Quantum criticality in the cuprate superconductors. Talk online: sachdev.physics.harvard.edu

Transcription

1 Quantum criticality in the cuprate superconductors Talk online: sachdev.physics.harvard.edu

2 The cuprate superconductors

3 Destruction of Neel order in the cuprates by electron doping, R. K. Kaul, M. Metlitksi, S. Sachdev, and C. Xu, Physical Review B 78, (2008). Paired electron pockets in the underdoped cuprates, V. Galitski and S. Sachdev, Physical Review B 79, (2009). Competition between spin density wave order and superconductivity in the underdoped cuprates, Eun Gook Moon and S. Sachdev, arxiv: HARVARD

4 1. Quantum criticality Outline Coupled dimer antiferromagnets vs. the cuprate superconductors 2. Fermi surfaces in the hole-doped cuprates Observations of quantum oscillations 3. Superconductivity 4. Competition between spin-density-wave order and superconductivity: phenomenological theory 5. Electronic theory of superconductivity and its competition with spin-density-wave order

5 Quantum criticality: coupled dimer antiferromagets vs. the cuprate supercondcutors

6 Square lattice antiferromagnet H = ij J ij Si S j Ground state has long-range Néel order Order parameter is a single vector field ϕ = η isi η i = ±1 on two sublattices ϕ 0 in Néel state.

7 Square lattice antiferromagnet H = ij J ij Si S j J J/λ Weaken some bonds to induce spin entanglement in a new quantum phase

8 Square lattice antiferromagnet H = ij J ij Si S j J J/λ Ground state is a quantum paramagnet with spins locked in valence bond singlets = 1 ( ) 2

9 λ c λ

10 Quantum critical Classical spin waves Dilute triplon gas Neel order

11 The cuprate superconductors

12 R. A. Cooper, Y. Wang, B. Vignolle, O. J. Lipscombe, S. M. Hayden, Y. Tanabe, T. Adachi, Y. Koike, M. Nohara, H. Takagi, Cyril Proust, N. E. Hussey, Science, 323, 603 (2009).

13 Crossovers in transport properties of hole-doped cuprates T* T coh? T (K) A F M (T) S-shaped upturns in (T) ~T d-wave SC Hole doping x ~T+ T 2 or ~T n (1 < n < 2) ~T 2 T FL? N. E. Hussey, J. Phys: Condens. Matter 20, (2008)

14 Crossovers in transport properties of hole-doped cuprates Strange metal T (K) A F M d-wave SC Hole doping x x m Strange metal: quantum criticality of optimal doping critical point at x = x m?

15 Only candidate quantum critical point observed at low T Strange metal T (K) x s A F M d-wave SC Hole doping x Spin and charge density wave order present below a quantum critical point at x = x s with x s 0.12 in the La series of cuprates

16 Fermi surfaces in the hole-doped cuprates

17 Γ Γ

18 Fermi surfaces in electron- and hole-doped cuprates Γ Hole states occupied Electron states occupied Effective Hamiltonian for quasiparticles: Γ H 0 = i<j t ij c iα c iα k ε k c kα c kα with t ij non-zero for first, second and third neighbor, leads to satisfactory agreement with experiments. The area of the occupied electron states, A e, from Luttinger s theory is A e = { 2π 2 (1 p) for hole-doping p 2π 2 (1 + x) for electron-doping x The area of the occupied hole states, A h, which form a closed Fermi surface and so appear in quantum oscillation experiments is A h =4π 2 A e.

19 Spin density wave theory In the presence of spin density wave order, ϕ at wavevector K = (π, π), we have an additional term which mixes electron states with momentum separated by K H sdw = ϕ c k,α σ αβ c k+k,β k,α,β where σ are the Pauli matrices. The electron dispersions obtained by diagonalizing H 0 + H sdw for ϕ (0, 0, 1) are E k± = ε k + ε k+k 2 ± (εk ε k+k 2 ) + ϕ 2 This leads to the Fermi surfaces shown in the following slides for hole doping.

