Influence of thermal phase fluctuations on the single particle Green function in a 2D d-wave superconductor
aa r X i v : . [ c ond - m a t . s up r- c on ] F e b Influence of thermal phase fluctuations on the spectral function in a 2D d-wavesuperconductor
M. Khodas and A.M. Tsvelik
Department of Condensed Matter Physics and Materials Science,Brookhaven National Laboratory, Upton, NY 11973-5000, USA
We study the spectral function of a two-dimensional superconductor in a regime of strong phasefluctuations. Although the developed approach is valid for any symmetry, we concentrate on d -wavesuperconductors. We obtain analytical expressions for the single electron Green function below thetransition temperature and have worked out a way to extrapolate it for a finite temperatures above T c . The suggested approach are easely generalizable for other models with critical fluctuations. PACS numbers: 71.10.Pm, 72.80.Sk
I. INTRODUCTION
According to a popular viewpoint the cuprate super-conductors in their underdoped regime are fundamen-tally different from the BCS ones due to the existence ofwide temperature range around T c dominated by phasefluctuations. In this range the order parameter ampli-tude is well defined but the global order has not yetemerged. The quantitative measure of this difference isthe ratio Q = 2 T c /πρ s (0) where ρ s (0) is zero tempera-ture phase stiffness. This ratio determines how close thetransition is to being mean-field-like. In BCS supercon-ductors Q ≪
1, in the underdoped cuprates Q ∼ T c [2] and from tempera-ture dependence of the thermal expansion coefficient [3].The analysis of temperature dependence of magnetiza-tion and London penetration depth for the high qualityunderdoped BiSCO crystals show that the superconduct-ing transition itself becomes closer to two-dimensionalBerezinskii-Kosterlitz-Thouless (BKT) one [4]. At last,there is a material (x=1/8 LSCO) where the couplingbetween the CuO planes is so weak that the transition isreal BKT [5].The problem of influence of the phase fluctuations onthe spectral function is a long standing one; its impor-tance being substanciated by the fact that the spec-tral function measured by Angle Resolved Photoemis-sion (ARPES) serves as one of the main probes of thestrong correlations in the cuprates. The experimentsshow that in the underdoped regime many features ofthe spectral function below T c survive above T c thoughin somewhat modified form. According to ARPES, theexcitation spectrum above T c gradually softens and losesits characteristic node-centered conical shape so that thenodal points gradually broaden into arcs [6],[7],[9]. Thequestion is whether the appearance of these arcs (or evenof the Fermi pockets) is due to superconducting fluctua-tions, as suggested, for instance, in [8]. To have a consis-tent understanding of the underdoped regime it is vital toobtain a correct detailed picture of the quasiparticle spec-tral function and establish its temperature dependence. The first step in this direction is to take into account su-perconducting fluctuations. This is possible to do evenwithout full microscopic theory. The problem has beenstudied by a number of authors ([13],[14],[15]), but, aswe argue in this paper, the approach taken is unreliablebeing based on uncontrolled approximations. II. FORMULATION OF THE MODEL
We consider a two-dimensional metal with a strongsuperconducting pairing in the state where the order pa-rameter amplitude is fixed, but the phase fluctuationsare strong. Our calculations allow for the SC (super-conducting) gap to have nodes on the Fermi surface. Forinstance, for d -wave SC the mean field quasiparticle spec-trum is given by E = ǫ ( k ) + ∆ [cos( ak x ) − cos( ak y )] (1)We will assume that v/a ≫ ∆ ( v is the Fermi velocity)and approximate the spectrum close to the node as E ≈ v k + 2∆ sin ( qa/
