Competing orders and quantum criticality in the cuprate superconductors

Transcription

1 Competing orders and quantum criticality in the cuprate superconductors Subir Sachdev Science 286, 2479 (1999). Quantum Phase Transitions Cambridge University Press Transparencies online at

2 Outline I. Coupled Ladder Antiferromagnet A. Ground states in limiting regimes B. Coherent state path integral C. Quantum field theory for critical point II. III. IV. Berry phases and duality in one dimension S=1/2 quantum XY model. Berry phases and duality in two dimensions Bond-centered charge ( spin-peierls ) order. Magnetic transitions in d-wave superconductors A. Theory of SC+SDW to SC quantum transition B. Phase diagrams of Mott insulators and superconductors in an applied magnetic field C. Comparison of predictions with experiments V. Conclusions Chapter 13 from Quantum Phase Transitions, and cond-mat/ cond-mat/

3 I.A Coupled Ladder Antiferromagnet N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 4529 (1994). J. Tworzydlo, O. Y. Osman, C. N. A. van Duin, J. Zaanen, Phys. Rev. B 59, 115 (1999). M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, cond-mat/ S=1/2 spins on coupled 2-leg ladders H = < ij> J ij! S i! S j 0 λ 1 J λ J

4 λ close to 1 Square lattice antiferromagnet Experimental realization: La CuO 2 4 Ground state has long-range magnetic (Neel) order! S i = ( ) i 1 + x iy N 0 0 Excitations: 2 spin waves k cx kx cy ky ε = +

5 λ close to 0 Weakly coupled ladders 1 = 2 ( ) Paramagnetic ground state! S = i Excitation: S=1 exciton (spin collective mode) Energy dispersion away from antiferromagnetic wavevector ε k = x x + cy ky c k 2 0

6 T=0 Spin gap λ c Neel order N0 0 1 Quantum paramagnet S! = 0 Neel state! S =± N 0 0 λ

7 I.B Coherent state path integral See Chapter 13 of Quantum Phase Transitions, S. Sachdev, Cambridge University Press (1999). Z = Path integral for a single spin ( H[ S] / T Tr e ) ( ) τ δ τ τ τ τ ( ) ( 2 = D N N 1) exp is A ( ) d d H S ( ) τ N ( ) Aτ τ dτ = Oriented area of triangle on surface of unit sphere bounded Action for lattice antiferromagnet ( τ) ( τ + dτ) 0 by N, N,and a fixed reference N ( τ) = η ( x, τ) + ( x, τ) N n L j j j j η =± 1 identifies sublattices j n and L vary slowly in space and time

8 ( ) ( 2 Z = D n x, τ δ n 1) exp is ηjaτ ( xj, τ) dτ j η Integrate out L and take the continuum limit j 1 d xdτ ( 2 2 ( ) ( ) ) τ n c xn g =± 1 identifies sublattices Small g: long-range Neel order and spin-wave theory Large g: duality mapping to the Quantum Dimer Model

9 I.C Quantum field theory for critical point φα λ close to λ c : neglect Berry phases and use soft spin field S b 1 g = d xdτ φ + c φ + rφ + φ 2 4! 3-component antiferromagnetic order parameter Oscillations of φ α about zero (for r > 0) spin-1 collective mode (( ) 2 ( ) 2 ) ( ) 2 x α τ α α α r > 0 λ < λ c r < 0 λ > λ c Imχ ( k, ω ) ε = + ck T=0 spectrum ω

10 Dynamic spectrum at the critical point Critical coupling ( λ = λ c ) ( k ω ) Im χ, ( ω ) (2 ) ~ ck η ck ω No quasiparticles --- dissipative critical continuum

11 Outline I. Coupled Ladder Antiferromagnet A. Ground states in limiting regimes B. Coherent state path integral C. Quantum field theory for critical point II. III. IV. Berry phases and duality in one dimension S=1/2 quantum XY model. Berry phases and duality in two dimensions Bond-centered charge ( spin-peierls ) order. Magnetic transitions in d-wave superconductors A. Theory of SC+SDW to SC quantum transition B. Phase diagrams of Mott insulators and superconductors in an applied magnetic field C. Comparison of predictions with experiments V. Conclusions

12 II. Quantum XY model in one dimension S. Sachdev and K. Park, cond-mat/

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16 Excitations of paramagnet with bond-charge-order Deconfined S=1/2 spinons λ Phase diagram Luttinger liquid (quasi-long-range XY order) J / J 2 1

