Quantum phase transitions of correlated electrons in two dimensions

Transcription

1 Quantum phase transitions of correlated electrons in two dimensions Subir Sachdev Science 86, 479 (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 in one dimension S=1/ quantum XY model. Berry phases in two dimensions Bond-centered charge ( spin-peierls ) order. Magnetic transitions in d-wave superconductors Effect of an applied magnetic field V. Transitions between BCS superconductors with distinct internal Cooper pair wavefunctions. A. Quantum field theory B. Photoemission and tunnelling experiments on the high temperature superconductors VI. Conclusions

3 I.A Coupled Ladder Antiferromagnet N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 459 (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/ spins on coupled -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 4 Ground state has long-range magnetic (Neel) order! S i = ( ) i 1 + x iy N 0 0 Excitations: spin waves k cx kx cy ky ε = +

5 λ close to 0 Weakly coupled ladders 1 = ( ) 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 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 ( ) ( = 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 ( ) ( Z = D n x, τ δ n 1) exp is ηjaτ ( xj, τ) dτ j 1 ( ( ) ( ) ) d d x τ n + c xn g η =± 1 identifies sublattices j

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

9 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 in one dimension S=1/ quantum XY model. Berry phases in two dimensions Bond-centered charge ( spin-peierls ) order. Magnetic transitions in d-wave superconductors Effect of an applied magnetic field V. Transitions between BCS superconductors with distinct internal Cooper pair wavefunctions. A. Quantum field theory B. Photoemission and tunnelling experiments on the high temperature superconductors VI. Conclusions

10 II. Quantum XY model in one dimension S. Sachdev and K. Park, cond-mat/ Write H XY = J 1 (S xj S x,j+1 + S yj S y,j+1 + λs zj S z,j+1 ) j + J (S xj S x,j+ + S yj S y,j+ + λs zj S z,j+ ) j n j = (cos(θ j ), sin(θ j ), 0) where j is now a discrete spacetime index upon a square lattice. Structure of Berry phase terms. S j η j A jτ = S j l j ɛ µν µ A jν For XY model, Sɛ µν µ A jν =(πs) vortex number Vortices in odd columns carry a factor ( 1) S. where l j =0(l j =1)on even (odd) columns.

11 S integer. Z XY = j dθ j exp 1 g cos( µ θ j ) j where µ = x, τ. This is the action for a classical XY model in D =. Displays Kosterlitz-Thouless transition. Dual height model for KT transition: Z XY = {m jµ } j dθ j exp 1 g j ( µ θ j πm jµ ) This is the Villain periodic Gaussian form. Poisson summation leads to Z XY = exp g {p j} j ( µ p j ) Height (or roughening) model on the square lattice All heights are integers.

12 For XY model, the A flux measures vortex number. In the periodic Gaussian formulation Sɛ µν µ A jν = πɛ µν µ m jν So with Berry phases partition function of S =1/ quantum XY model is Z XY = {m jµ } j dθ j exp 1 g j ( µ θ j πm jµ ) + iπ j l j ɛ µν µ m jν Vortices in odd columns contribute a factor ( 1) to the partition function Weights are not positive. Dual height model for Z XY Z XY = exp g {p j} j ( µ p j 1 ) µl j Height model in which the heights are integers (half-integers) on even (odd) columns.

13 Phases of height model Rough interface Tomonaga-Luttinger liquid with power-law spin correlations: n ix n jx = n iy n jy n iz n jz 1 i j g/(π) 1 i j π/g Smooth interface All correlations decay exponentially there is a gap to all excitations p j l j / has a definite value: any such definite value breaks a discrete symmetry of the Hamiltonian. p j l j / =0, 1/ (plus integer) are states with bond-centered charge order i.e. neighboring links have valence bonds with distinct probabilities p j l j / =1/4, 3/4 (plus integer) are the two states with Ising antiferromagnetic order.

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

15 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 in one dimension S=1/ quantum XY model. Berry phases in two dimensions Bond-centered charge ( spin-peierls ) order. Magnetic transitions in d-wave superconductors Effect of an applied magnetic field V. Transitions between BCS superconductors with distinct internal Cooper pair wavefunctions. A. Quantum field theory B. Photoemission and tunnelling experiments on the high temperature superconductors VI. Conclusions

16 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 (( ) ( ) ) ( ) x α τ α α S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 344 (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. 6, 1694 (1989).