2) (2)where k is the wave vector component perpendicular tothe Fermi surface and q is the one parallel to it. We takethe Fermi surface at the node for a flat one.In the mean field approximation fluctuations of the or-der parameter ∆ are ignored, and the resulting Greenfunction takes a familiar BSC form, G k ( ω ) = ( ω + ǫ k ) / ( ω − ǫ k − ∆ ( k )). Following our original assump-tion we will neglect the amplitude fluctuations of theorder parameter taking into account only phase fluc-tuations, ∆( x , t ) = ∆ei φ ( x ,t ) . At finite temperature T the long wavelength fluctuations are essentially classical(time independent). It is crucial for our considerationthat the superconducting fluctuations are space isotropicand the one-particle Green function close to the nodeis strongly anisotropic with the parameter of anisotropy∆ /ǫ F , where ∆ is maximal gap and ǫ F is the Fermienergy. Then when one considers a propagating quasi-particle, the fluctuations affect mostly its wave vectorcomponent perpendicular to the Fermi surface. The par-allel component can be considered as conserved. Theabove considerations allow us to consider one dimensionalfermions at a given Matsubara frequency ω n described bythe Lagrangian L = ¯ χ ω (cid:2) − i ω n ˆ1 − i v∂ x σ z +∆ σ + +∆ ∗ σ − (cid:3) χ ω + F φ . (3)In the last equation the first term is a standard pairingLagrangian written in terms of Nambu spinors, χ ω =( ψ ↑ ,ω n , ¯ ψ ↓ , − ω n ) T , and σ i are Pauli matrices. The secondterm gives the action for the classical phase fluctuationsin the form F φ T = ρ s T Z dxdy (cid:2) ( ∂ x φ ) + ( ∂ y φ ) (cid:3) . (4)Here the inverse temperature prefactor T − results fromthe integration over imaginary time. The discrete sym-metry of the lattice which includes C , the group ofin-plane rotations by π/ ψ ↑ , ψ †↓ ] = [Ψ ↑ , − iΨ †↓ ], and[ ψ †↑ , ψ ↓ ] = [iΨ ↑ , Ψ †↓ ]. Then, the Lagrangian Eq. (3), (4)after the analytic continuation, i ω n → ω +i0 is equivalentto the Hamiltonian H eff = v − i( ω + i0) τ + v − ∆( q )[ τ + ei φ (0) + τ − e − i φ (0) ]+ H bulk [ φ ] , (5)where τ a is the short hand notation for the fermionicbilinears: τ a ≡ Ψ + σ a Ψ. In this setting coordinate x playsthe role of Matsubara time. It is dual to the momen-tum component k k parallel to the Fermi velocity at thepoint of observation. Since at ∆ /ǫ F → ↑ , Ψ †↓ depends only on one coordi-nate x , though the phase field φ depends on both. Forconvenience we assign Ψ to coordinate y = 0. Since thepropagators of φ -fields are not supposed to be modifiedby insersions of fermionic loops, which would lead to over-counting, the fermionic number must be treated as con-served ψ + σ ψ σ = 1. Then the τ -operators become compo-nents of spin S=1/2. The Hamiltonian H bulk describesthe phase fluctuations. Their two point correlation func-tion is h ei φ ( x ) e − i φ ( x ) i = (cid:12)(cid:12)(cid:12)(cid:12) aξ ( T ) (cid:12)(cid:12)(cid:12)(cid:12) d F (cid:18) x ξ ( T ) (cid:19) , (6)where d = T / T BKT is the order parameter scaling di-mension, ξ ( T ) is the correlation length and T BKT = πρ s / F ( y ) is such that F ( y ≪
1) = y − d and F ( y > ∼ K ( y ) (more detailed infor-mation about this function can be extracted from [16]).Hence below the transition where ξ = ∞ the function (6)decays as a power law and above the transition it has the exponential asymptotics. The length scale a ∼ ( v/ǫ F ) isthe short distance cut-off. Below the BKT transition thebulk Hamiltonian is Gaussian: H bulk [ φ ; T < T
BKT ] = 18 πd Z ∞−∞ d y (cid:2) (4 πd ) Π + ( ∂ y φ ) (cid:3) , (7)where Π is the momentum density conjugated to the field φ , with an equal time commutator[Π( y ) , φ ( y )] − = − i δ ( y − y ) . (8)By construction integration over the momentum in thequantum action of our one-dimensional model, Eqs. (7),(8) produces the free energy of classical two-dimensionalthermal fluctuations, Eq. (4). Above the transition theHamiltonian Eq. (7) must include effects of vortices whichgenerate nonlinear terms. The rigorous discussion is pos-sible in the case T < T
BKT ; later we will present someextrapolation for temperatures above T BKT . We note inpassing that at
T < T
BKT model (5,7) is equivalent tothe anisotropic Kondo model in the imaginary magneticfield h = i ω −
