Damping of collective modes and quasiparticles in d-wave superconductors. Subir Sachdev M. Vojta. Yale University. C. Buragohain

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1 Damping of collective modes and quasiparticles in d-wave superconductors C. Buragohain Y. Zhang Subir Sachdev M. Vojta Transparencies on-line at Review article: cond-mat/ and references therein Quantum Phase Transitions, Cambridge University Press Yale University

2 Elementary excitations of a d-wave superconductor (A) S=0 Cooper pairs, phase fluctuations Negligible below T c except near a T=0 superconductor-insulator transition. Proliferate above T c due to free vortex density. (B) S=1/ Fermionic quasiparticles Ψ h : strongly paired fermions near (π,0), (0,π) have an energy gap ~ mev Ψ 1, : gapless fermions near the nodes of the superconducting gap at ( ±, ± ) with = K K K π (C) S=1 Bosonic, resonant collective mode S(Q,ω) Γ pure φ α Represented by, a vector field measuring the strength of antiferromagnetic spin fluctuations near Q ( ππ, ) Damping is small at T=0 Γpure 0 at T = 0 (Theory generalizes to the cases with incommensurate Q and Γ ) pure 0 res ω

3 Constraints from momentum conservation π Ψ h Ψ Ψ 1 φ α 0 Ψ h Q φ α Ψ h Ψ 1 Q Ψ π π Ψ h : strongly coupled to Ψ h 0 π and phase fluctuations (leads to strong damping above T c, and coherent pairing and gap formation below T c ) Ψ 1, : decoupled from φ α φ α (absence of damping and pairing?) and phase fluctuations

4 I. Zero temperature broadening of resonant collective mode by φ α impurities: comparison with neutron scattering experiments of Fong et al Phys. Rev. Lett. 8, 1939 (1999) II. Intrinsic inelastic lifetime of nodal quasiparticles Ψ 1, (Valla et al Science 85, 110 (1999) and Corson et al cond-mat/000343): critical survey of possible nearby quantum-critical points. Independent low energy quantum field theories for the φ α and the Ψ 1,

5 I. Zero temperature broadening of resonant collective mode by impurities Effect of arbitrary localized deformations ( impurities ) of density n imp Each impurity is characterized by an integer/half-odd-integer S As res 0 Γ imp res = n imp Dc res C S + O J res Correlation length ξ C S Universal numbers dependent only on S C = ; C / Zn impurities in YBCO have S=1/ Swiss-cheese model of quantum impurities (Uemura): Inverse Q of resonance ~ fractional volume of holes in Swiss cheese.

6 As res 0 there is a quantum phase transition to a magnetically ordered state (A) Insulating Neel state (or collinear SDW at wavevector Q) insulating quantum paramagnet (B) d-wave superconductor with collinear SDW at wavevector Q d-wave superconductor (paramagnet) Transition (B) is in the same universality class as (A) provided Ψ h fermions remain gapped at quantum-critical point.

7 Why appeal to proximity to a quantum phase transition? φ α ~ S=1 bound state in particle-hole channel at the antiferromagnetic wavevector (a) ~ φ α (b) (c) Quantum field theory of critical point allows systematic treatment of the strongly relevant multi-point interactions in (b) and (c).

8 1. (A) Paramagnetic and Neel ground states in two dimensions --- coupled-ladder antiferromagnet. Field theory of quantum phase transition.. Non-magnetic impurities (Zn or Li) in twodimensional paramagnets. 3. Application to (B) d-wave superconductors. Comparison with, and predictions for, expts

9 1. Paramagnetic and Neel states in insulators (Katoh and Imada; Tworzydlo, Osman, van Duin and Zaanen) S=1/ spins on coupled -leg ladders H = < ij> J J ij H S i H S Follow ground state as a function of λ 0 λ 1 λ J j

