Theory of the competition between spin density waves and d-wave superconductivity in the underdoped cuprates

Transcription

1 HARVARD Theory of the competition between spin density waves and d-wave superconductivity in the underdoped cuprates Talk online: sachdev.physics.harvard.edu

2 HARVARD Where is the quantum critical point in the cuprate superconductors? arxiv: Talk online: sachdev.physics.harvard.edu

3 Hole dynamics in an antiferromagnet across a deconfined quantum critical point, R. K. Kaul, A. Kolezhuk, M. Levin, S. Sachdev, and T. Senthil, Phys. Rev. B 75, (2007). Algebraic charge liquids R. K. Kaul, Yong-Baek Kim, S. Sachdev, and T. Senthil, Nature Physics 4, 28 (2008). Destruction of Neel order in the cuprates by electron doping, R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke 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, Physical Review B 80, (2009). Fluctuating spin density waves in metals S. Sachdev, M. Metlitski, Yang Qi, and Cenke Xu arxiv: HARVARD

4 The cuprate superconductors

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

6 Canonical quantum critical phase diagram of coupled-dimer antiferromagnet Classical spin waves Quantum critical Dilute triplon gas S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). Neel order Pressure in TlCuCl3 Christian Ruegg et al., Phys. Rev. Lett. 100, (2008)

7 Crossovers in transport properties of hole-doped cuprates Strange metal T (K) S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). C. M. Varma, Phys. Rev. Lett. 83, 3538 (1999). A F M d-wave SC Hole doping x x m Strange metal: quantum criticality of optimal doping critical point at x = x m?

8 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 density wave order present below a quantum critical point at x = x s with x s 0.12 in the La series of cuprates

9 Γ Γ

10 Theory of quantum criticality in the cuprates Fluctuating Fermi pockets Strange Metal Large Fermi surface R. Daou et al., Nature Physics 5, (2009) Spin density wave () Underlying ordering quantum critical point in metal at x = x m

11 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.

12 Spin density wave theory A spin density wave () is the spontaneous appearance of an oscillatory spin polarization. The electron spin polarization is written as S(r, τ) = ϕ (r, τ)e ik r where ϕ is the order parameter, and K is a fixed ordering wavevector. For simplicity we will consider the case of K =(π, π), but our treatment applies to general K.

13 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 αβc 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 electron and hole doping.

14 Spin density wave theory in hole-doped cuprates Increasing 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).

15 Spin density wave theory in hole-doped cuprates Increasing 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).

16 Spin density wave theory in hole-doped cuprates Increasing 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).

17 Spin density wave theory in hole-doped cuprates Increasing order Γ Γ Γ Γ Hole pockets order parameter is a vector, ϕ, whose amplitude vanishes at the transition to the Fermi liquid. 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).

18 Spin density wave theory in hole-doped cuprates Γ Γ Incommensurate order in YBa2Cu3O6+x A. J. Millis and M. R. Norman, Physical Review B 76, (2007). N. Harrison, Physical Review Letters 102, (2009).

19 Evidence for connection between linear resistivity and stripe-ordering in a cuprate with a low Tc Magnetic field of upto 35 T used to suppress superconductivity Linear temperature dependence of resistivity and change in the Fermi surface at the pseudogap critical point of a high-tc superconductor R. Daou, Nicolas Doiron-Leyraud, David LeBoeuf, S. Y. Li, Francis Laliberté, Olivier Cyr-Choinière, Y. J. Jo, L. Balicas, J.-Q. Yan, J.-S. Zhou, J. B. Goodenough & Louis Taillefer, Nature Physics 5, (2009)

20 Theory of quantum criticality in the cuprates Fluctuating Fermi pockets Strange Metal Large Fermi surface R. Daou et al., Nature Physics 5, (2009) Spin density wave () Underlying ordering quantum critical point in metal at x = x m

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

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

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

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

25 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange 3. Superconductivity in the underdoped regime U(1) gauge theory of fluctuating order

26 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange 3. Superconductivity in the underdoped regime U(1) gauge theory of fluctuating order

27 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order Write down a Landau-Ginzburg action for the quantum fluctuations of the 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); S. A. Kivelson, D.-H. Lee, E. Fradkin, and V. Oganesyan, Phys. Rev. B 66, (2002). ]

28 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order Write down a Landau-Ginzburg action for the quantum fluctuations of the 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); S. A. Kivelson, D.-H. Lee, E. Fradkin, and V. Oganesyan, Phys. Rev. B 66, (2002). ]

29 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order Write down a Landau-Ginzburg action for the quantum fluctuations of the 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); S. A. Kivelson, D.-H. Lee, E. Fradkin, and V. Oganesyan, Phys. Rev. B 66, (2002). ]

30 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

31 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC 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).

