑ k ; r c − ν ∗ n k ; r c c n ↓− k ; r c . Here c † n σ k ; r c = ∫ d r e ie A ( r c , t )· r ψ n σ k ; r c ( r ) c † σ r creates the local Bloch eigen-states of ˆ h c , whereas ( µ n k ; r c , ν n k ; r c ) T and E n k ; r c = (cid:113) ξ n k ; r c + | ∆ n k ; r c | are the Bogoliubov wavefunction in thislocal Bloch representation and the eigenenergy, respectively,and ∆ n k ; r c is the local momentum-space superconducting pair-ing function. The quasiparticle operators not only define theexcitations of the local Hamiltonian, but also determine theground state of the local Hamiltonian with annihilation opera-tors | G (cid:105) = N (cid:206) n σ k γ n σ k ; r c | (cid:105) . Here N is the normalizationfactor and | (cid:105) is the vacuum for electrons. a r X i v : . [ c ond - m a t . s up r- c on ] S e p Now we construct a quasiparticle wave packet centeredaround ( r c , k c ) with the local creation operators acting onthe superconducting ground state: | Ψ n ↑ ( r c , k c , t )(cid:105) = ∫ [ d k ] α ( k , t ) γ † n ↑ k ; r c | G (cid:105) , (2)where ∫ [ d k ] is shorthand for ∫ d m k /( π ) m with m the dimen-sion of the system. The envelope function α ( k , t ) is sharplydistributed in reciprocal space so that it makes sense to speakof the wave-vector k c = ∫ [ d k ]| α ( k , t )| k of the wave packet.We only demonstrate the spin-up wave packet, as the spin-down case can be easily extrapolated. Spin center and charge dipole of the wave packet. —For Bloch electrons the wave-packet center is simply thecharge center. However, superconducting quasiparticles aremomentum-dependent mixture of electrons and holes andthereby do not possess definite charges, rendering the chargecenter ill defined. On the other hand, spin is a conservedquantity in the absence of spin-orbit coupling, hence the spincenter serves physically as the center of a wave packet. Forthis purpose we consider the spin density operator ˆ S ( r ) = c †↑ , r c ↑ , r − c †↓ , r c ↓ , r , and calculate its wave-packet averaging S ( r ) = (cid:104) Ψ | ˆ S ( r )| Ψ (cid:105) − (cid:104) G | ˆ S ( r )| G (cid:105) . This gives the distribu-tion of spin on the wave packet, and its center, the spin center,is given by [20] r c ≡ ∫ d r S ( r ) r = ∂γ c ∂ k c + (cid:104) φ | i ∇ k c φ (cid:105) − ρ c ∇ k c θ c , (3)where θ c = arg ∆ n k c ; r c is related to the phase of the super-conducting order parameter, ρ c = ξ n k c ; r c / E n k c ; r c measuresthe non-conserved charge of the quasiparticles, | φ (cid:105) is the peri-odic part of the Bloch state | ψ n σ k c ; r c (cid:105) , and γ c = − arg α ( k c , t ) is the phase of the envelope function. The Berry connectionscontain not only the Bloch part A b k c = (cid:104) φ | i ∇ k c φ (cid:105) from thesingle-electron band structure, but also the superconductingpart A sc k c = − ρ c ∇ k c θ c from the momentum dependence ofthe superconducting order parameter.The spin center is not sufficient to describe the couplingof quasiparticles with electromagnetic fields, which would in-evitably involve information on the charge distribution uponthe spread of a wave packet. Since the charge distribution is notcentered at r c , there should be a charge dipole moment asso-ciated with a wave packet. Indeed one can consider the chargedensity operator ˆ Q ( r ) = e ( c †↑ r c ↑ r + c †↓ r c ↓ r ) , and its wave-packetaveraging Q ( r ) = (cid:104) Ψ | ˆ Q ( r )| Ψ (cid:105) − (cid:104) G | ˆ Q ( r )| G (cid:105) provides a properdefinition for the charge dipole moment [20] d ≡ ∫ d r Q ( r )( r − r c ) = e ( ρ c − ) ∂θ c ∂ k c . (4)It is nonzero only in the case of a momentum dependentphase of