Quantum transitions of d-wave superconductors in a magnetic field

Transcription

1 Quantum transitions of d-wave superconductors in a magnetic field Eugene Demler (Harvard) Kwon Park Anatoli Polkovnikov Subir Sachdev Matthias Vojta (Augsburg) Ying Zhang Science 86, 479 (1999). Transparencies online at

2 Further neighbor magnetic couplings La CuO4 T=0!" S 0 Experiments Magnetic order Universal properties of magnetic quantum phase transition do not change along this line!" S = 0 Concentration of mobile carriers δ in e.g. La Sr CuO δ δ 4 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 411 (199). A.V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994)

3 Outline I. Magnetic ordering transitions in two-dimensional quantum antiferromagnets: NMR measurements of Zn/Li impurities and neutron scattering measurements of phonon spectra. II. III. Effect of magnetic field on antiferromagnetic order in superconductor. Comparison of predictions with neutron scattering experiments. Conclusions

4 I. Magnetic quantum transition in the insulator (δ=0) φ α α =1,,3 Neel order parameter Action: S b 1 = d xdτ φ + c φ + V φ Missing: Spin Berry Phases (( ) ( ) ) ( ) x α τ α α S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 344 (1989). A isa e Berry phases induce bond charge order in quantum disordered phase with φ α = 0 ; e.g. spin-peierls or plaquette order (need not be quasi-1d) Dual order parameter N. Read and S. Sachdev, Phys. Rev. Lett. 6, 1694 (1989).

5 Square lattice with first(j 1 ) and second (J ) neighbor exchange interactions H = J S " S " i< j ij i j 1 = ( ) N. Read and S. Sachdev, Phys. Rev. Lett. 6, 1694 (1989). O. P. Sushkov, J. Oitmaa, and Z. Weihong, Phys. Rev. B 63, (001). M.S.L. du Croo de Jongh, J.M.J. van Leeuwen, W. van Saarloos, Phys. Rev. B 6, (000). Neel state Spin-Peierls (or plaquette) state Bond-centered charge order See however L. Capriotti, F. Becca, A. Parola, S. Sorella, cond-mat/ Onset of bond-charge order: Z 4 order parameter Co-existence Vanishing of spin gap and Neel order: φ 4 field theory J / J 1

6 Properties of paramagnet with bond-charge-order Stable S=1 spin exciton quanta of 3-component φα ε k = + ck + ck x x y y Spin gap S=1/ spinons are confined by a linear potential. S=0 holes can similarly be confined in pairs. E. Fradkin and S. Kivelson, Mod. Phys. Lett B 4, 5 (1990). S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 19 (1991).

7 Effect of static non-magnetic impurities (Zn or Li) Zn Zn Zn Zn Spinon confinement implies that free S=1/ moments form near each impurity χ impurity ( T 0) = SS ( + 1) kt 3 B A.M Finkelstein, V.E. Kataev, E.F. Kukovitskii, G.B. Teitel baum, Physica C 168, 370 (1990). J. Bobroff, H. Alloul, W.A. MacFarlane, P. Mendels, N. Blanchard, G. Collin, and J.-F. Marucco, Phys. Rev. Lett. 86, 4116 (001) D. L. Sisson, S. G. Doettinger, A. Kapitulnik, R. Liang, D. A. Bonn, and W. N. Hardy, Phys. Rev. B 61, 3604 (000).

8 Further neighbor magnetic couplings La CuO4 Magnetic order!" S 0 Experiments Concentration of mobile carriers δ in e.g. T=0 La!" S = 0 Sr CuO δ δ 4 Summary: Confined, paramagnetic Mott insulator has 1. Stable S=1 spin exciton φ α. (resonance mode). Broken translational symmetry:- bond charge order. (phonon spectra) 3. Pairing of holes. 4. S=1/ moments near non-magnetic impurities. (NMR experiments) Properties of paramagnetic insulator survive for a finite range of δ

9 Neutron scattering measurements of phonon spectra Discontinuity in the dispersion of a LO phonon of the O ions at wavevector k = π/ : evidence for bond-charge order with period a k = 0 k = π La 1.85 Sr 0.15 CuO 4 Oxygen Copper YBa Cu 3 CuO 6.95 R. J. McQueeney, T. Egami, J.-H. Chung, Y. Petrov, M. Yethiraj, M. Arai, Y. Inamura, Y. Endoh, C. Frost and F. Dogan, cond-mat/ R. J. McQueeney, Y. Petrov, T. Egami, M. Yethiraj, G. Shirane, and Y. Endoh, Phys. Rev. Lett. 8, 68 (1999). L. Pintschovius and M. Braden, Phys. Rev. B 60, R15039 (1999).

10 Computation of phonon damping by non-linear coupling to fluctuating spin-peierls order with period a K. Park and S. Sachdev, cond-mat/

11 Further neighbor magnetic couplings T=0!" S = 0 La CuO4!" S 0 Experiments Magnetic order Concentration of mobile carriers δ in e.g. La Sr CuO δ δ 4 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 411 (199). A.V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994) Universal properties of magnetic quantum phase transition do not change along this line

12 III. Effect of magnetic field on SDW order in SC phase T=0, H=0 Superconductor (SC) + Spin density wave (SDW) Superconductor (SC) r c r (some parameter in the Hamiltonian, possibly carrier concentration) Many experimental indications that the cuprate superconductors are not too far from such a quantum phase transition: G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 78, 143 (1997). Y. S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999). S. Katano, M. Sato, K. Yamada, T. Suzuki, and T. Fukase Phys. Rev. B 6, (000). B. Lake, G. Aeppli et al., Science 91, 1759 (001). Y. Sidis, C. Ulrich, P. Bourges, et al., cond-mat/ H. Mook, P. Dai, F. Dogan, cond-mat/ J.E. Sonier et al., preprint.

