Quantum disordering magnetic order in insulators, metals, and superconductors

Transcription

1 Quantum disordering magnetic order in insulators, metals, and superconductors Perimeter Institute, Waterloo, May 29, 2010 Talk online: sachdev.physics.harvard.edu HARVARD

2 Cenke Xu, Harvard arxiv: Max Metlitski, Harvard arxiv: Eun Gook Moon, Harvard arxiv: HARVARD

3 Outline 1. Quantum disordering magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases 2. Unified theory of spin liquids Majorana liquids 3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order

4 Outline 1. Quantum disordering magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases 2. Unified theory of spin liquids Majorana liquids 3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order

5 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.

6 Square lattice antiferromagnet H = ij J ij Si S j + + Add perturbations so ground state no longer has long-range Néel order

7 Square lattice antiferromagnet H = ij J ij Si S j + + Add perturbations so ground state no longer has long-range Néel order Describe the resulting state by an effective theory of fluctuations of the Néel order: R z (x, τ) Néel where R is a SU(2) spin rotation matrix related to the Néel order R z z z z z ; ϕ = z ασ αβ z β Ginzburg-Landau paradigm: Effective action for R z (x, τ), deduced by symmetries, describes quantum transitions and phases near the Néel state.

8 Order parameter description is incomplete Underlying electrons cannot be ignored even though charged excitations are fully gapped. They endow topological defects in the order parameter (hedgehogs, vortices...) with Berry phases: the defects acquire additional degeneracies and transform non-trivially under lattice space group e.g. with non-zero crystal momentum

9 Metals (in the 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.

10 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.

11 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 half-filling

12 Half-filled band 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 Half-filled band 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).

14 Half-filled band 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).

15 Half-filled band 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).

16 Half-filled band Increasing SDW order Insulator Γ Γ Γ Hot spots Insulator with Neel order has electrons filling a band, and no 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).

17 Half-filled band Increasing SDW order Insulator Γ Γ Γ Hot spots Insulator with Neel order has electrons filling a band, and no 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).

18 Square lattice antiferromagnet R z (x, τ) Néel Perform SU(2) rotation R z on filled band of electrons: c ψ+ c = z z z z ψ ψ ± states empty ψ ± states filled

19 Square lattice antiferromagnet R z (x, τ) Néel Perform SU(2) rotation R z on filled band of electrons: c ψ+ c = z z z z ψ This 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 ± occupying filled bands; an emergent U(1) gauge field A µ.

20 The Néel phase is the Higgs state with z α = 0. In the quantum disordered phase, with z α = 0 and z α excitations gapped, let us examine the theory for the ψ ± fermions. For simplicity, we focus on the honeycomb lattice, where this can be written in Dirac notation: L ψ = iψγ µ ( µ ia µ σ z ) ψ + mψρ y ψ where σ /ρ are Pauli matrices in spin/valley space.

21 Nature of quantum disordered phase The Néel phase is the Higgs state with z α = 0. In the quantum disordered phase, with z α = 0 and z α excitations gapped, let us examine the theory for the ψ ± fermions. For simplicity, we focus on the honeycomb lattice, where this can be written in Dirac notation: L ψ = iψγ µ ( µ ia µ σ z ) ψ + mψρ y σ z ψ where σ /ρ are Pauli matrices in spin/valley space.

22 Nature of quantum disordered phase The Néel phase is the Higgs state with z α = 0. In the quantum disordered phase, with z α = 0 and z α excitations gapped, let us examine the theory for the ψ ± fermions. For simplicity, we focus on the honeycomb lattice, where this can be written in Dirac notation: L ψ = iψγ µ ( µ ia µ σ z ib µ ρ y ) ψ + mψρ y σ z ψ where σ /ρ are Pauli matrices in spin/valley space. Introduce an external gauge field B µ to probe the structure of the gapped ψ ± phase

23 Nature of quantum disordered phase After integrating out the fermions, the quantum spin Hall physics implies a mutual Chern-Simons term between A µ and B µ L eff = i 2π µνλa µ ν B λ Changing the A µ flux (analog of electric field in QSHE), induces a B µ charge (analog of spin in QSHE).

24 Nature of quantum disordered phase After integrating out the fermions, the quantum spin Hall physics implies a mutual Chern-Simons term between A µ and B µ L eff = i 2π µνλa µ ν B λ Changing the A µ flux (analog of electric field in QSHE), induces a B µ charge (analog of spin in QSHE). Monopoles in A µ carry B µ charge. This endows A µ monopoles with non-zero crystal momentum.