superconducting order parameter. Furthermore, ifthe external-field-free system has either time-reversal (space-inversion) symmetry, d is an even (odd) function in momentumspace, as can be inspected from the semiclassical equations ofmotion proposed later. Berry curvatures and semiclassical dynamics. –The distinc-tive properties of the wave packet are anticipated to stronglyaffect its semiclassical dynamics determined by the Lagrangian L = (cid:104) Ψ | i ddt − ˆ H c | Ψ (cid:105) − (cid:104) G | i ddt − ˆ H c | G (cid:105) [9], and should be em-bodied in various Berry curvatures characterizing the dynam-ical structure. Adopting the circular gauge A ( r c ) = B × r c ,which is suitable for the approximately uniform magnetic fieldin regions far away from vortex lines, after some algebra weget [20] (hereafter the wave packet center label c is omitted forsimplicity): L = − E + k · (cid:219) r + (A b r − ρ v s + B × ˜ d ) · (cid:219) r + (A b k − ρ ∇ k θ ) · (cid:219) k . (5)Here the coupling of the wave packet to the magnetic fieldinvolves the charge dipole and gives B × ˜ d , with ˜ d = d / v s = ∇ r θ − e A is half of the gauge invariant super-current velocity, and A b r = (cid:104) φ | i ∇ r φ (cid:105) is the real-space Berryconnection of the single-electron wave function.The structure of the Lagrangian implies that the total Berryconnections in the momentum and real space take the formsof A k = A b k − ρ ∇ k θ and A r = A b r − ρ v s + B × ˜ d , re-spectively. Various Berry curvatures are then formed as Ω λ α λ β = ∂ λ α A λ β − ∂ λ β A λ α , where λ = r , k , and α and β are Cartesian indices. In particular, Ω k α k β and Ω r α r β areanti-symmetric tensors with respect to ( α , β ) , whose vectorforms read respectively Ω k = i (cid:104)∇ k φ | × |∇ k φ (cid:105) − ∇ k ρ × ∇ k θ (6)and Ω r = i (cid:104)∇ r φ | × |∇ r φ (cid:105) + e ρ B − ∇ r ρ × v s + ∇ r × ( B × ˜ d ) . (7)One can readily verify that the above Ω k coincides with thatobtained from the Bogoliubov-de Gennes equation [7]. Thefirst terms in these two equations are the familiar Berry curva-tures from the single-electron band structure [9], while otherterms involves superconductivity. Moreover, the characteris-tics of superconductors, i.e., the charge non-conservation andthe resultant charge dipole of wave packet and the real-spacesupercurrent, are embedded in the last three terms of Ω r .Regarding the phase-space Berry curvature Ω kr , there areremarkable qualitative differences from that for Bloch elec-trons, namely Ω kr = Ω kr (cid:44) Ω kr will play a vital role in anumber of experimental measurables [9]. For example, in thepresence of pure magnetic perturbations, its trace readsTr [ Ω kr ] = −∇ k ρ · v s − e ρ B · (∇ k ρ × ∇ k θ ) . (8)As will be shown later, this trace of the Berry-curvature tensorplays an important role in the geometric modulations to thequasiparticle local density of states [9].With the above Berry curvatures, the Euler-Lagrange equa-tions of motion for superconducting quasiparticles possess thesame noncanonical structure as for Bloch electrons [8, 9]. Hav- ing realized this, we neglect the Berry curvatures from Blochband structures for simplicity and focus on those originatedfrom superconductivity. Thus the equations of motion read: (cid:219) r = ∇ k E + (cid:219) k × (∇ k ρ × ∇ k θ ) + ∇ k ( ρ v s − B × ˜ d ) · (cid:219) r − (cid:219) r · ∇ r ( ρ ∇ k θ ) , (9) (cid:219) k = −∇ r E + (cid:219) r × ( e ρ B − ∇ r ρ × v s + ∇ r × ( B × ˜ d )) − ∇ r ( ρ ∇ k θ ) · (cid:219) k + (cid:219) k · ∇ k ( ρ v s − B × ˜ d ) . In the absence of superconductivity, ρ = ˜ d =
0, and θ =