10 λ close to 1 Square lattice antiferromagnet Experimental realization: La CuO 4 Ground state has long-range magnetic (Neel) order H S i = ( ) i x + iy N 0 Excitations: spin waves Quasiclassical wave dynamics at low T (Chakravarty et al, 1989; Tyc et al, 1989)

11 λ close to 0 Weakly coupled ladders 1 = ( ) Paramagnetic ground state H S = i Excitation: S=1, φ α particle (collective mode) Energy dispersion away from antiferromagnetic wavevector ε = res + c k res res Spin gap

12 Neel order N0 Spin gap res 0 1 λ Quantum paramagnet S H = 0 λ c Neel Hstate S N 0

13 Nearly-critical paramagnets λ is close to λ c Quantum field theory: [ 1 ( S b = d d xdτ ( x φ α ) + c ( τ φ α ) + rφα) + g ] 4! (φ α) φα r > 0 r < 0 3-component antiferromagnetic order parameter λ < λ c λ > λ c Oscillations of φ α about zero (for r > 0) spin-1 collective mode Imχ ( k, ω ) T=0 spectrum ω

14 Coupling g approaches fixed-point value under renormalization group flow: beta function (ε = 3-d) : β(g) = ɛg + 11g 6 3g3 1 + O(g4 ) Only relevant perturbation r strength is measured by the spin gap res and c completely determine entire spectrum of quasi-particle peak and multiparticle continua, the S matrices for scattering between the excitations, and T > 0 modifications.

15 . Quantum impurities in nearly-critical paramagnets Make any localized deformation of antiferromagnet; e.g. remove a spin X Y χ b Susceptibility χ = Aχ + χ In paramagnetic phase as T 0 = D c res e π res / k χ For a general impurity imp B T ; χ b imp (A = area of system) imp = S( S 3k + 1) T defines the value of S B

16 Orientation of impurity spin -- n α (τ ) (unit vector) Action of impurity spin S imp = dτ [ isa α (n) dn α dτ γsn α(τ)φ α (x =0,τ) ] (n) A α Dirac monopole function Boundary quantum field theory: S b + S imp Recall - S b = d d xdτ [ 1 ( ( x φ α ) + c ( τ φ α ) + rφ α) + g 4! (φ α) ]

17 Coupling γ approaches also approaches a fixed-point value under the renormalization group flow (Sengupta, 97 Beta function: β(γ) = ɛγ Sachdev+Ye, 93 + γ3 γ 5 + 5g γ Smith+Si 99) 144 ( + π S(S +1) 1 ) gγ 3 + O ( (γ, g) 7) 3 3 No new relevant perturbations on the boundary; All other boundary perturbations are irrelevant e.g. λ dτφα ( x = 0, τ ) (This is the simplest allowed boundary perturbation for S=0 its irrelevance implies C 0 = 0) res and c completely determine spin dynamics near an impurity No new parameters are necessary! Finite density of impurities nimp Relevant perturbation strength determined by only energy scale that is linear in n imp and contains only bulk parameters Γ n Dc imp ( ) res

18 Fate of collective mode peak Without impurities χ ( G, ω) = res A ω A Dω Γ With impuritiesχ ( G, ω ) = Φ, res res res Φ Universal scaling function. We computed it in a self-consistent, non-crossing approximation Predictions: Half-width of line Γ Universal asymmetric lineshape

19 3. Application to d-wave superconductors (YBCO) Zn impurity in YBa Moments measured by analysis of Knight shifts Cu3O6.7 M.-H. Julien, T. Feher, M. Horvatic, C. Berthier, O. N. Bakharev, P. Segransan, G. Collin, and J.-F. Marucco, Phys. Rev. Lett. 84, 34 (000); also earlier work of the group of H. Alloul Berry phases of precessing spins do not cancel between the sublattices in the vicinity of the impurity: net uncancelled phase of S=1/ Pepin and Lee: Modeled Zn impurity as a potential scatterer in the unitarity limit, and obtained quasi-bound states at the Fermi level. Our approach: Each bound state captures only one electron and this yields a Berry phase of S=1/; residual potential scattering of quasiparticles is not in the unitarity limit.