17 H = J S! S! i< j ij i j Square lattice with first (J 1 ) and second (J ) neighbor exchange interactions 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 state Bond-centered charge order S. Sachdev and K. Park, cond-mat/ Co-existence? 0.4 J / J 1 1 = ( )

18 Quantum dimer model D. Rokhsar and S. Kivelson Phys. Rev. Lett. 61, 376 (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 4, 4568 (1990).

19 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.

20 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).

21 Field theory of bond order Discretize coherent state path integral on a cubic lattice in spacetime: Z = ( ) dn j δ(n j 1) exp 1 n j n j+ˆµ i η j A jτ g j j,µ j where µ = x, y, τ, and we assume henceforth that S =1/. For large g, perform a high temperature expansion to obtain an effective action for the A jµ. This action must be invariant under the gauge transformation A jµ A jµ µ γ j associated with the change in choice of n 0 (γ j is the oriented area of the spherical triangle formed by n j and the two choices for n 0 ). Also, it should be invariant under A jµ A jµ +4π because area of triangle is uncertain modulo 4π. Simplest large g effective model Z = ( 1 ( ) ) 1 da jµ exp cos e ɛ µνλ ν A jλ i η j A jτ j j with e g. This is compact QED in +1 dimensions with Berry phases.

22 Exact duality transform on periodic Gaussian ( Villain ) action for compact QED yields Z = ( ) exp e ( µ h j µ X j ) {h j} with h j integer. Height model in +1 dimensions with offsets X j =0, 1/4, 1/, 3/4 onthe four dual sublattices. j

23 For large e, low energy height configurations are in exact one-toone correspondence with dimer coverings of the square lattice +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.

24 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 in one dimension S=1/ quantum XY model. Berry phases in two dimensions Bond-centered charge ( spin-peierls ) order. Magnetic transitions in d-wave superconductors Effect of an applied magnetic field V. Transitions between BCS superconductors with distinct internal Cooper pair wavefunctions. A. Quantum field theory B. Photoemission and tunnelling experiments on the high temperature superconductors VI. Conclusions

25 IV. Magnetic transitions in d-wave superconductors 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.

26 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

27 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. 87, 0670 (001).

28 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!

29 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

30 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).

31 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.

32 ( 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

33 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

34 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

35 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)

36 Neutron scattering off La Sr CuO -x x 4 ( x = 0.163, SC phase) in H=0 ( red dots) and H=7.5T ( blue dots). B. Lake, G. Aeppli et al., Science 91, 1759 ( 001) Elastic neutron scattering off La Sr CuO -x x 4 ( x = 0.10, SC + SDW phase) in H=0 (blue dots) and H=5T ( red do ts). B. Lake, H. Ronnow et al., cond-mat/010406

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

38 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 in one dimension S=1/ quantum XY model. Berry phases in two dimensions Bond-centered charge ( spin-peierls ) order. Magnetic transitions in d-wave superconductors Effect of an applied magnetic field V. Transitions between BCS superconductors with distinct internal Cooper pair wavefunctions. A. Quantum field theory B. Photoemission and tunnelling experiments on the high temperature superconductors VI. Conclusions

39 V. Quantum transitions between BCS superconductors From numerical/analytical/rg studies of the square lattice Hubbard model we know that ground state has d x y superconductivity near half filling C.J. Halboth and W. Metzner Phys. Rev. Lett. 85, 516 (000). d xy superconductivity for small electron density M.A. Baranov and M. Yu Kagan, Z. Phys. B 86, 37 (199). M.A. Baranov, A.V. Chubukov, and M. Yu Kagan, Int. J. Mod. Phys. B 6, 471 (199). We model this phenomenologically: H = ε k c kσ c kσ + J 1 S j S j+ˆµ + J S j S j+ˆν k j,µ j,ν where the sum on µ is over x, y, thatonν is over x + y, x y, S j 1 c jσ σ σσ c jσ with σ the Pauli matrices, and the dispersion ε k = t 1 (cos(k x ) + cos(k y )) t (cos(k x + k y ) + cos(k x k y )) µ

40 BCS mean-field theory We choose H BCS = k ε k c kσ c kσ J 1 J j,µ j,ν µ (c j c j+ˆµ, c j c j+ˆµ, )+h.c. ν (c j c j+ˆν, c j c j+ˆν, )+h.c. x = y x y x+y = x y xy The complex numbers x y and xy are to be determined by minimizing the ground state energy per site E BCS = J 1 x y + J xy where the fermionic quasiparticle dispersion is d k 4π [E k ε k ] E k = [ ε k + J 1 x y (cos k x cos k y )+J xy sin k x sin k y ] 1/ The energy only depends upon the relative phase between x y and xy.