0, hence the equations of motion reduce to the usual onesfor electrons [8]. It is also worthwhile to mention that, fortrivial superconducting pairing, the momentum-space Berryconnection vanishes but the real-space one may still survivedue to the supercurrent velocity: A r = − ρ v s . The resultingBerry curvature in real space is given by Ω r = e ρ B + ∇ r ρ × v s .The equations of motion describe the quasiparticle dynamicssubjected to background super-flow, and take a similar form tothose for bosonic Bogoliubov quasiparticles in a Bose-Einsteincondensate with a vortex [21].Equation (9) is the central result of this work. It provides aframework to understand quasiparticle dynamics in supercon-ductors subjected to various perturbations. In the following,we apply this semiclassical theory to calculate several proper-ties of superconductors. Density of states. –A most direct consequence of the Berrycurvatures appearing in the equations of motion is the break-down of the phase-space volume conservation. As a result, thephase-space measure D( r , k ) is modified by Berry curvatures[11], which to the first order of the spatial inhomogeneity canbe expressed as D( r , k ) = + Tr Ω k r − Ω r · Ω k . (10)The modification may originate from various perturbations,such as the supercurrent and magnetic field. We note that ∂ D/ ∂ B = Ω k r and Ω r · Ω k cancel each other, in sharp contrast to the case of Bloch elec-trons [11]. D would influence the quasiparticle local density of states n ( r , ω ) , which is just the integration of the phase-space volumewith the fixed quasiparticle energy E , n ( r , ω ) = ∫ [ d k ]D( r , k )(| µ | δ ( ω − E r , k ) + | ν | δ ( ω + E r , k )) . (11)This quasiparticle density of states is proportional to the differ-ential conductance which can be directly measured by scanningtunneling microscopy [22]. For instance, in the case of a smallsupercurrent, we have D = − ∇ k ρ · v s according to Eq. (8),which gives the modulation part as δ n ( r , ω ) = − ∫ [ d k ] v s ·∇ k ρ (| µ | δ ( ω − E r , k ) + | ν | δ ( ω + E r , k )) , (12)where δ n = n − n with n = ∫ [ d k ]((| µ | δ ( ω − E r , k ) + | ν | δ ( ω + E r , k )) being the local density of states given by the conven- tional formula. This modification to the density of states de-pends on the direction of the supercurrent, hence could beexperimentally verified by injecting supercurrent on differentdirections. Thermal Hall transport. – The semiclassical theory can alsobe employed to study the transport properties in superconduc-tors such as the intrinsic thermal Hall effect. Compared to theGreen’s function method [23, 24], the semiclassical theory hasan advantage of subtracting conveniently the circulating mag-netization current [25] without a detailed calculation of theenergy magnetization [26]. Here we sketch the key steps fromthe semiclassical equations towards the thermal Hall trans-port. We start from the semiclassical expression for the localenergy current density j Q = ∫ [ d k ]D( k ) f ( E k , T ) E k (cid:219) r [27]where f ( E k , T ) is the Fermi-Dirac distribution at temperature T . Then we substitute the equation of motion for (cid:219) r [28], andfind j Q = −∇ T × ∂∂ T ∫ [ d k ] h Ω k + ∇ × ∫ [ d k ] h Ω k where we in-troduce the auxiliary function h ( E k , T ) = − ∫ ∞ E k d η f ( η, T ) η .Now the second term is a circulating current which should bediscounted, leaving the transport current j Q,tr = ∫ [ d k ] ∂ h ∂ T Ω k ×∇ T . The Hall response of this current is given by κ Q xy = T ∫ [ d k ]( Ω k ) z ∫ ∞ E k d ηη f (cid:48) ( η, T ) , (13)where the factor 2 denotes the spin degeneracy.The above formula only accounts