20 Additional low-energy spin fluctuations in a d-wave superconductor Nodal quasiparticles Ψ There is a Kondo coupling between moment around impurity and Ψ: J K S n α Ψ * σ α Ψ However, because density of states vanishes linearly at the Fermi level, there is no Kondo screening for any finite J K (below a finite J K ) with (without) particle-hole symmetry (Withoff+Fradkin, Chen+Jayaprakash, Buxton+Ingersent)

21 YBa Cu3O % Zn H. F. Fong, P. Bourges, Y. Sidis, L. P. Regnault, J. Bossy, A. Ivanov, D.L. Milius, I. A. Aksay, and B. Keimer, Phys. Rev. Lett. 8, 1939 (1999) Γ = n imp res 5 mev, Γ = = 40 mev Dc = 0. ev res = 0.15 Quoted half-width = 4.5 mev

22 II. Intrinsic inelastic lifetime of nodal quasiparticles Ψ 1, Photoemission on BSSCO (Valla et al Science 85, 110 (1999)) Quantum-critical damping of quasiparticles along (1,1) Quasi-particles sharp along (1,0)

23 ImΣ ~ kb T for Dω < k B T ImΣ ~ Dω for Dω > k B T Marginal Fermi liquid (Varma et al 1989) but only for nodal quasi-particles strong k dependence at low temperatures Origin of inelastic scattering? In a Fermi liquid ImΣ ~ T In a BCS d-wave superconductor ImΣ ~ T 3

24 THz conductivity of BSCCO (Corson et al cond-mat/000343) Quantum-critical damping of quasi-particles

25 Proximity to a quantum-critical point Superconducting T c T T X Superconducting state X Quantum critical g c d-wave superconductor g (Crossovers analogous to those near quantum phase transitions in boson models Weichmann et al 1986, Chakravarty et al 1989) Relaxational dynamics in quantum critical region (Sachdev+Ye, 199) G F ( k ω ) Λ η, = 1 F Φ ck k Dω ( ) ηf k T BT kbt B, Nodal quasiparticle Green s function k wavevector separation from node

26 Necessary conditions 1. Quantum-critical point should be below its upper-critical dimension and obey hyperscaling.. Critical field theory should not be free required to obtain damping in the scaling limit. Combined with (1) this implies that characteristic relaxation times ~ D / k B T 3. Nodal quasi-particles should be part of the critical-field theory. 4. Quasi-particles along (1,0), (0,1) should not couple to critical degrees of freedom.

27 1. d-wave superconductors. Candidates for X: a) Staggered-flux (or orbital antiferromagnet) order + d-wave superconductivity (breaks T timereversal symmetry). b) Superconductivity + charge density order (charge stripes) c) (d+is)-wave superconductivity (breaks T) d) d + id x y wave superconductivity (breaks T) xy

28 1. d-wave superconductors Gapless Fermi Points in a d-wave superconductor at wavevectors K=0.391π = Ψ * 3 1 * f f f f = Ψ * 4 * 4 f f f f x y matrices in Nambu space are Pauli, z τ x τ S Ψ = d d k (π) d T ω n Ψ 1 ( iω n + v F k x τ z + v k y τ x )Ψ 1 + d d k (π) d T ω n Ψ ( iω n + v F k y τ z + v k x τ x )Ψ. ), ( K ±K ±

29 a. Orbital antiferromagnet Checkerboard pattern of spontaneous currents: (Affleck+Marston 1988, Schulz 1989, Wang, Kotliar, Wang, 1990, Wen+Lee, 1996) T-breaking Ising order parameter φ c k+g,a c k,a = iφ(cos k x cos k y ) ; G =(π, π) S φ = d d xdτ (Nayak, 000) ] [ 1 ( τφ) + c ( φ) + s 0 φ + u 0 4 φ4 For K=π/, only coupling to nodal quasiparticles d is ~ ζ d xdτφ ΨΨ; ζ is irrelevant and leads to d + 1 / ν 1.83 Ising Im Σ ~ T ~ T