17 Outline I. Coupled Ladder Antiferromagnet A. Ground states in limiting regimes B. Coherent state path integral C. Quantum field theory for critical point II. III. IV. Berry phases and duality in one dimension S=1/2 quantum XY model. Berry phases and duality in two dimensions Bond-centered charge ( spin-peierls ) order. Magnetic transitions in d-wave superconductors A. Theory of SC+SDW to SC quantum transition B. Phase diagrams of Mott insulators and superconductors in an applied magnetic field C. Comparison of predictions with experiments V. Conclusions

18 III. Berry phases and the square lattice antiferromagnet H = J S! S! Action: i< j S Missing: Spin Berry Phases A b ij i j 1 = d xdτ φ + c φ + V φ 2 (( ) 2 ( ) 2 ) ( ) x α τ α α S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989). isa e Berry phases induce bond charge order in quantum disordered phase with φ α = 0 ; Dual order parameter N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).

19 H = J S! S! i< j ij i j Square lattice with first (J 1 ) and second (J 2 ) neighbor exchange interactions N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). O. P. Sushkov, J. Oitmaa, and Z. Weihong, Phys. Rev. B 63, (2001). M.S.L. du Croo de Jongh, J.M.J. van Leeuwen, W. van Saarloos, Phys. Rev. B 62, (2000). Neel state Spin-Peierls state Bond-centered charge order S. Sachdev and K. Park, cond-mat/ Co-existence? 0.4 J 2 / J 1 1 = 2 ( )

20 Quantum dimer model D. Rokhsar and S. Kivelson Phys. Rev. Lett. 61, 2376 (1988) Quantum entropic effects prefer one-dimensional striped structures in which the largest number of singlet pairs can resonate. The state on the upper left has more flippable pairs of singlets than the one on the lower left. These effects lead to a broken square lattice symmetry near the transition to the Neel state. N. Read and S. Sachdev Phys. Rev. B 42, 4568 (1990).

21 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 2 Spin gap S=1/2 spinons are confined by a linear potential.

22 Effect of static non-magnetic impurities (Zn or Li) Zn Zn Zn Zn Spinon confinement implies that free S=1/2 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 (2001).

23 Field theory of bond order

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25 For large e 2, low energy height configurations are in exact one-toone correspondence with dimer coverings of the square lattice 2+1 dimensional height model is the path integral of the quantum dimer model There is no roughening transition for three dimensional interfaces, which are smooth for all couplings There is a definite average height of the interface Ground state has bond-charge order.

26 Outline I. Coupled Ladder Antiferromagnet A. Ground states in limiting regimes B. Coherent state path integral C. Quantum field theory for critical point II. III. IV. Berry phases and duality in one dimension S=1/2 quantum XY model. Berry phases and duality in two dimensions Bond-centered charge ( spin-peierls ) order. Magnetic transitions in d-wave superconductors A. Theory of SC+SDW to SC quantum transition B. Phase diagrams of Mott insulators and superconductors in an applied magnetic field C. Comparison of predictions with experiments V. Conclusions

27 Parent compound of the high temperature superconductors: La CuO 2 4 Mott insulator: square lattice antiferromagnet H = < ij> J ij! S i! S j Ground state has long-range magnetic (Néel) order Néel order parameter: ( ) φ α i + i x y = 1 S ; α = x, y, z iα φα 0

28 Introduce mobile carriers of density δ by substitutional doping of out-of-plane ions e.g. La 2 δsrδcuo 4 Exhibits superconductivity below a high critical temperature T c Superconductivity in a doped Mott insulator " BCS superconductor obtained by the Cooper instability of a metallic Fermi liquid Quantum numbers of ground state and low energy quasiparticles are the same, but characteristics of the Mott insulator are revealed in the vortices. S. Sachdev, Phys. Rev. B 45, 389 (1992); K. Park and S. Sachdev Phys. Rev. B 64, (2001).? STM measurement of J.E. Hoffman et al., Science, Jan 2002.

29 Neel LRO Zero temperature phases of the cuprate superconductors as a function of hole density SDW along (1,1) +localized holes SC+SDW SC ~0.05 ~0.12 Theory for a system with strong interactions: describe SC and SC+SDW phases by expanding in the deviation from the quantum critical point between them. H δ B. Keimer et al. Phys. Rev. B 46, (1992). S. Wakimoto, G. Shirane et al., Phys. Rev. B 60, R769 (1999). G. Aeppli, T.E. Mason, S.M. Hayden, H.A. Mook, J. Kulda, Science 278, 1432 (1997). Y. S. Lee, R. J. Birgeneau, M. A. Kastner et al., Phys. Rev. B 60, 3643 (1999). J. E. Sonier et al., cond-mat/ C. Panagopoulos, B. D. Rainford, J. L. Tallon, T. Xiang, J. R. Cooper, and C. A. Scott, preprint.