13 Main results (extreme Type II superconductivity) T=0 Elastic scattering intensity I( H) H 3H 1 a ln I(0) H H c = + c H ( r rc ) ~ ln 1/ ( ( r rc )) S = 1 exciton energy ε ( H) ε(0) H 3H 1 b ln H c = c H All functional forms are exact. Similar results apply to other competing orders e.g. SC + staggered flux E. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. in press, cond-mat/

14 Structure of quantum theory Charge-order is not critical: can neglect Berry phases. Generically, momentum conservation prohibits decay of S=1 exciton φ α into S=1/ fermionic excitations at low energies. Virtual pairs of fermions only renormalize parameters in the effective action for φ α. Zeeman coupling only leads to corrections at order H Simple Landau theory couplings between φ α and superconducting order ψ are allowed (S.-C. Zhang, Science 75, 1089 (1997)), e.g.: V φ V φ + λφ ψ S b ( ) ( ) α α α 1 = d xdτ φ + c φ + V φ (( ) ( ) ) ( ) x α τ α α

15 Theory should account for quantum spin fluctuations All effects are ~ H except those associated with H induced superflow. Can treat SC order in a static Ginzburg-Landau theory Action F / T + S + S GL b c 4 ψ FGL = d x ψ + + x ia ( ) ψ v Sc = d xdτ φα ψ S b 1/ T 1 g 0 ( ( ) ( ) ) ( ) x α τ α α α = d x dτ φ + c φ + rφ + φ 4!

16 Self-consistent Hartree theory of quantum spin fluctuations (large N limit) ( xx,, ) = ( x, ) ( x, ) χ ω δ φ ω φ ω n αβ α n β n ( ω ) n c x + V ( x) χ( x, x, ωn) = δ( x x ) V ( x) ( ) ( / 6 ) (,, ) = r+ v ψ x + NgT χ x x ω ( ) ( ) " 1 + ψ x x ia ψ( x) + ( NvT / ) χ( x, x, ωn) ψ( x) = 0 V ( x) local classical energy of spin fluctuations; can become negative in vortex cores for v > 0. However, spin gap remains finite because of quantum fluctuations ω ω n n n

17 Energy Lowest energy spin-exciton eigenmode V ( x) # Spin gap 0 x Vortex cores As 0, #, because of self interaction, g, of spin excitations. A.J. Bray and M.A. Moore, J. Phys. C 15, L765 (198). J.A. Hertz, A. Fleishman, and P.W. Anderson, Phys. Rev. Lett. 43, 94 (1979).

18 Dominant effect: uniform softening of spin excitations by superflow kinetic energy r v s 1 r Spatially averaged superflow kinetic energy H 3H v ln c s Hc H See D. P. Arovas et al., Phys. Rev. Lett. 79, 871 (1997) for a different viewpoint.

19 ( x) Influence of ψ on extended spin eigenmodes: ψ ψ 1 x ( x) = 1 outside each vortex core because ( x) In SC phase, spin gap obeys: of superflow kinetic energy 3.0H c = 1 ln Hc H 3.0H c ( H ) = ( 0) ln 3v Hc H Ng 1 g In SC+SDW phase, intensity of elastic scattering obey s: H 4π c v 6v H 3.0H = + 3v Hc H g 1 g c ( ) I( 0) ln I H H

20 Intensity (counts/min) Elastic neutron scattering off La CuO 4+ y B. Khaykovich, Y. S. Lee, S. Wakimoto, K. J. Thomas, M. A. Kastner, and R.J. Birgeneau, preprint. H = 7.5 T k H = 0 T h H c I(H)/I(0) h (r.l.u.) Solid line --- fit to : ( ) ( 0) I H a is the only fitting parameter Best fit value - a =.4 with H I H (T) H 3.0H 1 a ln c = + Hc H c = 60T

21 Neutron scattering of La -xsrxcuo 4 at x=0.1 B. Lake, G. Aeppli, et al. H Hc Solid line - fit to : I( H) a ln H H = c

22 Presence of vortex lattice leads to supermodulation in the spin exciton spectrum Computation of spin susceptibility χ ( k, ω) in self-consistent large N theory of φ fluctuations in a vortex lattice α r = 0.1, H/H c = 0.04 Intensity 0 ω k π / ( vortex lattice spacing)

23 Consequences of a finite London penetration depth (finite κ)

24 Further neighbor magnetic couplings La CuO4!" S 0 Conclusions Experiments Magnetic order!" S = 0 T=0 Concentration of mobile carriers δ Confined, paramagnetic Mott insulator has 1. Stable S=1 spin exciton. φ α. Broken translational symmetry:- bondcentered charge order. 3. Pairing of holes. 4. S=1/ moments near non-magnetic impurities Theory of magnetic ordering quantum transitions in antiferromagnets and superconductors leads to quantitative theories for Spin correlations in a magnetic field Effect of Zn/Li impurities on collective spin excitations