Quantum criticality and the phase diagram of the cuprates

Transcription

1 Quantum criticality and the phase diagram of the cuprates Talk online: sachdev.physics.harvard.edu HARVARD

2 Victor Galitski, Maryland Ribhu Kaul, Harvard Kentucky Max Metlitski, Harvard Eun Gook Moon, Harvard Cenke Xu, Harvard Santa Barbara HARVARD

3 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)

4 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)

5 Crossovers in transport properties of hole-doped cuprates Strange metal S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). T (K) A. J. Millis, Phys. Rev. B 48, 7183 (1993). A F M d-wave SC Hole doping x x m Strange metal: quantum criticality of optimal doping critical point at x = x m? C. M. Varma, Phys. Rev. Lett. 83, 3538 (1999).

6 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

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

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

9 Large Fermi surfaces in cuprates Hole states occupied Γ Electron states occupied Γ H 0 = i<j t ij c iα c iα k ε k c kα c kα The area of the occupied electron/hole states: { 2π 2 (1 x) for hole-doping x A e = 2π 2 (1 + p) for electron-doping p A h = 4π 2 A e

10 Spin density wave theory The electron spin polarization obeys S(r, τ) = ϕ (r, τ)e ik r where ϕ is the spin density wave () order parameter, and K is the ordering wavevector. For simplicity, we consider K =(π, π).

11 Spin density wave theory Spin density wave Hamiltonian H sdw = ϕ c k,α σ αβc k+k,β k,α,β Diagonalize H 0 + H sdw for ϕ = (0, 0, ϕ) E k± = ε k + ε k+k 2 ± (εk ε k+k 2 ) + ϕ 2

12 Spin density wave theory Increasing 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). D. Senechal and A.-M. S. Tremblay, Phys. Rev. Lett. 92, (2004)

13 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).

14 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)

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

16 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

17 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.

18 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.

19 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.

20 Theory of quantum criticality in the cuprates Small Fermi pockets with pairing fluctuations Strange Metal Classical spin waves Large Fermi surface Quantum critical Dilute triplon gas Thermally fluctuating Magnetic quantum criticality Spin gap d-wave superconductor Neel order Criticality of the coupled dimer antiferromagnet at x=xs Spin density wave () Competition between order and superconductivity moves the actual quantum critical point to x = x s <x m.

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

22 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Overdoped vs. underdoped pairing Electronic theory of competing orders 3. Theory of quantum critical point Dominance of planar graphs

23 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Overdoped vs. underdoped pairing Electronic theory of competing orders 3. Theory of quantum critical point Dominance of planar graphs

24 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). B. Kyung, J.-S. Landry, and A.-M. S. Tremblay, Phys. Rev. B 68, (2003). ]

25 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. Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order 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). B. Kyung, J.-S. Landry, and A.-M. S. Tremblay, Phys. Rev. B 68, (2003). ]

26 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. Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order 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). B. Kyung, J.-S. Landry, and A.-M. S. Tremblay, Phys. Rev. B 68, (2003). ]

27 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).

28 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).

29 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).

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 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).

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 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).

32 B. Lake, H. M. Rønnow, N. B. Christensen, G. Aeppli, K. Lefmann, D. F. McMorrow, P. Vorderwisch, P. Smeibidl, N. Mangkorntong, T. Sasagawa, M. Nohara, H. Takagi, and T. E. Mason, Nature 415, 299 (2002) 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).

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 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).

34 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).

35 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).

36 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)

37 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H ncreasing "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:

38 Quantum oscillations Nature 450, 533 (2007)

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

40 Increasing order Nd 2 x Ce x CuO 4 T. Helm, M. V. Kartsovnik, M. Bartkowiak, N. Bittner, M. Lambacher, A. Erb, J. Wosnitza, and R. Gross, Phys. Rev. Lett. 103, (2009).

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

42 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).

43 Phenomenological quantum theory of competition between superconductivity (SC) and spin-density wave () order H "Normal" (Large Fermi surface) ncreasing (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.

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

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

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

47 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)

48 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Overdoped vs. underdoped pairing Electronic theory of competing orders 3. Theory of quantum critical point Dominance of planar graphs

49 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Overdoped vs. underdoped pairing Electronic theory of competing orders 3. Theory of quantum critical point Dominance of planar graphs

50 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)

51 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)

52 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)

53 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)

54 Approaching the onset of antiferromagnetism in the spin-fluctuation theory : T c increases upon approaching the transition. Pairing from fluctuations leads to κ < 0: 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).

55 Approaching the onset of antiferromagnetism in the spin-fluctuation theory : T c increases upon approaching the transition. Pairing from fluctuations leads to κ < 0: 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).

56 Approaching the onset of antiferromagnetism in the spin-fluctuation theory : T c increases upon approaching the transition. Pairing from fluctuations leads to κ < 0: 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 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)

58 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)

59 T Physics of competition: d-wave SC and eat up same pieces of the large Fermi surface. Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Thermally fluctuating Magnetic quantum criticality Spin gap SC+ Strange Metal d-wave SC Large Fermi surface SC B. Kyung, J.-S. Landry, and A.-M. S. Tremblay, Phys. Rev. B 68, (2003). (Small Fermi pockets) H M "Normal" (Large Fermi surface)

60 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 H. J. Schulz, Physical Review Letters 65, 2462 (1990)

61 Theory of underdoped cuprates ) With R = ( z z z z the theory is invariant under z α e iθ z α ; ψ + e iθ ψ + ; ψ e iθ ψ. We obtain a U(1) gauge theory of bosonic neutral spinons z α ; spinless, charged fermions ψ ± ; an emergent U(1) gauge field A µ.