20 Spin density wave theory in hole-doped cuprates Increasing SDW order Γ S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

21 Spin density wave theory in hole-doped cuprates Increasing SDW order Γ Γ S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

22 Spin density wave theory in hole-doped cuprates Increasing SDW order Electron pockets Γ Γ Γ Hole pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

23 Spin density wave theory in hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

24 Quantum critical Classical spin waves Dilute triplon gas Neel order

25 Theory of quantum criticality in the cuprates Small Fermi pockets Strange Metal Large Fermi surface Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = x m

26 Quantum oscillations Quantum oscillations and the Fermi surface in an underdoped high T c superconductor (ortho-ii ordered YBa 2 Cu 3 O 6.5 ). The period corresponds to a carrier density N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J.-B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy, and L. Taillefer, Nature 447, 565 (2007)

27 Quantum oscillations Onsager-Lifshitz relation: Frequency of oscillations in 1/H = hc 2e (Area of Fermi surface) 2π 2 Luttinger relation: Area of Fermi surface = 4π 3 (density of fermions)

28 Quantum oscillations Nature 450, 533 (2007)

29 Quantum oscillations Nature 450, 533 (2007) Quantum oscillations can be explained by SDW theory, but electron pockets are not seen in photoemission. Clear hole pockets are not seen either.

30 Superconductivity in hole-doped cuprates

31 Γ Γ δ

32 Overdoped SC State: Momentum-dependent Pair Energy Gap Shen et al PRL 70, 3999 (1993) Ding et al PRB (1996) Mesot et al PRL (1999) The SC energy gap has four nodes.

33 Theory of quantum criticality in the cuprates Small Fermi pockets Strange Metal Large Fermi surface Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = x m

34 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Onset of d-wave superconductivity hides the critical point x = x m

35 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

36 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

37 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave superconductor Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

38 Competition between SDW order and superconductivity: phenomenological theory

39 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave (SDW) order Write down a Landau-Ginzburg action for the quantum fluctuations of the SDW order ( ϕ ) and superconductivity (ψ): [ S = d 2 1 rdτ 2 ( τ ϕ ) 2 + c2 2 ( x ϕ ) 2 + r 2 ϕ 2 + u ( ϕ 2 ) 2 4 ] + κ ϕ 2 ψ 2 + d 2 r [ ( x i(2e/ c)a)ψ 2 ψ 2 + ψ 4 2 where κ > 0 is the repulsion between the two order parameters, and A = H is the applied magnetic field. E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). see also E. Demler, W. Hanke, and S.-C. Zhang, Rev. Mod. Phys. 76, 909 (2004) ]

40 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave (SDW) order Write down a Landau-Ginzburg action for the quantum fluctuations of the SDW order ( ϕ ) and superconductivity (ψ): [ S = d 2 1 rdτ 2 ( τ ϕ ) 2 + c2 2 ( x ϕ ) 2 + r 2 ϕ 2 + u ( ϕ 2 ) 2 4 ] + κ ϕ 2 ψ 2 + d 2 r [ ( x i(2e/ c)a)ψ 2 ψ 2 + ψ 4 2 where κ > 0 is the repulsion between the two order parameters, and A = H is the applied magnetic field. E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). see also E. Demler, W. Hanke, and S.-C. Zhang, Rev. Mod. Phys. 76, 909 (2004) ]

41 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave (SDW) order Write down a Landau-Ginzburg action for the quantum fluctuations of the SDW order ( ϕ ) and superconductivity (ψ): [ S = d 2 1 rdτ 2 ( τ ϕ ) 2 + c2 2 ( x ϕ ) 2 + r 2 ϕ 2 + u ( ϕ 2 ) 2 4 ] + κ ϕ 2 ψ 2 + d 2 r [ ( x i(2e/ c)a)ψ 2 ψ 2 + ψ 4 2 where κ > 0 is the repulsion between the two order parameters, and A = H is the applied magnetic field. E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). see also E. Demler, W. Hanke, and S.-C. Zhang, Rev. Mod. Phys. 76, 909 (2004) ]

42 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave (SDW) order C "Normal" H H c2 M D SDW SC+ SDW H sdw SC B A r c r cm r x E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

43 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave (SDW) order C "Normal" H H c2 M D SDW SC+ SDW H sdw SC B A r c x s r cm x m r x SDW order is more stable in the metal than in the superconductor: x m >x s. E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