30 b. Charge stripe order Charge density δρ ~ Re Φ [ ] igx igy e + Φ e x y If G K fermions do not couple efficiently to the order parameter and are not part of the critical theory G Action for quantum fluctuations of order parameter S Φ = d d xdτ [ τ Φ x + τ Φ y + Φ x + Φ y ( + s 0 Φx + Φ y ) + u ( 0 Φx 4 + Φ y 4) + v 0 Φ x Φ y ] Coupling to fermions ~ λ d d xdτ Φ and λ is irrelevant at the critical point a ΨΨ ImΣ ~ ~ T + 1 / ν T d (between and 3) for /3 < ν < 1

31 c. (d+is)-wave superconductivity T-breaking Ising order parameter φ (Kotliar, 1989) c k c k = 0 (cos k x cos k y )+iφ(cos k x +cosk y ). Effective action: [ 1 S φ = d d xdτ ( τφ) + c ( φ) + s 0 φ + u 0 4 φ4 Efficient coupling to nodal quasi-particles (generically) S Ψφ = d d xdτ [ λ 0 φ ( Ψ 1τ y Ψ 1 +Ψ τ y Ψ )]. Coupling λ 0 takes a non-zero fixed-point value in the critical field theory Strong inelastic scattering of nodal-quasiparticles in the scaling limit ] Nodal quasiparticle lifetime ~ D / k B T However: strong scattering of quasi-particles also along (1,0), (0,1) directions

32 d. d + id xy -wave superconductivity x y T-breaking Ising order parameter φ (Rokhsar 1993, Laughlin 1994) c k c k = 0 (cos k x cos k y )+iφ sin k x sin k y. Effective action: [ 1 S φ = d d xdτ ( τφ) + c ( φ) + s 0 φ + u 0 4 φ4 Efficient coupling to nodal quasi-particles (generically) S Ψφ = d d xdτ [ λ 0 φ ( Ψ 1τ y Ψ 1 Ψ τ y Ψ )]. Coupling λ 0 takes a non-zero fixed-point value in the critical field theory Strong inelastic scattering of nodal-quasiparticles in the scaling limit ] Nodal quasiparticle lifetime ~ D / k B T Moreover: no scattering of quasi-particles along (1,0), (0,1) directions!

33 Large N (Sp(N)) phase diagram for timereversal symmetry breaking and chargeordering in a d-wave superconductor.

34 Gapped quasiparticles: Below T c : negligible damping Above T c : damping from strong coupling to superconducting phase and SDW fluctuations. π Brillouin zone k y 0 0 k x π Nodal quasiparticles: Below T c : damping from fluctuations to order d + id Above T c : same mechanism applies as long as quantum-critical length < superconducting phase coherence length. Quasiparticles do not couple to phase or SDW fluctuations. x y xy

35 Conclusions: Part I 1. Universal T=0 damping of S=1 collective mode by non-magnetic impurities. nimp ( Dc) Linewidth: Γ res independent of impurity parameters.. New interacting boundary conformal field theory in +1 dimensions 3. Universal irrational spin near the impurity at the critical point.

36 Conclusions: Part II Classification of quantum-critical points leading to critical damping of quasiparticles in superconductor Most attractive possibility: T breaking transition from a d superconductor to x y a d + id superconductor x y xy Leads to quantum-critical damping along (1,1), and no damping along (1,0), with no unnatural fine-tuning. Note: stable ground state of cuprates can always be a d superconductor; only need x y thermal/quantum fluctuations to d + id x y xy order in quantum-critical region. Experimental update: Tafuri+Kirtley (cond-mat/ ) claim signals of T breaking near non-magnetic impurities in YBCO films