Quantum criticality of Fermi surfaces in two dimensions

Transcription

1 Quantum criticality of Fermi surfaces in two dimensions Talk online: sachdev.physics.harvard.edu HARVARD

2 Yejin Huh, Harvard Max Metlitski, Harvard HARVARD

3 Outline 1. Quantum criticality of Fermi points: Dirac fermions in d-wave superconductors 2. Quantum criticality of Fermi surfaces: Onset of spin density wave order in the cuprates

4 Outline 1. Quantum criticality of Fermi points: Dirac fermions in d-wave superconductors 2. Quantum criticality of Fermi surfaces: Onset of spin density wave order in the cuprates

5 The cuprate superconductors

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

8 d-wave superconductivity in cuprates Γ H 0 = i<j Hole states occupied Electron states occupied t ij c iα c iα k Γ ε k c kα c kα Begin with free electrons.

9 d-wave superconductivity in cuprates - + Γ ε k c kα c kα + k c k c k + c.c. Γ + H = k Begin with free electrons. Add d-wave pairing interaction k cos k x cos k y which vanishes along diagonals

10 d-wave superconductivity in cuprates - + Γ Γ + ε k c kα c kα + k c k c k + c.c. H = k Begin with free electrons. Add d-wave pairing interaction k which vanishes along diagonals Obtain Bogoliubov quasiparticles with dispersion ε 2 k + 2 k

11 d-wave superconductivity in cuprates S Ψ = 4 two-component Dirac fermions d 2 k (2π) 2 T Ψ 1a ( iω n + v F k x τ z + v k y τ x ) Ψ 1a ω n + d 2 k (2π) 2 T ω n Ψ 2a ( iω n + v F k y τ z + v k x τ x ) Ψ 2a.

12 Nematic order in YBCO V. Hinkov, D. Haug, B. Fauqué, P. Bourges, Y. Sidis, A. Ivanov, C. Bernhard, C. T. Lin, and B. Keimer, Science 319, 597 (2008)

13 Broken rotational symmetry in the pseudogap phase of a high-tc superconductor R. Daou, J. Chang, David LeBoeuf, Olivier Cyr- Choiniere, Francis Laliberte, Nicolas Doiron- Leyraud, B. J. Ramshaw, Ruixing Liang, D. A. Bonn, W. N. Hardy, and Louis Taillefer arxiv: S. A. Kivelson, E. Fradkin, and V. J. Emery, Nature 393, 550 (1998).

14 d-wave superconductivity in cuprates Now consider a discrete spontaneous symmetry breaking, with Ising symmetry, described by a real scalar field φ. Two cases of experimental interest are: Break 4-fold lattice rotation symmetry to 2-fold lattice rotations: leads to a superconductor with nematic order: equivalent to d x 2 y 2 + s pairing. Time-reversal symmetry breaking: leads to a d x2 y 2 + id xy superconductor, in ε H = H φ + which k c the kα c kα Dirac + fermions k c k c k are + massive c.c. k H φ = φ k c k c k + c.c.

15 d-wave superconductivity in cuprates Now consider a discrete spontaneous symmetry breaking, with Ising symmetry, described by a real scalar field φ. Two cases of experimental interest are: Break 4-fold lattice rotation symmetry to 2-fold lattice rotations: leads to a superconductor with nematic order: equivalent to d x 2 y 2 + s pairing. Time-reversal symmetry breaking: leads to a d x2 y 2 + id xy superconductor, in which the Dirac fermions are massive H = H φ + k ε k c kα c kα + k c k c k + c.c. H φ = iφ k sin k x sin k y c k c k + c.c.

16 Lattice rotation symmetry breaking d x 2 y 2 superconductor + nematic order d x 2 y 2 superconductor φ =0 φ =0 r c r

17 Time-reversal symmetry breaking d x 2 y 2 ± id xy superconductor d x 2 y 2 superconductor φ =0 φ =0 r c r

18 T T c T n Quantum Critical φ/v 1/λ c 1/λ M. Vojta, Y. Zhang, and S. Sachdev, Phys. Rev. Lett. 85, 4940 (2000) E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson, Phys. Rev. B 77, (2008).