for the contribution fromquasiparticles beyond the superconducting condensate. It isphysically reasonable to make the connection κ + κ Q xy = κ BdGxy between this “quasiparticle plus condensate” description andthe Bogoliubov-de Gennes (BdG) one [24], Here κ is thethermal Hall conductivity contributed by the condensate and κ BdGxy = T ∫ [ d k ]( Ω k ) z (cid:16)∫ ∞ E k − ∫ ∞− E k (cid:17) d ηη f (cid:48) ( η, T ) is the con-ductivity obtained using the particle-hole symmetric BdGbands. In κ BdGxy the spin degeneracy and the particle-hole re-dundancy cancel out, and − E k means the BdG ”valence band”whose Berry curvature is −( Ω k ) z . Therefore, the condensatecontribution reads κ = − T ∫ [ d k ]( Ω k ) z ∫ ∞−∞ d ηη f (cid:48) ( η, T ) = π C k B T (cid:126) , (14)where the summation over momentum gives exactly the Chernnumber C . This recovers the quantized thermal Hall conduc-tance given by edge-state analysis [17, 18]. Having clarifiedthe above relationship, in the following we use the simplifiednotation κ + κ Q xy → κ xy to represent the total thermal Hallconductivity. Model illustration: twisted-bilayer graphene with d + idsuperconductivity. – To illustrate the application of the semi-classical theory, we consider the twisted-bilayer graphene sys-tem which has been proposed to support a topological chirald-wave superconducting state [29–32]. We take the effectivefour-band tight-binding Hamiltonian to describe the system[19], ˆ H = − µ (cid:213) i ˜ c † i ˜ c i + t (cid:213) (cid:104) i , j (cid:105) ˜ c † i ˜ c j + t (cid:213) [ i , j ] ˜ c † i ˜ c j + t (cid:213) [ i , j ] ˜ c † i [ i σ y ⊗ σ ] ˜ c j + h . c . (15)where ˜ c † i ≡ ( c † i , x , ↑ , c † i , y , ↑ , c † i , x , ↓ , c † i , y , ↓ ) is the electron creationoperator with two distinct orbitals α = ( p x , p y ) , σ y is thePauli matrix, σ is the identity matrix, t i ( i = , ,
3) arehopping parameters, and (cid:104) i , j (cid:105) and [ i , j ] represent the sum-mations over the three nearest neighbor lattice vectors andover the second-nearest neighbors within the same sublat-tice, respectively. We diagonalize this Hamiltonian and takethe band with the dispersion function ξ ( k ) = −| t h ( k )| + t h ( k ) + t h ( k ) − µ, where h ( k ) = + e i k x cos ( √ k y ) is from the nearest-neighbor hopping, h ( k ) = cos ( k x ) + cos (− k x + √ k y ) + cos (− k x − √ k y ) and h ( k ) = sin ( k x ) + sin (− k x + √ k y ) + sin (− k x − √ k y ) are from the nextnearest-neighbor hopping. Superconductivity in twisted bi-layer graphene with d x − y + id xy pairing symmetry canbe described by the superconducting gap function in theform of [2, 33, 34] ∆ ( k ) = (cid:205) i = ∆ i cos ( k · R i − ϕ k ) , where ( ∆ , ∆ , ∆ ) ≡ (cid:113) ( ∆ , − ∆ + i √ ∆ (cid:48) , − ∆ − i √ ∆ (cid:48) ) with ∆ and ∆ (cid:48) being the superconducting gap amplitudes for d x − y and d xy pairing, respectively, ϕ k = arg [ h ( k )] is the phase ofthe nearest-neighbor hopping, and R i are the three nearest-neighbor lattice vectors.Now we can calculate the momentum-space Berry curvatureby Eq. (6) for this tight-binding model. In Fig. 1a, we demon-strate the Berry curvature with typical band parameters givenin Ref. [19] and symmetric d x − y and d xy gaps. The bandstructure of the tight-binding model has trivial topology, andthe Berry curvatures are entirely contributed by the supercon-ducting gap function. Because of the particle-hole symmetryin superconductors, the Berry curvatures concentrate aroundthe Fermi surface. This is clearly shown in Fig. 1a., where theBerry curvature has symmetric peaks reflecting the D sym-metry of the lattice structure and the gap function. In Fig. 1bwe show the result with asymmetric d and id pairing gaps. Forthis case the superconducting gap breaks the rotational sym-metry, leaving only the reflectional symmetry with respect tothe k x -axis for the Berry curvature distribution.As a simple example, we study the density of states mod-ulation due to a supercurrent which could originate from theinjected current or a superconducting vortex. As shown inFig. 