32 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC 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).

33 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC For doping with x s < x < x m, order appears at a quantum phase transition at H = H sdw > 0. E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

34 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC 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).

35 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).

36 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC Neutron scattering on YBa2Cu3O6.45 D. Haug et al., Phys. Rev. Lett. 103, (2009). E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

37 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 / ~12 min) H = 0T (a) 100 H = 14.9T 80 T = 2K Q H (r.l.u.) 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) Intensity (a.u.) (b) H = 0T H = 13.5T T = 2K Energy E (mev) 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, Phys. Rev. Lett. 103, (2009)

38 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC 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). S. E. Sebastian, N. Harrison, C. H. Mielke, Ruixing Liang, D. A. Bonn, W.~N.~Hardy, and G. G. Lonzarich, arxiv:

39 Quantum oscillations Nature 450, 533 (2007)

40 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC Change in frequency of quantum oscillations in electron-doped materials identifies xm = 0.165

41 Increasing order Nd 2 x Ce x CuO 4 T. Helm, M. V. Kartsovni, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J. Wosnitza, R. Gross, arxiv:

42 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC Neutron scattering at H=0 in same material identifies xs = 0.14 < xm

43 Nd 2 x Ce x CuO 4 Spin correlation length!/a x=0.038 x=0.075 x=0.106 x=0.129 x=0.134 x=0.145 x=0.150 x= Temperature (K) E. M. Motoyama, G. Yu, I. M. Vishik, O. P. Vajk, P. K. Mang, and M. Greven, Nature 445, 186 (2007).

44 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC Experiments on Nd 2 x Ce x CuO 4 show that at low fields x s =0.14, while at high fields x m =0.165.

45 H "Normal" (Large Fermi surface) (Small Fermi pockets) H c2 M SC+ H sdw SC

46 SC+ SC (Small Fermi pockets) H M "Normal" (Large Fermi surface)

47 Fluctuating, paired Fermi pockets SC+ SC (Small Fermi pockets) H M "Normal" (Large Fermi surface)

48 T Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Thermally fluctuating Magnetic quantum criticality Spin gap d-wave SC SC+ SC (Small Fermi pockets) H M "Normal" (Large Fermi surface)

49 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange 3. Superconductivity in the underdoped regime U(1) gauge theory of fluctuating order

50 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange 3. Superconductivity in the underdoped regime U(1) gauge theory of fluctuating order

51 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Thermally fluctuating Magnetic quantum criticality Spin gap d-wave SC SC+ SC (Small Fermi pockets) H M "Normal" (Large Fermi surface)

52 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Theory of the onset of d-wave superconductivity from a large Fermi surface Thermally fluctuating H Magnetic quantum criticality Spin gap SC+ (Small Fermi pockets) d-wave SC M SC "Normal" (Large Fermi surface)

53 Spin-fluctuation exchange theory of d-wave superconductivity in the cuprates Increasing order Γ Γ Γ ϕ Γ Fermions at the large Fermi surface exchange fluctuations of the order parameter ϕ. D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)

54 d-wave pairing of the large Fermi surface K ϕ _ + Γ _ + c k c k k = 0 (cos(k x ) cos(k y )) D. J. Scalapino, E. Loh, and J. E. Hirsch, Phys. Rev. B 34, 8190 (1986) K. Miyake, S. Schmitt-Rink, and C. M. Varma, Phys. Rev. B 34, 6554 (1986)

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

56 Approaching the onset of antiferromagnetism in the spin-fluctuation theory T c increases upon approaching the transition. 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).

57 Approaching the onset of antiferromagnetism in the spin-fluctuation theory T c increases upon approaching the transition. 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).