19 Discrete symmetry breaking in d-wave superconductors Field theory for transition with Ising order described by a real scalar field φ: S = S Ψ + S φ + S Ψφ S Ψ = + 4 two-component Dirac fermions d 2 k (2π) 2 T Ψ 1a ( iω n + v F k x τ z + v k y τ x ) Ψ 1a ω n d 2 k (2π) 2 T Ψ 2a ( iω n + v F k y τ z + v k x τ x ) Ψ 2a. ω n

20 Discrete symmetry breaking in d-wave superconductors Field theory for transition with Ising order described by a real scalar field φ: S = S Ψ + S φ + S Ψφ S Ψ = + 4 two-component Dirac fermions d 2 k (2π) 2 T Ψ 1a ( iω n + v F k x τ z + v k y τ x ) Ψ 1a ω n d 2 k (2π) 2 T Ψ 2a ( iω n + v F k y τ z + v k x τ x ) Ψ 2a. ω n S φ = d 2 xdτ Ising field theory 1 2 ( τ φ) 2 + c2 2 ( φ)2 + r 2 φ2 + u 0 24 φ4 ;

21 Ising order and Dirac fermions couple via a Yukawa term. S Ψφ = d 2 xdτ λ 0 φ Ψ 1a τ x Ψ 1a + Ψ 2a τ x Ψ 2a, Nematic ordering S Ψφ = d 2 xdτ λ 0 φ Ψ 1a τ y Ψ 1a + Ψ 2a τ y Ψ 2a Time reversal symmetry breaking M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

22 Ising order and Dirac fermions couple via a Yukawa term. S Ψφ = d 2 xdτ λ 0 φ Ψ 1a τ x Ψ 1a + Ψ 2a τ x Ψ 2a, Nematic ordering S Ψφ = d 2 xdτ λ 0 φ Ψ 1a τ y Ψ 1a + Ψ 2a τ y Ψ 2a Time reversal symmetry breaking For the latter case only, with v F = v = c, theory reduces to relativistic Gross-Neveu model M. Vojta, Y. Zhang, and S. Sachdev, Physical Review Letters 85, 4940 (2000)

23 Expansion in number of fermion spin components Nf Integrating out the fermions yields an effective action for the scalar order parameter S φ = N f v v F Γ + N f 2 λ 0 φ(x, τ); v v F d 2 xdτ + irrelevant terms rφ 2 (x, τ) where Γ is a non-local and non-analytic functional of φ. The theory has only 2 couplings constants: r and v /v F. Y. Huh and S. Sachdev, Physical Review B 78, (2008).

24 Expansion in number of fermion spin components Nf Integrating out the fermions yields an effective action for the nematic order parameter S φ = N f 2 k,ω + λ2 0 8v F v φ(k, ω) 2 r ω 2 + vf 2 k2 x ω 2 + vf 2 k2 x + v 2 k2 y +(x y) +higher order terms which cannot be neglected E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson, arxiv:

25 Expansion in number of fermion spin components Nf Integrating out the fermions yields an effective action for the T-breaking order parameter S φ = N f 2 k,ω + λ2 0 8v F v φ(k, ω) 2 r ω 2 + v 2 F k2 x + v 2 k2 y +(x y) +higher order terms which cannot be neglected E.-A. Kim, M. J. Lawler, P. Oreto, S. Sachdev, E. Fradkin, S.A. Kivelson, arxiv:

26 Expansion in number of fermion spin components Nf Integrating out the fermions yields an effective action for the scalar order parameter S φ = N f v v F Γ + N f 2 λ 0 φ(x, τ); v v F d 2 xdτ + irrelevant terms rφ 2 (x, τ) where Γ is a non-local and non-analytic functional of φ. The theory has only 2 couplings constants: r and v /v F. Y. Huh and S. Sachdev, Physical Review B 78, (2008).

27 Expansion in number of fermion spin components Nf Integrating out the fermions yields an effective action for the scalar order parameter S φ = N f v v F Γ + N f 2 λ 0 φ(x, τ); v v F d 2 xdτ + irrelevant terms rφ 2 (x, τ) where Γ is a non-local and non-analytic functional of φ. There is a systematic expansion in powers of The theory has only 2 couplings constants: r and v /v F. 1/N f for renormalization group equations and all critical properties. Y. Huh and S. Sachdev, Physical Review B 78, (2008).

28 Outline 1. Quantum criticality of Fermi points: Dirac fermions in d-wave superconductors 2. Quantum criticality of Fermi surfaces: Onset of spin density wave order in the cuprates

29 Outline 1. Quantum criticality of Fermi points: Dirac fermions in d-wave superconductors 2. Quantum criticality of Fermi surfaces: Onset of spin density wave order in the cuprates

30 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: A e = 2π 2 (1 x) 2π 2 (1 + p) for hole-doping x for electron-doping p A h = 4π 2 A e

31 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 (SDW) order parameter, and K is the ordering wavevector. For simplicity, we consider K =(π, π).

32 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

33 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).

34 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).

35 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).

36 Electron-doped cuprates Increasing SDW order Γ Γ Γ Γ Electron pockets Hole pockets Large Fermi surface breaks up into electron and hole pockets D. Senechal and A.-M. S. Tremblay, Physical Review Letters 92, (2004) J. Lin, and A. J. Millis, Physical Review B 72, (2005).