44 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave (SDW) order C "Normal" H H c2 M D SDW SC+ SDW H sdw SC B A r c x s r cm x m r x SDW order is more stable in the metal than in the superconductor: x m >x s. E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

45 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave (SDW) order C "Normal" H H c2 M D SDW SC+ SDW H sdw SC B A r c x s r cm x m r x For doping with x s < x < x m, SDW order appears at a quantum phase transition at H = H sdw > 0. E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

46 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave (SDW) order C "Normal" H H c2 M D SDW SC+ SDW H sdw SC B A r c x s r cm x m r x Neutron scattering on La1.9Sr0.1CuO4 B. Lake et al., Nature 415, 299 (2002) E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

47 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave (SDW) order C "Normal" H H c2 M D SDW SC+ SDW H sdw SC B A r c x s r cm x m r x Neutron scattering on La1.855Sr0.145CuO4 J. Chang et al., Phys. Rev. Lett. 102, (2009). E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

48 J. Chang, N. B. Christensen, Ch. Niedermayer, K. Lefmann, H. M. Roennow, D. F. McMorrow, A. Schneidewind, P. Link, A. Hiess, M. Boehm, R. Mottl, S. Pailhes, N. Momono, M. Oda, M. Ido, and J. Mesot, Phys. Rev. Lett. 102, (2009). J. Chang, Ch. Niedermayer, R. Gilardi, N.B. Christensen, H.M. Ronnow, D.F. McMorrow, M. Ay, J. Stahn, O. Sobolev, A. Hiess, S. Pailhes, C. Baines, N. Momono, M. Oda, M. Ido, and J. Mesot, Physical Review B 78, (2008).

49 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave (SDW) order C "Normal" H H c2 M D SDW SC+ SDW H sdw SC B A r c x s r cm x m r x Neutron scattering on YBa2Cu3O6.45 D. Haug et al., arxiv: E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

50 Intensity (counts / ~10 min) (a) H = 0T Q H (r.l.u.) (c) I(14.9T,2K)! I(0T,2K) Q H (r.l.u.) Intensity (counts / ~10 min) Intensity (counts / ~10 min) (b) H = 14.9T Q H (r.l.u.) (d) Q H Q K H = 14.9T 2K 50K Q K (r.l.u.) Intensity (counts / ~1 min) D. Haug, V. Hinkov, A. Suchaneck, D. S. Inosov, N. B. Christensen, Ch. Niedermayer, P. Bourges, Y. Sidis, J. T. Park, A. Ivanov, C. T. Lin, J. Mesot, and B. Keimer, arxiv:

51 Phenomenological quantum theory of competing orders Competition between superconductivity (SC) and spin-density wave (SDW) order C "Normal" H H c2 M D SDW SC+ SDW H sdw SC B A r c x s r cm x m r x Quantum oscillations without Zeeman splitting N. Doiron-Leyraud, C. Proust, D. LeBoeuf, J. Levallois, J.-B. Bonnemaison, R. Liang, D. A. Bonn, W. N. Hardy, and L. Taillefer, Nature 447, 565 (2007)

52 Electronic theory of superconductivity and its competition with spin-density wave order

53 Theory of quantum criticality in the cuprates Small Fermi pockets Strange Metal Large Fermi surface Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = x m

54 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Onset of d-wave superconductivity hides the critical point at x = x m and moves it to x = x s <x m

55 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Theory of the onset of d-wave superconductivity from a large Fermi surface

56 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Theory of the onset of d-wave superconductivity from a large Fermi surface

57 Spin-fluctuation exchange theory of d-wave superconductivity in the cuprates Increasing SDW order Γ Γ Γ ϕ Γ Fermions at the large Fermi surface exchange fluctuations of the SDW order parameter ϕ. David Pines, Douglas Scalapino

58 Pairing by SDW fluctuation exchange We now allow the SDW field ϕ to be dynamical, coupling to electrons as H sdw = ϕ q c k,α σ αβc k+k+q,β. k,q,α,β Exchange of a ϕ quantum leads to the effective interaction H ee = 1 2 q p,γ,δ k,α,β where the pairing interaction is V αβ,γδ (q)c k,α c k+q,βc p,γc p q,δ, V αβ,γδ (q) = σ αβ σ γδ χ 0 ξ 2 +(q K) 2, with χ 0 ξ 2 the SDW susceptibility and ξ the SDW correlation length.