58 Nodal/anti-nodal dichotomy in STM measurements E (0,π) (π,0) k y k x J.C. Davis and collaborators

59 STM in BSCCO A. Pushp, C. V. Parker, A. N. Pasupathy, K. K. Gomes, S. Ono, J. Wen, Z. Xu, G. Gu, and A. Yazdani, Science 324, 1689 (2009)

60 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange 3. Superconductivity in the underdoped regime U(1) gauge theory of fluctuating order

61 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Superconductivity in the overdoped regime BCS pairing by spin fluctuation exchange 3. Superconductivity in the underdoped regime U(1) gauge theory of fluctuating order

62 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Thermally fluctuating Magnetic quantum criticality Spin gap d-wave SC SC+ SC (Small Fermi pockets) H M "Normal" (Large Fermi surface)

63 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Theory of the onset of d-wave superconductivity from small Fermi pockets Thermally fluctuating (Small Fermi pockets) H Magnetic quantum criticality Spin gap SC+ d-wave SC M SC "Normal" (Large Fermi surface)

64 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Physics of competition: d-wave SC and eat up same pieces of the large Fermi surface. Thermally fluctuating (Small Fermi pockets) H Magnetic quantum criticality Spin gap SC+ d-wave SC M SC "Normal" (Large Fermi surface)

65 Theory of underdoped cuprates Increasing order Γ Γ Γ Γ Begin with ordered state, and rotate to a frame polarized along the local orientation of the order ˆ ϕ ) ) ( c c = R ( ψ+ ψ ; R ˆ ϕ σ R = σ z ; R R =1

66 Theory of underdoped cuprates ) With R = ( z z z z the theory is invariant under the U(1) gauge transformation z α e iθ z α ; ψ + e iθ ψ + ; ψ e iθ ψ and the order is given by ˆ ϕ = z α σ αβ z β

67 Theory of underdoped cuprates Starting from the -fermion model with Lagrangian L = k c kα ( τ ε k ) c kα E sdw c iα ˆ ϕ i σ αβ c iβ e ik r i + 1 2t i ( µ ˆ ϕ ) 2

68 Theory of underdoped cuprates we obtain a U(1) gauge theory of fermions ψ p with U(1) charge p = ±1 and pocket Fermi surfaces, relativistic complex scalars z α with charge 1, representing the orientational fluctuations of the order Shraiman-Siggia term L = k,p=± [ ψ kp ( τ ipa τ + ε k pa ) ψ kp ] E sdw ψ kp pψ k+k,p

69 Theory of underdoped cuprates we obtain a U(1) gauge theory of fermions ψ p with U(1) charge p = ±1 and pocket Fermi surfaces, relativistic complex scalars z α with charge 1, representing the orientational fluctuations of the order Shraiman-Siggia term + 1 t [ ] ( τ ia τ )z α 2 + v 2 ia)z α 2 + iλ( z α 2 1)

70 Theory of underdoped cuprates we obtain a U(1) gauge theory of fermions ψ p with U(1) charge p = ±1 and pocket Fermi surfaces, relativistic complex scalars z α with charge 1, representing the orientational fluctuations of the order Shraiman-Siggia term k,p,q z q p/2, z q+p/2, ( p ε(k) k ) ψ k+q, ψ k q,+ + c.c.

71 Normal state above T c Increasing f ±v Γ ψ ± = f ±v near (±π/2, ±π/2). These are spinless fermions, and are invisible to photoemission.

72 Normal state above T c Increasing Gauge forces lead to Γ bound states of f ±v and z α which form banana Fermi surfaces of spinful fermions visible to photoemission. R. K. Kaul, Yong-Baek Kim, S. Sachdev, and T. Senthil, Nature Physics 4, 28 (2008).

73 ncreasing Pairing across T c Electron c 2α, spinless fermion g ± Γ Electron c 1α, spinless fermion g ± Focus on pairing near (π, 0), (0, π), where ψ ± g ±, and the electron operators are ) ( ) ( ) ( ) g+ c2 g+ = R c z ; = R 1 g c z 2 g ( z z ) R z z z. ( c1

74 ncreasing Pairing across T c Electron c 2α, spinless fermion g ± Γ Electron c 1α, spinless fermion g ± Focus on pairing near (π, 0), (0, π), where ψ ± g ±, and the electron operators are ) ( ) ( ) ( ) g+ c2 g+ = R c z ; = R 1 g c z 2 g ( z z ) R z z z. ( c1

75 Fluctuating pocket theory for electrons near (0, π) and (π, 0) Attractive gauge forces lead to simple s-wave pairing of the g ± g + g = For the physical electron operators, this pairing implies c 1 c 1 = z α 2 c 2 c 2 = z α 2 i.e. d-wave pairing!