0. We notice this equivalency though wehave not been able to make use of it in our calculations.The setting of the problem as given by Eqs. (3), (4) isnot different from the one in [13],[14],[15]. However, inthese previous attempts to solve the problem the authorsused the gauge transformation method which we claim isinadequate. The bosonic exponent present in Eqs. (3),(5) has been absorbed into the definition of the fermionicfield. As a result the term ψ + σ ψ σ ∂ x φ (0) appeared in theHamiltonian. The problem with this approach is that thepath integral expression for the electron Green’s functionis dominated by the field configurations with large φ gra-dients. The same effect appears when one attempts todevelop a perturbation theory in ∂ x φ : each diagram di-verges at small distances. This fact has been overlooked.Here the equivalence of the current problem to the Kondomodel is helpful since as is well known the latter cannotbe treated by the methods employed in [14],[15]. III. SOLUTION BY PERTURBATION THEORYIN ∆ In the present section we calculate the Green func-tion by the perturbative expansion in ∆ for the modelEqs. (3), (4). It has a crucial advantage of being free ofultraviolet divergencies. The infrared divergencies are re-moved at finite external frequency and momentum (as wewill show below, the integrals diverge only at ω + vk = 0).The infrared behavior is controlled by the long wave-length fluctuations of the order parameter which justifiesthe use of the effective low energy action for the phasefluctuations Eq. (4). The above arguments indicate thatwe can study the model Eqs. (3), (4) in perturbation the-ory expanding the Green function in powers of ∆ by ac-counting for most infrared singular contributions at eachorder.Below for definiteness we consider positive energies, ω >
0. In other words we study the “particle” partof the spectrum. For negative frequencies the spectralfunction can be obtained by particle-hole transformation, ω → − ω , and k → − k . Most of the spectral weight is ex-pected to be found close to the particle mass shell, k = ω .In general, the Green function G rω ( k ) is peaked for wavevectors close to the particle (hole) mass shell, k ∼ ± ω .Here the superscript r ( l ) designates right (left) movingparticles. In the former case the most singular contribu-tions can be resummed in a usual fashion, by introducingthe self energy Σ ω ( k ), and writing the Green function inthe standard form, G ω ( k ) = [ ω − k − Σ ω ( k )] − . The selfenergy can be written as (we set v = 1 and drop theexplicit q -dependence of ∆)Σ k ( ω ) = ∆ ( ω + k ) − d (9) × " C ( d ) + ∞ X n =1 C n (cid:18) d, k + ω ω (cid:19) (cid:18) ∆ ( ω + k ) − d (cid:19) n The coefficients C n ( d, ρ ) have singularities only at ρ = 0which means that the self energy is a smooth function ofits arguments in the broad vicinity of the particle massshell, k ∼ ω making the resummation scheme justified.The direct calculation yields: C ( d, x ) = e − i πd Γ(1 − d ) , C ( d, / ≈ d e − i πd . (10)These considerations and the fact that away from x =0( ω = − k ) C n ( d, ρ ) vanish at d → G ω ( k ) = (cid:20) ω − k − ∆ ( q ) a d ( ω + k ) − d (1 − i πd ) (cid:21) − . (11)It follows from equation (11) that the quasi-particle dis-persion relation, E ( k ) as determined by the relation E ( k ) = k + ReΣ( E ( k ) , k ) differs from the mean fieldBSC result. In particular, the spectral gap identical tothe Kondo scale in the magnetic impurity problem. Thelatter one is given by T K = ∆( q ) [∆( q ) a ] d/ (1 − d ) (12)The gap magnitude for a given q is suppressed with re-spect to its mean field value ∆( q ). In addition, the spec-tral line acquires a finite width, proportional to the spec-trum renormalization Γ( k ) ≈ πd ( E ( k ) − k ) at d ≪ k ∼ − ω . The self energy is not a useful quantity inthis case as it diverges at k = − ω and becomes moresingular with increasing order of the perturbation the-ory. We therefore sum the most singular contributionsto the Green function. For the “amputated” Green func-tion, ¯ G rω ( k ) = [ G ω ( k ) − G ,ω ( k )][ G r ,ω ( k )] − , we obtainthe expansion in even powers of ∆ similar to (9). Thecoefficients of this expansion are denoted as ˜ C n . The co-efficients with n ≥ k = − ω . To find the momentum dependence of the Green function at k ∼ − ω we have calculated the whole set of constants˜ C n ( d,