1c, the obtained modulation δ n is quite considerable in (b)(c) (d) 𝛿𝛿𝛿𝛿 ( 𝜔𝜔 ) ) 𝛿𝛿 ( 𝜔𝜔 ) ) (a) FIG. 1. Berry curvatures of the tight-binding model for twisted-bilayer graphene with (a) symmetric d x − y and id xy superconduct-ing gaps ∆ = ∆ (cid:48) and (b) asymmetric superconducting gaps ∆ = ∆ (cid:48) .Model parameters are taken as t / t = . t / t = . µ = − . t and ∆ / t = .
1. (c) Berry curvature modification to the quasiparticledensity of states δ n ( ω ) (solid line) with a constant supercurrent of ∇ r θ = √ π ˆ x . The conventional density of states n ( ω ) is demon-strated for comparison (dashed line). (d) The thermal Hall conductiv-ity as a function of temperature. Parameters for (c) and (d) are takenthe same as those for (a). comparison with the non-perturbed density of states n . Wealso note that this modulation depends on both the amplitudesand the direction of the supercurrent, and can have much richerpattern if other perturbations are introduced.We also calculate the intrinsic thermal Hall transport ofthe toy model, demonstrating the temperature dependence ofthe thermal Hall conductivity, with κ as its zero temperaturevalue. As shown in Fig. 1d, the ratio of the thermal Hallconductivity to κ has a near exponential dependence on thetemperature at the low temperature regime, and becomes anapproximated linear function at higher temperatures. Thesefeatures would be helpful for identifying the d + id paring intwisted-bilayer graphene systems.Finally, we note that the results shown in Fig. 1 are obtainedwith a single band, while for the tight-binding model thereare two bands intersecting with the chemical potential. Thethermal conductance from the two bands are exactly the same,while the modulations to the local density of states have asign reversal and a resultant cancellation. In order to observethe modulation to the local density of states in twisted-bilayergraphene system, band or momentum resolved tunneling ex-periments are required.In summary, we derived the semiclassical equations of mo-tion for superconducting quasiparticle wave packets, and iden-tified various Berry curvature contributions in momentumspace, real space as well as phase space. We demonstratedthe power of the theory with examples such as the density ofstates modulation and the thermal Hall transport, and appliedthe theory to study the twisted-bilayer graphene system. Ourtheory opens up a new route to study rich Berry-phase effectson equilibrium and transport properties of superconductingquasiparticles. Acknowledgments.—
We thank Zhongbo Yan, Tianxing Ma,Huaiming Guo, and Jihang Zhu for very valuable discussions.This work was supported by NKRDPC-2017YFA0206203,2017YFA0303302, 2018YFA0305603, NSFC (Grant No.11774435), and Guangdong Basic and Applied Basic ResearchFoundation (Grant No. 2019A1515011620). The work at TheUniversity of Texas at Austin was supported by NSF (EFMA-1641101) and Robert A. Welch Foundation (F-1255). ∗ These authors contributed equally to this work. † Corresponding author: [email protected][1] Xiao-Liang Qi and Shou-Cheng Zhang, “Topological insulatorsand superconductors,” Rev. Mod. 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