76 ncreasing Strong pairing of the g ± electron pockets _ + Γ _ + g ± g + g =

77 ncreasing Weak pairing of the f ± hole pockets _ f ±v + + Γ _ f +1 (k)f 1 ( k) (k x k y )J g + g ; f +2 (k)f 2 ( k) (k x + k y )J g + g ; f +1 (k)f 2 ( k) = 0,

78 Features of superconductivity d-wave superconductivity. Nodal-anti-nodal dichotomy: strong pairing near (π, 0), (0, π), and weak pairing near zone diagonals. T c decreases as spin correlation increases (competing order effect). Shift in quantum critical point of ordering: gauge fluctuations are stronger in the superconductor. After onset of superconductivity, monopoles condense and lead to confinement and nematic and/or valence bond solid (VBS) order. ncreasing _ + Γ _ +

79 Features of superconductivity d-wave superconductivity. Nodal-anti-nodal dichotomy: strong pairing near (π, 0), (0, π), and weak pairing near zone diagonals. T c decreases as spin correlation Strong increases (competing order effect). s-wave pairing Shift in quantum critical point of ordering: gauge fluctuations are stronger in the superconductor. Weak After onset of superconductivity, p-wave monopoles condense and lead to confinement pairingand nematic and/or valence bond solid (VBS) order. ncreasing _ + Γ _ +

80 Features of superconductivity d-wave superconductivity. Nodal-anti-nodal dichotomy: strong pairing near (π, 0), (0, π), and weak pairing near zone diagonals. T c decreases as spin correlation increases (competing order effect). Shift in quantum critical point of ordering: gauge fluctuations are stronger in the superconductor. After onset of superconductivity, monopoles condense and lead to confinement and nematic and/or valence bond solid (VBS) order.

81 Features of superconductivity d-wave superconductivity. Nodal-anti-nodal dichotomy: strong pairing near (π, 0), (0, π), and weak pairing near zone diagonals. T c decreases as spin correlation increases (competing order effect). Shift in quantum critical point of ordering: gauge fluctuations are stronger in the superconductor. After onset of superconductivity, monopoles condense and lead to confinement and possible valence bond solid (VBS) order.

82 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Thermally fluctuating Magnetic quantum criticality Spin gap d-wave SC SC+ SC (Small Fermi pockets) H M "Normal" (Large Fermi surface)

83 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Thermally fluctuating Magnetic quantum criticality Spin gap d-wave SC SC+ SC (Small Fermi pockets) H M "Normal" (Large Fermi surface)

84 Features of superconductivity d-wave superconductivity. Nodal-anti-nodal dichotomy: strong pairing near (π, 0), (0, π), and weak pairing near zone diagonals. T c decreases as spin correlation increases (competing order effect). Shift in quantum critical point of ordering: gauge fluctuations are stronger in the superconductor. After onset of superconductivity, monopoles condense and lead to confinement and nematic and/or valence bond solid (VBS) order.

85 Features of superconductivity d-wave superconductivity. Nodal-anti-nodal dichotomy: strong pairing near (π, 0), (0, π), and weak pairing near zone diagonals. T c decreases as spin correlation increases (competing order effect). Shift in quantum critical point of ordering: gauge fluctuations are stronger in the superconductor. After onset of superconductivity, monopoles condense and lead to confinement and nematic and/or valence bond solid (VBS) order. R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, (2008).

86 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Thermally fluctuating Magnetic quantum criticality Spin gap d-wave SC SC+ SC (Small Fermi pockets) H M "Normal" (Large Fermi surface)

87 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Thermally fluctuating Magnetic quantum criticality Spin gap d-wave SC SC+ SC (Small Fermi pockets) H M "Normal" (Large Fermi surface)

88 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface T sdw VBS and/or nematic d-wave SC SC+ SC R. K. Kaul, M. Metlitksi, S. Sachdev, and Cenke Xu, Physical Review B 78, (2008). (Small Fermi pockets) H M "Normal" (Large Fermi surface)

89 Include metal-insulator transition H insulator (Small Fermi pockets) "Normal" (Large Fermi surface) H c2 M SC+ H sdw VBS and/or nematic SC

90 Conclusions Theory of competition between spin density wave order and superconductivity: Spin density waves and d-wave superconductivity compete for the same portion of the Fermi surface: expressed by U(1) gauge theory of pocket fermions and CP 1 model

91 Conclusions Identified quantum criticality in cuprate superconductors with a critical point at optimal doping associated with onset of spin density wave order in a metal Elusive optimal doping quantum critical point has been hiding in plain sight. It is shifted to lower doping by the onset of superconductivity

92 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface T sdw VBS and/or nematic d-wave SC SC+ SC (Small Fermi pockets) H M "Normal" (Large Fermi surface)