0) and summed over the corresponding series. Asa result of interaction with the field created by the fluctu-ating order parameter the right moving particle is scat-tered off as a left moving hole, see Fig. 1. When the (a) l x x R lr y y x lr r rr r rl l x (b) FIG. 1: (Color online) Propagation of the right-moving par-ticle. (a) k ∼ + ω and (b) k ∼ − ω . Solid straight arrowed(black) lines labeled with letters r ( l ) designates propagatorsof the right-moving particle (left-moving hole). Pairs of out-going (red) and incoming (blue) dashed wavy lines designatesthe insertion of ∆ei φ and ∆e − i φ phase factors with close ar-guments. In the panel (b) a pair of distant solid wavy linesdepicts the two uncompensated phase factors. order parameter is fixed these processes give rise to ausual BCS-like quasi-particle spectrum. In our case theorder parameter has different phases at different collisionevents. Our goal is to study the effect of these fluctua-tions to the leading order in the parameter d . For ω ∼ k the right-moving particle propagates along much longerdistance in between the consecutive scatterings than theleft moving hole. This is shown in Fig. 1(a). As a re-sult the space arguments in the phase factors attachedto the propagator of the hole merge. For that reasonthe integration over the short distances of propagation ofthe hole leads to effective fusion of its propagator. Thisexplains why the Green function is determined by theself energy to the second order in ∆, i.e. the result ofEq. (11). Indeed, corrections to Eq. (11) results fromthe interaction between distant dipoles and are small inparameter d ∼ T .The situation at ω ∼ − k is different, see Fig. 1(b).In this case the left-moving hole propagates much longerdistances than the right-moving particle. As a result thearguments of the two outermost phase factors are sepa-rated by large distance which substantially modifies theinfrared behavior of the propagator. The two uncom-pensated phase factors lead to a power law dependencein Eq. (17) absent in Eq. (11). We now derive the resultof Eq. (17) analytically. To illustrate the calculation weconsider the order ∆ , (see Fig. 1(b)): δ ¯ G rω ( R ) = ( − i) ∆ G l , − ω ( y − R ) G r ,ω ( y − x ) × G l , − ω ( y − x ) G r ,ω ( y − x ) G l , − ω ( − x ) ×h e − i φ ( R ) ei φ ( y ) e − i φ ( x ) ei φ ( y ) ei φ ( x ) e − i φ (0) i . (13)Here the bare retarded Green functions ± i G r,l , ± ω ( x ) = θ ( x )ei ωx . The infrared singularity in the Fourier trans-formation of Eq. (13) at k ∼ − ω accumulates at largedistances, R ∝ −| ω + k | − . In terms of rescaled vari-ables, x , = − Rξ , , y , − x , = − Rη , it means thatthe leading contribution comes from η , ≪
1. Integrat-ing over negative R ’s we obtain δ (6) ¯ G rω ( k ) ≈ ∆ Γ(5 − d )(2 ω ) − d Z dx Z x dx × Z ∞ dη dη η − d η − d ( ρ + η + η ) − d . (14)As the integrals over η variables are convergent allowingus to replace the upper integration limit by infinity. Theremaining integration in Eq. (14) yields δ (6) ¯ G rω ( k ) ≈ ∆ Γ (1 − d )Γ(3 − d )2!(2 ω ) − d ρ − d . (15)The arguments leading to Eq. (15) can be generalized toobtain the most singular contribution to arbitrary order∆ n as follows δ (2 n ) ¯ G rω ( k ) ≈ ∆ n Γ n − (1 − d )Γ( n − d )( n − ω ) n (1 − d ) − ρ n − d . (16)Summing all leading singularities Eq. (16) we obtain G ω ( k ) ≃ ω ( q )Γ(1 − d )e − i πd ω (cid:18) ω + k − ∆ ( q )Γ(1 − d ) e − π i d (2 ω ) − d (cid:19) − d . (17)At d → ξ − ( T ) ex-ceeds the Kondo scale T K given in Eq. (12). Since theBKT correlation length is exponentially large ξ − ( T ) ∼ ∆ exp (cid:2) − C ( T /T
BKT − − / (cid:3) , there is a range of tem-peratures and q where this condition is fulfilled. In theopposite limit ξ − ≫ ∆( q ) one can use Eq. (6)) to cal-culate the leading order contribution to the self energy,The following formula provides an interpolation between T > T
BKT and
T < T
BKT region:Σ (2) ( q, k, ω ) = ∆ ( q ) ξ ( ξ/a ) − d (1 + ( ξ ( ω + k )) ) − / (18) × (cid:2) exp (cid:8) − i πd + 2 d sinh − [ ξ ( ω + k )] (cid:9) − (1 + ( ξ ( ω + k )) ) − / (cid:3) . ¯ q ¯ ω A (a) ¯ q ¯ ω A (b) FIG. 2: (Color online) The spectral function at k = 0 asa function of dimensionless parameters ¯ q = ∆( qaξ/v ) and¯ ω = ( ωξ/v ) as found form Eq. (18) for a) d= 0.125 and b) d=0.5. ¯ q ¯ ω m a x FIG. 3: (Color on line) The dependence of the frequency ¯ ω max maximizing the spectral function (18) on the dimensionlessmomentum ¯ q for d = 0 . d = 0 .
