The phase diagram of the cuprates and the quantum phase transitions of metals in two dimensions

Transcription

1 The phase diagram of the cuprates and the quantum phase transitions of metals in two dimensions Niels Bohr Institute, Copenhagen, May 6, 2010 Talk online: sachdev.physics.harvard.edu HARVARD

2 Max Metlitski, Harvard arxiv: Eun Gook Moon, Harvard Phys. Rev. B 80, (2009) HARVARD

3 Outline 1. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity 2. Theory of spin density wave ordering in a metal Strong-coupling in d=2 3. Instabilities near SDW critical point d-wave pairing and bond density wave

4 Outline 1. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity 2. Theory of spin density wave ordering in a metal Strong-coupling in d=2 3. Instabilities near SDW critical point d-wave pairing and bond density wave

5 Central ingredients in cuprate phase diagram: antiferromagnetism, superconductivity, and change in Fermi surface Γ Γ

6 d-wave Antiferromagnetism superconductivity Fermi surface

7 d-wave Antiferromagnetism superconductivity Fermi surface

8 Fermi surface+antiferromagnetism Γ Hole states occupied Electron states occupied + Γ The electron spin polarization obeys S(r, τ) = ϕ (r, τ)e ik r where K is the ordering wavevector.

9 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) 2π 2 (1 + x) for hole-doping p 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.

10 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 electron and hole doping.

11 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron 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).

12 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron 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).

13 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron pockets Hot spots 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).

14 Hole-doped cuprates Increasing SDW order Γ Hole pockets Γ Γ Γ Hole pockets Electron pockets Hot spots Fermi surface breaks up at hot spots into electron and 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).

15 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron pockets Hot spots Fermi surface breaks up at hot spots into electron and 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).

16 Evidence for small Fermi pockets Fermi liquid behaviour in an underdoped high Tc superconductor Suchitra E. Sebastian, N. Harrison, M. M. Altarawneh, Ruixing Liang, D. A. Bonn, W. N. Hardy, and G. G. Lonzarich arxiv:

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

18 d-wave Antiferromagnetism superconductivity Fermi surface

19 Spin density wave d-wave superconductivity Fermi surface

20 Spin density wave d-wave superconductivity Fermi surface

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

22 Theory of quantum criticality in the cuprates T * Fluctuating, Small Fermi paired pockets Fermi with pairing pockets 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

23 Theory of quantum criticality in the cuprates T * Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). d-wave superconductor E. G. Moon and S. Sachdev, Phy. Rev. B 80, (2009) Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

24 Theory of quantum criticality in the cuprates T * Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). d-wave superconductor E. G. Moon and S. Sachdev, Phy. Rev. B 80, (2009) Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

25 Theory of quantum criticality in the cuprates T * Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave superconductor E. G. Moon and S. Sachdev, Phy. Rev. B 80, (2009) Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

26 Theory of quantum criticality in the cuprates T * Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface V. Galitski and S. Sachdev, Phys. Rev. B 79, (2009). Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave superconductor E. G. Moon and S. Sachdev, Phys. Rev. B 80, (2009) Spin density wave (SDW) Physics of competition: d-wave SC and SDW eat up same pieces of the large Fermi surface.

27 T T * Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave SC

28 T T * Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface T sdw d-wave SC SC+ SDW SC SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

29 T T * Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface T sdw d-wave SC SC+ SDW SC Quantum oscillations SDW (Small Fermi pockets) M H "Normal" (Large Fermi surface)

30 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). 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).

31 T T * Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface T sdw d-wave SC SC+ SDW SC SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

32 Similar phase diagram for CeRhIn5 G. Knebel, D. Aoki, and J. Flouquet, arxiv:

33 Similar phase diagram for the pnictides Ort Tet Ishida, Nakai, and Hosono arxiv: v1 50 AFM / Ort 0 SC S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. K. Pratt, A. Thaler, N. Ni, S. L. Bud'ko, P. C. Canfield, J. Schmalian, R. J. McQueeney, A. I. Goldman, arxiv:

34 Outline 1. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity 2. Theory of spin density wave ordering in a metal Strong-coupling in d=2 3. Instabilities near SDW critical point d-wave pairing and bond density wave

35 Outline 1. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity 2. Theory of spin density wave ordering in a metal Strong-coupling in d=2 3. Instabilities near SDW critical point d-wave pairing and bond density wave

36 T T * Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface T sdw d-wave SC SC+ SDW SC SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

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

38 Hole-doped cuprates Increasing SDW order Γ Γ Γ Γ Hole pockets Electron pockets Large Fermi surface breaks up into electron and 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).