41 Evolution of ground state x 0 xy 0 x y xy = 0 ( ) ( ) π arg xy = arg ± x y d x y superconductor d + id superconductor x y y 0 xy = x xy d xy 0 y 0 superconductor J / J 1 BCS theory fails near quantum critical points

42 Field theory for transition from d to d id superconductivity + x y x y xy y x 1 Gapless Fermi Points in a d-wave superconductor at wavevectors ( ±K, ± K) K=0.391π 3 4 Ψ 1 = f f f f3 Ψ = f f f 4 f4 dk ( π ) + ( π ) n ( z x ω ) n F xτ yτ S = T Ψ i + v k + v k Ψ Ψ ω 1 1 dk ( ) T z x Ψ i ωn + v F k yτ + v k xτ Ψ ω n

43 Ising order parameter for transition φ ~ i Coupling to low energy fermions ( ) iλ d xdτ φ f f f f f f f f ( y y Ψ ) 1 Ψ1 Ψ Ψ = d xdτ λφ τ τ Action for low energy fluctuations near critical point S 1 c s u = d xd τ ( τφ) ( φ) φ φ xy ( z x) τ ivf xτ iv yτ + Ψ + + Ψ 1 1 h.c. ( z x) τ ivf yτ iv xτ + Ψ + + Ψ ( y y ) + λφ Ψ1τ Ψ1 Ψτ Ψ

44 {For v = v terms with fermions F = ( ) Ψγ Ψ +Ψ γ Ψ + λφ Ψ Ψ Ψ Ψ 1 µ µ 1 µ µ 1 1 Chiral symmetry breaking in the Higgs-Yukawa model} Momentum shell renormalization group equations dλ (3 d) 3 = λ C1λ d" du 4 = (3 du ) Cu + C3λ C4λ u d" where C are functions of v / v ; similar flow equation f or v F 1 4 F / v. Critical point is controlled by fixed point u = u λ = λ v v * *,, F =, which obeys hyperscaling, and is Lorentz invariant.

45 Finite temperature crossovers near a quantum critical point: Quantum Ising Chain ( x z z gσ + σ σ ) H = I J i i i+ 1 i Quasiclassical dynamics Quasiclassical dynamics i χω ( ) = dt σ j () t, σk ( 0) e = T k 7/4 $ 0 A ( 1 iω / Γ +...) π kt B Γ R = tan 16 $ z z iωt S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 411 (199). S. Sachdev and A.P. Young, Phys. Rev. Lett. 78, 0 (1997). R σσ z j z k = = ~ j k P. Pfeuty Annals of Physics, 57, 79 (1970) ( + ) ( ) 1/4

46 In quasiparticle regimes of Ising chain, spin relaxation rate ~ In a Fermi liquid, quasiparticle relaxation rate ~ T In a BCS d-wave superconductor, quasiparticle relaxation rate ~ T e / T 3 In quantum critical region associated with a second-order phase transition with hyperscaling properties Relaxation dynamics with rate ~ kt B / $ χ ω C = η Φ T ( ) $ ω kbt

47 Crossovers near transition in d-wave superconductor T Superconducting T c d x y superconductivity Quantum critical s c d x + y superconductivity s id xy S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 411 (199). M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (000).

48 Damping of Nodal Quasiparticles Photoemission on BSSCO (Valla et al Science 85, 110 (1999)) Im Σ ~ kt B

49 Yoram Dagan and Guy Deutscher, cond-mat/ Observations of splitting of the ZBCP Spontaneous splitting (zero field) Magnetic field splitting Covington, M. et al. Observation of Surface-Induced Broken Time-Reversal Symmetry in YBa Cu 3 O 7-δ Tunnel Junctions, Phys. Rev. Lett. 79, (1997)

50 Zero Field splitting and χ -1 versus [ max - ] 1/ All YBCO samples g [mv] /kt c /kt c

51 Conclusions I: Phase transitions of BCS superconductors Examined general theory of all possible candidates for zero momentum, spin-singlet order parameters which can induce a second-order quantum phase transitions in a d-wave superconductor Only cases (A) d d + is x y x y (B) d d + id x y x y pairing and pairing have renormalization group fixed points with a non-zero interaction strength between the bosonic order parameter mode and the nodal fermions, and so are candidates for producing damping ~ k B T of nodal fermions. xy Independent evidence for (B) from tunneling experiments.

52 Further neighbor magnetic couplings La CuO4 Conclusions II: 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. Broken translational symmetry:- bondcentered charge order. 3. S=1/ 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