25 dashedline (blue), and d = 0 .
125 dashed dotted line (red).
In Fig. 2 we present graphically the spectral function A ω ( k ) = − (1 /π )Im G ω ( k ). The quasiparticle dispersioncan be identified as the energy ω max where the spectralfunction is at its maximum. The dispersion relation ob-tained in this way is depicted in Fig. 3 for different val-ues of parameter d . Below T < T
BKT where ξ − = 0one should use Eq. (17) instead of Eq. (18) close to thesingularity line ω = − k . IV. CONCLUSIONS
In summary, we have studied the electron Green func-tion (and the related spectral function) in the regimeof strong superconducting fluctuations. As is evidentfrom Figs. 2 and 3, these fluctuations affect the nor-mal state dispersion in such a way that the maximumof the spectral function is shifted down in energy in com-parison with its mean field value. Naturally, the effectis more pronounced for larger temperatures. This in-deed may create the impression that the system developsa “Fermi arc”. We hope that the advances in the ex-perimental techniques will allow for a detailed compar-ison with the present theory. We argue that the quasi-classical approximation employed in the previous publi-cations [13],[14],[15] cannot provide a quantitative infor-mation about the spectral density along the arcs. Thisapproximation is justified only if Green function changeson the scale smaller than the variation scale of the pairingpotential. In general the former is set by the particle’smass. As the quasi-particles become massless at the nodethe quasi-classics is not justified when ∆( k ) /v < ξ − ( T ).More specifically the inverse square root singularity re-ported in Ref. [15] is an artifact of the quasi-classicalapproximation and is in fact smeared. The typical scaleof the spectral function is temperature dependent andcan be estimated as ∼ d ( ξ/v )∆ ( v/ξ ) close to the node.Our approach is easely generalizable for other problemswhere quasiparticles coexist with critical or almost crit- ical collective excitations such as magnetic fluctuationsat the onset of antiferromagnetic order. Acknowledgments
We are grateful to A. Chubukov and Z. Tesanovic forencouraging discussions and interest to the work. A. M.T. was supported by the Center for Emerging Supercon-ductivity funded by the U.S. Department of Energy, Of-fice of Science. M. Khodas acknowledges support fromBNL LDRD grant 08-002. [1] V. J. Emery and S. A. Kivelson, Nature , 434 (1995).[2] N. P. Ong and Y. Wang, Physica 408-410C, 11 (2004);Y. Wang et. al. , Phys. Rev. Lett. , 247002 (2005); L.Li et.al. , arXiv:0906.1823.[3] C. Meingast et.al. , Phys. Rev. Lett , 1606 (2001).[4] S. Weyeneth, T. Schneider and E. Giannini, Phys. Rev.B , 214504 (2009).[5] Q.Li et. al. , Phys. Rev. Lett , 067001 (2007).[6] A. Kanigel et.al. , Nature Physics , 447 (2007).[7] J. Meng et.al. , arXiv:0906.2682.[8] P. W. Anderson, arXiv; 0807.0578[9] H.-B. Yang et. al. , Nature , 77 (2008).[10] Kai-Yu Yang, T. M. Rice and Fu-Chun Zhang,Phys. Rev. B , 174501 (2006).[11] Kai-Yu Yang, H. B. Yang, P. D. Johnson, T. M. Rice,Fu-Chun Zhang, EPL, , 37002 (2009).[12] A. Caldeira and A. J. Leggett, Phys. Rev. Lett. , 211(1981).[13] M. Franz and A. J. Millis, Phys. Rev. B , 14572 (1998).[14] H.-J. Kwon and A. T. Dorsey, Phys. Rev. B , 6438(1999).[15] E. Berg and E. Altman, Phys. Rev. Lett , 247001(2007).[16] S. Lukyanov and A. Zamolodchikov, Nucl.Phys. B607