30 Insulator with localized holes Further neighbor magnetic couplings La 2CuO4 Magnetic order T=0 #! S 0 SC+SDW Experiments Universal properties of magnetic quantum phase transition change little along this line. #! S = 0 Superconductor (SC) Concentration of mobile carriers δ in e.g. La Sr CuO 2 δ δ 4 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). A.V. Chubukov, S. Sachdev, and J. Ye, Phys. Rev. B 49, (1994)

31 Magnetic ordering transitions in the insulator Square lattice with first(j 1 ) and second (J 2 ) neighbor exchange interactions (say) H = J S! S! i< j ij i j 1 = 2 ( ) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). O. P. Sushkov, J. Oitmaa, and Z. Weihong, Phys. Rev. B 63, (2001). M.S.L. du Croo de Jongh, J.M.J. van Leeuwen, W. van Saarloos, Phys. Rev. B 62, (2000). Neel state Spin-Peierls (or plaquette) state Bond-centered charge order See however L. Capriotti, F. Becca, A. Parola, S. Sorella, cond-mat/ J 2 / J 1

32 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 2 Spin gap S=1/2 spinons are confined by a linear potential. Transition to Neel state Bose condensation of φ α Develop quantum theory of SC+SDW to SC transition driven by condensation of a S=1 boson (spin exciton)

33 Doping the paramagnetic Mott insulator Large N theory in region with preserved spin rotation symmetry S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999). M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. B 62, 6721 (2000). See also J. Zaanen, Physica C 217, 317 (1999), S. Kivelson, E. Fradkin and V. Emery, Nature 393, 550 (1998), S. White and D. Scalapino, Phys. Rev. Lett. 80, 1272 (1998).

34 IV.A Theory of SC+SDW to SC quantum transition Spin density wave order parameter for general ordering wavevector Φ α S α ikr ( r) =Φ ( r) e + c.c. α ( ) ( ikr r is a comple x field except for =(, ) when ( 1) ) x + r r K ππ e = y Associated charge density wave order δρ ( r) S ( r) ( r) = Φ e + c.c. 2 2 i2kr α α α Wavevector K=(3π/4,π) Exciton wavefunction Φ α (r) describes envelope of this order. Phase of Φ α (r) represents sliding degree of freedom J. Zaanen and O. Gunnarsson, Phys. Rev. B 40, 7391 (1989). H. Schulz, J. de Physique 50, 2833 (1989). O. Zachar, S. A. Kivelson, and V. J. Emery, Phys. Rev. B 57, 1422 (1998).

35 Action for SDW ordering transition in the superconductor S ( ) r α τ α α = drdτ Φ + c Φ + V Φ Similar terms present in action for SDW ordering in the insulator Coupling to the S=1/2 Bogoliubov quasiparticles of the d-wave superconductor Trilinear Yukawa coupling 2 drd τ Φ ΨΨ is prohibited unless ordering wavevector is fine-tuned. α κ α 2 2 τ Φα drd Ψ Ψ is allowed Scaling dimension of κ = (1/ ν - 2) < 0 irrelevant.

36 Neutron scattering measurements of dynamic spin correlations of the superconductor (SC) in a magnetic field B. Lake, G. Aeppli, K. N. Clausen, D. F. McMorrow, K. Lefmann, N. E. Hussey, N. Mangkorntong, M. Nohara, H. Takagi, T. E. Mason, and A. Schröder, Science 291, 1759 (2001). Peaks at (0.5, 0.5) ± (0.125, 0) and (0.5,0.5) ± (0,0.125) dynamic SDW of period 8 Neutron scattering off La 2-δSrδCuO 4 ( δ = 0.163, SC phase) at low temperatures in H=0 ( red dots) and H=7.5T ( blue d ots)

37 S. Sachdev, Phys. Rev. B 45, 389 (1992), and N. Nagaosa and P.A. Lee, Phys. Rev. B 45, 966 (1992), suggested an enhancement of dynamic spin-gap correlations (as in a spin-gap Mott insulator) in the cores of vortices in the underdoped cuprates. In the simplest mean-field theory, this enhancement appears most easily for vortices with flux hc/e. D. P. Arovas, A. J. Berlinsky, C. Kallin, and S.-C. Zhang, Phys. Rev. Lett. 79, 2871 (1997) suggested static Néel order in the cores of vortices (SC order rotates into Néel order in SO(5) picture). Using a picture of dynamically fluctuating spins in the vortices, the amplitude of the field-induced signal, and the volume-fraction of vortex cores (~10%), Lake et al. estimated that in such a model each spin in the vortex core would have a lowfrequency moment equal to that in the insulating state at δ=0 (0.6 µ B ). Observed field-induced signal is much larger than anticipated.