39 Hole-doped cuprates Increasing SDW order Γ Γ Γ ϕ Γ Hole pockets Electron pockets ϕ fluctuations act on the large Fermi surface 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).

40 Start from the spin-fermion model Z = Dc α Dϕ exp ( S) S = dτ k λ c kα dτ i τ ε k c kα c iα ϕ i σ αβ c iβ e ik r i + dτd 2 r 1 2 ( r ϕ ) 2 ζ + 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4

41 =3 =2 =4 =1 Low energy fermions ψ 1α, ψ 2α =1,...,4 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α v =1 1 =(v x,v y ), v =1 2 =( v x,v y )

42 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α v 1 v 2 ψ 1 fermions occupied ψ 2 fermions occupied

43 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α v 1 v 2 Hot spot Cold Fermi surfaces

44 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4

45 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ Yukawa coupling: L c = λϕ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ 1α σ αβψ2β + ψ 2α σ αβψ1β

46 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: Hertz theory L ϕ = 1 2 ( r ϕ ) 2 + ζ Yukawa coupling: L c = λϕ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Integrate out fermions and obtain non-local corrections to L ϕ L ϕ = 1 2 ϕ 2 q 2 + γ ω /2 ; γ = 2 πv x v y Exponent z = 2 and mean-field criticality (upto logarithms)

47 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ Yukawa coupling: L c = λϕ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Integrate out fermions and obtain non-local corrections to L ϕ L ϕ = 1 2 ϕ 2 q 2 + γ ω /2 ; γ = 2 πv x v y Exponent z = 2 and mean-field criticality (upto logarithms) OK in d =3, but higher order terms contain an infinite number of marginal couplings in d =2 Hertz theory Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, (2004).

48 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = λϕ ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Perform RG on both fermions and ϕ, using a local field theory.

49 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ Yukawa coupling: L c = λϕ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Under the rescaling x = xe, τ = τe z, the spatial gradients are fixed if the fields transform as ϕ = e (d+z 2)/2 ϕ ; ψ = e (d+z 1)/2 ψ. Then the Yukawa coupling transforms as λ = e (4 d z)/2 λ For d = 2, with z = 2 the Yukawa coupling is invariant, and the bare time-derivative terms ζ, ζ are irrelevant.

50 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = ϕ ψ 1α σ αβψ2β + ψ 2α σ αβψ1β With z = 2 scaling, ζ is irrelevant. So we take ζ 0 ( watch for dangerous irrelevancy).

51 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = ϕ ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Set ϕ wavefunction renormalization by keeping co-efficient of ( r ϕ ) 2 fixed (as usual).

52 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = ϕ ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Set fermion wavefunction renormalization by keeping Yukawa coupling fixed. Y. Huh and S. Sachdev, Phys. Rev. B 78, (2008).

53 L f = ψ 1α ζ τ iv 1 r ψ 1α + ψ 2α ζ τ iv 2 r ψ 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + ζ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = ϕ ψ 1α σ αβψ2β + ψ 2α σ αβψ1β We find consistent two-loop RG factors, as ζ 0, for the velocities v x, v y, and the wavefunction renormalizations. Consistency check: the expression for the boson damping constant, γ = 2 πv x v y, is preserved under RG.

54 RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. Dynamical Nesting v 1 v 2 Bare Fermi surface

55 RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. Dynamical Nesting Dressed Fermi surface

56 RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. Dynamical Nesting Bare Fermi surface

57 RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. Dynamical Nesting Dressed Fermi surface

58 RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. In ϕ SDW fluctuations, characteristic q and ω scale as q ω 1/2 exp 3 64π 2 ln(1/ω) N 3 However, 1/N expansion cannot be trusted in the asymptotic regime..