37 Increasing SDW order Quantum oscillations 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).

38 Quantum oscillations Nature 450, 533 (2007)

39 Quantum oscillations Nature 450, 533 (2007)

40 Theory of quantum criticality in the cuprates 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

41 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 Identifies x m 0.24 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)

42 Theory of quantum criticality in the cuprates 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

43 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 (SDW) Onset of d-wave superconductivity hides the critical point x = x m

44 Theory of quantum criticality in the cuprates 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.

45 Theory of quantum criticality in the cuprates 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.

46 Theory of quantum criticality in the cuprates 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.

47 Theory of quantum criticality in the cuprates Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Thermally fluctuating SDW Magnetic quantum criticality Spin gap Strange Metal d-wave Classical spin waves superconductor Neel order Large Fermi surface Quantum critical E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, Dilute (2001). triplon gas E. G. Moon and S. Sachdev, Phy. Rev. B 80, (2009) Criticality of the coupled dimer antiferromagnet at x=xs Spin density wave (SDW) Competition between SDW order and superconductivity moves the actual quantum critical point to x = x s <x m.

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

49 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

50 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 E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001).

51 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 Hc2 SC+ SDW SC E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

52 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 Hc2 SC+ SDW SC Quantum oscillations E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

53 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 SC+ SDW SC Hsdw E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

54 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 SC+ SDW SC Hsdw E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

55 T Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Neutron scattering & muon resonance Thermally fluctuating SDW Magnetic quantum criticality Spin gap SC+ SDW d-wave SC SC Hsdw E. Demler, S. Sachdev and Y. Zhang, Phys. Rev. Lett. 87, (2001). SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

56 T Neutron scattering experiments on Nd 2 x Ce x CuO 4 show that at low fields x s =0.14, while quantum oscillations at high fields show that x m = Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Thermally fluctuating SDW Magnetic quantum criticality Spin gap SC+ SDW Strange Metal d-wave SC Large Fermi surface SC SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

57 T Physics of competition: d-wave SC and SDW eat up same pieces of the large Fermi surface. Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Thermally fluctuating SDW Magnetic quantum criticality Spin gap SC+ SDW Strange Metal d-wave SC Large Fermi surface SC V. Galitski and S. Sachdev, Physical Review B 79, (2009). Eun Gook Moon and S. Sachdev, Physical Review B 80, (2009). SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

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

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

60 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Theory of SDW quantum phase transition in Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave SC metal SC+ SDW SC SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

61 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Theory of SDW quantum phase transition in Thermally fluctuating SDW Magnetic quantum criticality Spin gap d-wave SC metal SC+ SDW SC SDW (Small Fermi pockets) H M "Normal" (Large Fermi surface)

62 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

63 =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 )

64 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

65 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

66 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

67 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 Hertz-Moriya-Millis (HMM) theory 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)

68 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 Hertz-Moriya-Millis (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 But, higher order terms contain an infinite number of marginal couplings ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 πv x v y Exponent z = 2 and mean-field criticality (upto logarithms) ψ 1α σ αβψ2β + ψ 2α σ αβψ1β Ar. Abanov and A.V. Chubukov, Phys. Rev. Lett. 93, (2004).

69 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.

70 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.

71 Two approaches: A Fix λ = 1 and perform RG in a 1/N expansion, where N is the number of fermion flavors B Make λ part of the bare fermion dispersion by transforming electrons to a rotating reference frame determined by the local orientation of the SDW order ϕ.

72 Two approaches: A Fix λ = 1 and perform RG in a 1/N expansion, where N is the number of fermion flavors B Make λ part of the bare fermion dispersion by transforming electrons to a rotating reference frame determined by the local orientation of the SDW order ϕ.

73 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:

74 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).

75 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).

76 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 = ϕ 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4 ψ 1α σ αβψ2β + ψ 2α σ αβψ1β

77 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).

78 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).

79 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).

80 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.

81 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 d = 3 πn α 2 1+α 2

82 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 d = 3 πn α 2 1+α 2 The velocities flow as 1 dv x v x d A(α) B(α) = A(α)+B(α) 2 3 α πn 1+α πN α α 1+ ; 1 dv y v y d = A(α)+B(α) 2 1 α α tan 1 1 α

83 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 d = 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 = πN α α 1+ α α tan 1 1 α

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

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

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

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

88 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..

89 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

90 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

91 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

92 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

93 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).

94 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

95 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:

96 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

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

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

99 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Naturally formulated in route B: theory of fluctuating Fermi pockets Thermally fluctuating SDW SDW (Small Fermi pockets) H Magnetic quantum criticality Spin gap SC+ SDW d-wave SC M SC "Normal" (Large Fermi surface)

100 T Fluctuating, Small Fermi pockets paired Fermi with pairing pockets fluctuations Strange Metal Large Fermi surface Naturally formulated in route B: theory of fluctuating Fermi pockets Thermally fluctuating SDW SDW (Small Fermi pockets) H Magnetic quantum criticality Spin gap SC+ SDW d-wave SC M SC "Normal" (Large Fermi surface)

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