38 IV.B Phase diagrams in a magnetic field. Insulator with localized holes Further neighbor magnetic couplings La 2CuO4 Magnetic order T=0 #! S = #! 0 S 0 A. Effect of magnetic field on onset of SDW on insulator B. Effect of magnetic field on SC+SDW to SC transition SC+SDW Superconductor (SC) Concentration of mobile carriers δ in e.g. La Sr CuO 2 δ δ 4

39 IV.B Phase diagrams in a magnetic field. A. Effect of magnetic field on onset of SDW in the insulator H couples via the Zeeman term ( iε H )( iε H ) 2 * τ α τ α ασρ σ ρ τ α αβγ β γ Φ Φ Φ Φ Φ H SDW Spin singlet state with a spin gap J 2 /J 1 Characteristic field gµ B H =, the spin gap 1 Tesla = mev Related theory applies to spin gap systems in a field and to double layer quantum Hall systems at ν=2

40 IV.B Phase diagrams in a magnetic field. B. Effect of magnetic field on SDW+SC to SC transition S Theory should account for dynamic quantum spin fluctuations All effects are ~ H 2 except those associated with H induced superflow. Can treat SC order in a static Ginzburg-Landau theory b 1/ T g g2 τ r α τ α α α α ( ) = dr d Φ + c Φ + sφ + Φ + Φ v Sc = drdτ Φα ψ ψ FGL = d r ψ + + r ia Infinite diamagnetic susceptibility of non-critical superconductivity leads to a strong effect. 2 ( ) ψ 2 (extreme Type II superconductivity) () = Φ(, τ) δ ln Z ψ() r = δψ () r Z ψ r D r e 0 S S F GL b c

41 Envelope of spin-exciton eigenmode in potential V 0 (x) Envelope of lowest energy spin-exciton eigenmode Φ ( r) = ( r) + Φ ( r) 2 after including exciton interactions: V V g 0 α α 0 ( r) = + v ( r) 2 Potential V s ψ acting on excitons Energy Spin gap 0 x Vortex cores Strongly relevant repulsive interactions between excitons imply that low energy excitons must be extended. A.J. Bray and M.A. Moore, J. Phys. C 15, L7 65 (1982). J.A. Hertz, A. Fleishman, and P.W. Anderson, Phys. Rev. Lett. 43, 942 (1979).

42 Dominant effect: uniform softening of spin excitations by superflow kinetic energy r v s 1 r Spatially averaged superflow kinetic energy v H 3H ln 2 c2 s Hc2 H H 3Hc2 Tuning parameter s replaced by seff ( H) = s C ln H H c2

43 Main results T=0 Elastic scattering intensity H 3Hc I( H) I(0) a ln H H 2 = + c2 H ( s sc ) ~ ln 1/ ( ( s sc )) S = 1 exciton energy ε H 3H H = b H H c2 ( ) ε( 0) ln c2 All functional forms are exact. E. Demler, S. Sachdev, and Y. Zhang, Phys. Rev. Lett. 87, (2001).

44 Φα Structure of long-range SDW order in SC+SDW phase ( r) Computation in a self-consistent large N theory Y. Zhang, E. Demler, and S. Sachdev, cond-mat/ s = s c s s c = -0.3 Dynamic structure factor 3 2 S( k, ω) = ( 2π) δ( ω) fg δ( k G) + % G G reciprocal lattice vectors of vortex lattice. k measures deviation from SDW ordering wavevector K

45 R.J. Birgeneau, cond-mat/ Elastic neutron scattering off La CuO 2 4+ y B. Khaykovich, Y. S. Lee, S. Wakimoto, K. J. Thomas, M. A. Kastner, and Neutron scattering measurements of static spin correlations of the superconductor+spin-density-wave (SC+SDW) in a magnetic field Solid line --- fit to : ( ) ( 0) I H a is the only fitting parameter Best fit value - a = 2.4 with H I c2 = 1+ ln Hc2 H c2 a H = 60T 3.0H

46 Neutron scattering of La 2-xSrxCuO 4 at x=0.1 B. Lake, G. Aeppli, et al., Nature, Jan H Hc Solid line - fit to : I( H) a ln H H 2 = c2

47 Dynamic SDW fluctuations in the SC phase Computation of spin susceptibility χ ( k, ω) in self-consistent large N theory of Φ fluctuations α Field H chosen to place the system close to boundary to SC+SDW phase 2 π / ( vortex lattice spacing)

48 Prediction of static CDW order by vortex cores in SC phase, with dynamic SDW correlations Spin gap state in vortex core appears by a local quantum disordering transition of magnetic order: by our generalized phase diagram, charge order should appear in this region. K. Park and S. Sachdev Physical Review B 64, (2001).