59 d-wave pairing of the large Fermi surface K ϕ _ + Γ _ + c k c k k = 0 (cos(k x ) cos(k y ))

60 Approaching the onset of antiferromagnetism in the spin-fluctuation theory Ar. Abanov, A. V. Chubukov and J. Schmalian, Advances in Physics 52, 119 (2003).

61 Approaching the onset of antiferromagnetism in the spin-fluctuation theory T c increases upon approaching the SDW transition. SDW and SC orders do not compete, but attract each other. No simple mechanism for nodal-anti-nodal dichotomy. Ar. Abanov, A. V. Chubukov and J. Schmalian, Advances in Physics 52, 119 (2003).

62 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Onset of d-wave superconductivity hides the critical point at x = x m and moves it to x = x s <x m

63 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Theory of the onset of d-wave superconductivity from small Fermi pockets

64 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Theory of the onset of d-wave superconductivity from small Fermi pockets

65 Fermi pockets in hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Begin with SDW ordered state, and focus on fluctuations in the orientation of ϕ, by using a unit-length bosonic spinor z α ϕ = z α σ αβ z β

66 ncreasing SDW Electron pockets Hole Γ pockets Charge carriers in the lightly-doped cuprates with Neel order

67 ncreasing SDW Electron operator c 1α Γ For a spacetime dependent SDW order, ϕ = z α σ αβ z β, ( c1 c 1 ) = R z ( g+ g ) ; R z ( z z z z ). So g ± are the up/down electron operators in a rotating reference frame defined by the local SDW order

68 ncreasing SDW Electron operator c 2α Γ For a spacetime dependent SDW order, ϕ = z α σ αβ z β, ( c2 c 2 ) = R z ( g+ g ) ; R z ( z z z z ). Same SU(2) matrix also rotates electrons in second pocket.

69 Low energy theory for spinless, charge e fermions g ±, and spinful, charge 0 bosons z α : L = L z + L g L z = 1 [ ] ( τ ia τ )z α 2 + v 2 ia)z α 2 + iλ( z α 2 1) t + Berry phases of monopoles in A µ. CP 1 field theory for z α and an emergent U(1) gauge field A µ. Coupling t tunes the strength of SDW orientation fluctuations. L g = g + [ ( τ ia τ ) 1 + g ] 2m ( ia)2 µ Two Fermi surfaces coupled to the emergent U(1) gauge field A µ with opposite charges g + [ ( τ + ia τ ) 1 2m ( + ia)2 µ ] g

70 ncreasing SDW _ f ±v + + Γ _ g ± d-wave pairing of the electrons is associated with Strong s-wave pairing of g ± Weak p-wave pairing of f ±v.

71 Conclusions

72 Crossovers in transport properties of hole-doped cuprates T* T coh? T (K) A F M (T) S-shaped upturns in (T) ~T d-wave SC Hole doping x ~T+ T 2 or ~T n (1 < n < 2) ~T 2 T FL? N. E. Hussey, J. Phys: Condens. Matter 20, (2008)

73 Crossovers in transport properties of hole-doped cuprates Strange metal T (K) A F M d-wave SC Hole doping x x m Strange metal: quantum criticality of optimal doping critical point at x = x m?

74 Only candidate quantum critical point observed at low T Strange metal T (K) x s A F M d-wave SC Hole doping x Spin and charge density wave order present below a quantum critical point at x = x s with x s 0.12 in the La series of cuprates

75 Theory of quantum criticality in the cuprates Small Fermi pockets Strange Metal Large Fermi surface Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = x m

76 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

77 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

78 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Large Fermi surface Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave superconductor Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

79 Conclusions Gauge theory for pairing in the underdoped cuprates, describing angular fluctuations of spin-density-wave order Natural route to d-wave pairing with strong pairing at the antinodes and weak pairing at the nodes Explains characteristic competing order features of fielddoping phase diagram: SDW order is more stable in the metal than in the superconductor.