59 Outline 1. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity 2. Theory of spin density wave ordering in a metal Strong-coupling in d=2 3. Instabilities near SDW critical point d-wave pairing and bond density wave

60 Outline 1. Phase diagram of the cuprates Quantum criticality of the competition between antiferromagnetism and superconductivity 2. Theory of spin density wave ordering in a metal Strong-coupling in d=2 3. Instabilities near SDW critical point d-wave pairing and bond density wave

61 =3 =2 =4 =1 Low energy fermions ψ 1α, ψ 2α =1,...,4

62 ψ 3 2 Hot spots have strong instability to d-wave pairing near SDW critical point. This instability is ψ 3 1 stronger than the BCS instability of a Fermi liquid. Pairing order parameter: ε αβ ψ 3 1αψ 1 1β ψ 3 2αψ 1 2β

63 + ϕ ψ 3 1 ψ 3 2 ψ 3 1 ψ 3 2 ψ 3 1 d-wave Cooper pairing instability in particle-particle channel

64 Emergent Pseudospin symmetry Continuum theory of hotspots in invariant under: ψ ψ U ψ ψ where U are arbitrary SU(2) matrices which can be different on different hotspots.

65 + ϕ ψ 3 1 ψ 3 2 ψ 3 1 ψ 3 2 ψ 3 1 d-wave Cooper pairing instability in particle-particle channel

66 + ϕ ψ 3 1 ψ 3 2 ψ 3 1 ψ 3 2 ψ 3 1 Bond density wave (with local Ising-nematic order) instability in particle-hole channel

67 ψ 3 2 d-wave pairing has a partner instability in the particle-hole channel ψ 3 1 Density-wave order parameter: ψ 3 1α ψ1 1α ψ 3 2α ψ1 2α

68 ψ 3 2 ψ 3 1

69 ψ 3 2 Q ψ 3 1 ψ 3 1 Q ψ 3 2 Single ordering wavevector Q: c k Q/2,α c k+q/2,α = Φ(cos k x cos k y )

70 1 Bond density measures amplitude for electrons to be in spin-singlet valence bond: VBS order 1 No modulations on sites. Modulated bond-density wave with local Ising-nematic ordering: c k Q/2,α c k+q/2,α = Φ(cos k x cos k y )

71 1 Bond density measures amplitude for electrons to be in spin-singlet valence bond: VBS order 1 No modulations on sites. Modulated bond-density wave with local Ising-nematic ordering: c k Q/2,α c k+q/2,α = Φ(cos k x cos k y )

72 STM measurements of Z(r), the energy asymmetry in density of states in Bi 2 Sr 2 CaCu 2 O 8+δ. M. J. Lawler, K. Fujita, Jhinhwan Lee, A. R. Schmidt, Y. Kohsaka, Chung Koo Kim, H. Eisaki, S. Uchida, J. C. Davis, J. P. Sethna, and Eun-Ah Kim, preprint

73 STM measurements of Z(r), the energy asymmetry in density of states in Bi 2 Sr 2 CaCu 2 O 8+δ. M. J. Lawler, K. Fujita, Jhinhwan Lee, A. R. Schmidt, Y. Kohsaka, Chung Koo Kim, H. Eisaki, S. Uchida, J. C. Davis, J. P. Sethna, and Eun-Ah Kim, preprint A C D B

74 STM measurements of Z(r), the energy asymmetry in density of states in Bi 2 Sr 2 CaCu 2 O 8+δ. M. J. Lawler, K. Fujita, Jhinhwan Lee, A. R. Schmidt, Y. Kohsaka, Chung Koo Kim, H. Eisaki, S. Uchida, J. C. Davis, J. P. Sethna, and Eun-Ah Kim, preprint A C D B Strong anisotropy of electronic states between x and y directions: Electronic Ising-nematic order

75 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

76 Conclusions Theory for the onset of spin density wave in metals is strongly coupled in two dimensions For the cuprate Fermi surface, there are strong instabilities near the quantum critical point to d-wave pairing and bond density waves with local Ising-nematic ordering