49 Pinning of static CDW order by vortex cores in SC phase, with dynamic SDW correlations A.Polkovnikov, S. Sachdev, M. Vojta, and E. Demler, cond-mat/ Y. Zhang, E. Demler, and S. Sachdev, cond-mat/ Superflow reduces energy of dynamic spin exciton, but action so far does not lead to static CDW order because all terms are invariant under the sliding symmetry: Φ α ( r) Φ ( r) i Small vortex cores break this sliding symmetry on the lattice scale, and lead to a pinning term, which picks particular phase of the local CDW order α e θ S pin ( r ) 1/ T All rv where ψ v = 0 0 ( r ) 2 i = ζ dτ Φ v e θ α + c.c. With this term, SC phase has static CDW but dynamic SDW δρ ( ) ( ) 2 α 2 i2 Kr Φ α e + α ( r) ( r) Φ 0 ; Φ = 0 r r c.c. ; S α α ikr ( r) =Φ ( r) e + c.c. Friedel oscillations of a doped spin-gap antiferromagnet α

50 Pinning of CDW order by vortex cores in SC phase Computation in self-consistent large N theory ( r, τ) ζ dτ ( r, τ) ( r, τ ) 2 Φ Φ Φ 2 * α 1 α α v 1 & ' low magnetic field high magnetic field near the boundary to the SC+SDW phase Φ 2 α ( r, τ )

51 Simplified theoretical computation of modulation in local density of states at low energy due to CDW order induced by superflow and pinned by vortex core A. Polkovnikov, S. Sachdev, M. Vojta, and E. Demler, cond-mat/ ( ) ( r ) H = t c c + c c, + hc.. + v µ c c, ij iσ jσ ij iσ j σ i iσ iσ ij i 2 3/4 r r0 / ξc 1 cos 0 cos cx cy { } ( ) ( ) ( ) ( ) v r = v K r r + K r r e r r +

52 STM around vortices induced by a magnetic field in the superconducting state J. E. Hoffman, E. W. Hudson, K. M. Lang, V. Madhavan, S. H. Pan, H. Eisaki, S. Uchida, and J. C. Davis, Science, Jan Local density of states Diffe re n tialc onductance (ns ) Regular QPSR Vortex 1Å spatial resolution image of integrated LDOS of Bi 2 Sr 2 CaCu 2 O 8+δ ( 1meV to 12 mev) at B=5 Tesla Sam ple B ias (m V ) S.H. Pan et al. Phys. Rev. Lett. 85, 1536 (2000).

53 Vortex-induced LDOS integrated from 1meV to 12meV 7 pa b 0 pa 100Å J. Hoffman et al, Science, Jan 2002.

54 Fourier Transform of Vortex-Induced LDOS map K-space locations of vortex induced LDOS K-space locations of Bi and Cu atoms J. Hoffman et al Science, Jan Distances in k space have units of 2π/a 0 a 0 =3.83 Å is Cu-Cu distance

55 Doping the paramagnetic Mott insulator Large N theory in region with preserved spin rotation symmetry S. Sachdev and N. Read, Int. J. Mod. Phys. B 5, 219 (1991). M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999). M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. B 62, 6721 (2000). See also J. Zaanen, Physica C 217, 317 (1999), S. Kivelson, E. Fradkin and V. Emery, Nature 393, 550 (1998), S. White and D. Scalapino, Phys. Rev. Lett. 80, 1272 (1998).

56 Effect of magnetic field on SDW+SC to SC transition Main results (extreme Type II superconductivity) T=0 Neutron scattering observation of SDW order enhanced by superflow. Neutron scattering observation of SDW fluctuations enhanced by superflow. STM observation of CDW fluctuations enhanced by superflow and pinned by vortex cores. Prospects for studying quantum critical point between SC and SC+SDW phases by tuning H?

57 Further neighbor magnetic couplings La 2CuO4 Conclusion: Framework for spin/charge order in cuprate superconductors #! S 0 Experiments Magnetic order #! S = 0 T=0 Confined, paramagnetic Mott insulator has 1. Stable S=1 spin exciton. φ α 2. Broken translational symmetry:- bondcentered charge order. 3. S=1/2 moments near non-magnetic impurities Concentration of mobile carriers δ 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