62 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. V. Galitski and S. Sachdev, Physical Review B 79, (2009).

63 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. Eun Gook Moon and S. Sachdev, Physical Review B 80, (2009).

64 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, Phys. Rev. B 78, (2008).

65 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)

66 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)

67 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Overdoped vs. underdoped pairing Electronic theory of competing orders 3. Theory of quantum critical point Dominance of planar graphs

68 Outline 1. Phenomenological quantum theory of competition between superconductivity and order Survey of recent experiments 2. Overdoped vs. underdoped pairing Electronic theory of competing orders 3. Theory of quantum critical point Dominance of planar graphs

69 Max Metlitski M. Metlitski and S. Sachdev, to appear Ar. Abanov, A.V. Chubukov, and J. Schmalian, Advances in Physics 52, 119 (2003) Sung-Sik Lee, arxiv:

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

71 Hertz-Millis-Moriya (HMM) theory: mean field theory + Gaussian fluctuations of overdamped paramagnons.

72 Theory for the onset of spin density wave order in metals is strongly coupled in two dimensions

73 Start from the spin-fermion model Z = Dc α D ϕ exp ( S) S = dτ k c kα ( τ ε k ) c kα λ dτ c iα ϕ i σ αβ c iβ e ik r i i + dτd 2 r [( r ϕ ) 2 + 1c 2 ( τ ϕ ) 2 ]

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

75 L f = ψ l 1α ( ζ τ iv l 1 r ) ψ l 1α + ψ l 2α ( ζ τ iv l 2 r ) ψ l 2α v 1 v 2 ψ 1 fermions occupied ψ 2 fermions occupied

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

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

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

79 L f = ψ l 1α ( ζ τ iv l 1 r ) ψ l 1α + ψ l 2α ( ζ τ iv l 2 r ) ψ l 2α Order parameter: L ϕ = 1 2 ( r ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 Yukawa coupling: L c = ϕ ( ) ψ l 1α σ αβψ2β l + ψ l 2α σ αβψ1β l HMM theory 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) But, higher order terms contain an infinite number of marginal couplings Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, (2004).

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

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

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

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

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

85 RG-improved Migdal-Eliashberg theory RG flow can be computed a 1/N expansion (with N fermion species) in terms of a single dimensionless coupling α = v y /v x whose flow obeys dα dl = 3 πn α 2 1+α 2

86 RG-improved Migdal-Eliashberg theory RG flow can be computed a 1/N expansion (with N fermion species) in terms of a single dimensionless coupling α = v y /v x whose flow obeys dα dl = 3 πn α 2 1+α 2 The velocities flow as 1 dv x v x dl A(α) B(α) = A(α)+B(α) 2 3 α πn 1+α 2 ( )( 1 1 2πN α α 1+ ; 1 dv y v y dl = A(α)+B(α) 2 ( 1 α α ) tan 1 1 α )

87 RG-improved Migdal-Eliashberg theory RG flow can be computed a 1/N expansion (with N fermion species) in terms of a single dimensionless coupling α = v y /v x whose flow obeys dα dl = 3 πn α 2 1+α 2 The anomalous dimensions of ϕ and ψ are ( ( ) η ϕ = 2πN α α + α 2 + α2 tan 1 1 ) α η ψ = 1 ( )( ( ) 1 1 4πN α α 1+ α α tan 1 1 α )

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

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

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

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

92 RG-improved Migdal-Eliashberg theory α = v y /v x 0 logarithmically in the infrared. In ϕ 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..

93 New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) ϕ propagator 1 N 1 (q 2 + γ ω ) fermion propagator v q + iζω + i 1 1 N γv ωf ( v 2 q 2 ω )

94 New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) ϕ propagator 1 N 1 (q 2 + γ ω ) fermion propagator v q + iζω + i 1 1 N γv ωf ( v 2 q 2 ω ) Dangerous

95 New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) Ignoring fermion self energy: 1 N 2 1 ζ 2 1 ω

96 New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) Ignoring fermion self energy: 1 N 2 1 Actual order 1 N 0 ζ 2 1 ω

97 Double line representation A way to compute the order of a diagram. Extra powers of N come from the Fermi-surface What are the conditions for all propagators to be on the Fermi surface? Concentrate on diagrams involving a single pair of hot-spots Any bosonic momentum may be (uniquely) written as R. Shankar, Rev. Mod. Phys. 66, 129 (1994). S. W. Tsai, A. H. Castro Neto, R. Shankar, and D. K. Campbell, Phys. Rev. B 72, (2005).

98 New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) = Singularities as ζ 0 appear when fermions in closed blue and red line loops are exactly on the Fermi surface Actual order 1 N 0

99 New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) = Actual order 1 N 0 Graph is planar after turning fermion propagators also into double lines by drawing additional dotted single line loops for each fermion loop Sung-Sik Lee, arxiv:

100 New infra-red singularities as ζ 0 at higher loops (Breakdown of Migdal-Eliashberg) = Actual order 1 N 0 A consistent analysis requires resummation of all planar graphs

101 T Fluctuating, paired Fermi pockets T* Strange Metal Large Fermi surface T c Fluctuating, paired Fermi pockets T sdw d-wave SC VBS and/or Ising nematic order SC+ SC H c2 H Insulator M (Small Fermi pockets) "Normal" (Large Fermi surface) 96

102 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

103 Conclusions Theory for the onset of spin density wave order in metals is strongly coupled in two dimensions