25 Nature of quantum disordered phase z α =0 z α =0 Néel state Valence bond solid (VBS) s zc s z N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)

26 Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) C. Xu and S. Sachdev, Phys. Rev. B 79, (2009) Valence bond solid Z 2 spin liquid (VBS) M Neel antiferromagnet Spiral antiferromagnet

27 Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) C. Xu and S. Sachdev, Phys. Rev. B 79, (2009) Valence bond solid Z 2 spin liquid (VBS) M Neel antiferromagnet Spiral antiferromagnet

28 Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) C. Xu and S. Sachdev, Phys. Rev. B 79, (2009) Z 2 spin liquid Valence bond solid (VBS) M Quantum disordering spiral order leads to a Z2 spin liquid Neel antiferromagnet Spiral antiferromagnet

29 Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) C. Xu and S. Sachdev, Phys. Rev. B 79, (2009) Valence bond solid Z 2 spin liquid (VBS) M Neel antiferromagnet Spiral antiferromagnet

30 Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) C. Xu and S. Sachdev, Phys. Rev. B 79, (2009) Z 2 spin liquid Valence bond solid (VBS) Neel antiferromagnet M Described by a deconfined Z 2 gauge theory, with topological degeneracy on a torus, and gapped spinon and vison excitations with mutual semionic statistics Spiral antiferromagnet N. Read and S. Sachdev, Phys. Rev. Lett. 63, 1773 (1991). (also X.-G. Wen, Phys. Rev. B 44, 2664 (1991))

31 Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) C. Xu and S. Sachdev, Phys. Rev. B 79, (2009) Valence bond solid Z 2 spin liquid (VBS) M M Neel antiferromagnet Spiral antiferromagnet

32 Phase diagram of frustrated antiferromagnets N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991) C. Xu and S. Sachdev, Phys. Rev. B 79, (2009) Z 2 spin liquid Valence bond solid (VBS) Neel M M Spiral Multicritical point M described by a doubled Chern-Simons theory; non-supersymmetric analog of the ABJM model antiferromagnet antiferromagnet

33 Phase diagram of J1-J2-J3 antiferromagnet on the square lattice ~1/S N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

34 Phase diagram of J1-J2-J3 antiferromagnet on the square lattice Collinear magnetic order (Neel) ~1/S N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

35 Phase diagram of J1-J2-J3 antiferromagnet on the square lattice ~1/S Valence Bond Solids N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

36 Phase diagram of J1-J2-J3 antiferromagnet on the square lattice ~1/S N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

37 Phase diagram of J1-J2-J3 antiferromagnet on the square lattice Spiral magnetic order ~1/S N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

38 Phase diagram of J1-J2-J3 antiferromagnet on the square lattice ~1/S Z2 spin liquids N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

39 Phase diagram of J1-J2-J3 antiferromagnet on the square lattice M ~1/S M N. Read and S. Sachdev Phys. Rev. Lett. 63, 1773 (1991)

40 Outline 1. Quantum disordering magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases 2. Unified theory of spin liquids Majorana liquids 3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order

41 Outline 1. Quantum disordering magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases 2. Unified theory of spin liquids Majorana liquids 3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order

42 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin.

43 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin. Quantum disordering magnetic order c = z z ψ+ c z z ψ

44 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin. Quantum disordering magnetic order c = z z ψ+ c z z ψ SU(2) spin

45 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin. Quantum disordering magnetic order c = z z ψ+ c z z ψ U(1) charge

46 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin. Quantum disordering magnetic order c = z z ψ+ c z z ψ U U 1 SU(2) s;gauge S. Sachdev, M. A. Metlitski, Y. Qi, and S. Sachdev Phys. Rev. B 80, (2009)

47 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin. Projected fermion wavefunctions (Fisher, Wen, Lee, Kim) c b 1 b 2 f1 c = b 2 b 1 f 2

48 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin. Projected fermion wavefunctions (Fisher, Wen, Lee, Kim) c b 1 b 2 f1 c = b 2 b 1 f 2 neutral fermionic spinons

49 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin. Projected fermion wavefunctions (Fisher, Wen, Lee, Kim) c b 1 b 2 f1 c = b 2 b 1 f 2 charged slave boson/rotor

50 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin. Projected fermion wavefunctions (Fisher, Wen, Lee, Kim) c b 1 b 2 f1 c = b 2 b 1 f 2 SU(2) pseudospin

51 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin. Projected fermion wavefunctions (Fisher, Wen, Lee, Kim) c b 1 b 2 f1 c = b 2 b 1 f 2 SU(2) spin

52 Global symmetry operations: Spin rotations, SU(2) spin Combine electromagnetic charge (electron number) U(1) charge with particle-hole transformations to obtain SU(2) pseudospin. Projected fermion wavefunctions (Fisher, Wen, Lee, Kim) c b 1 b 2 f1 c = b 2 b 1 f 2 U U 1 SU(2) p;gauge

53 Unified spin liquid theory Decompose electron operator into real fermions, χ: c = χ 1 + iχ 2 ; c = χ 3 + iχ 4 Introduce a 4-component Majorana fermion ζ i, i =1...4 and a SO(4) matrix R, and decompose: χ = R ζ

54 Unified spin liquid theory Decompose electron operator into real fermions, χ: c = χ 1 + iχ 2 ; c = χ 3 + iχ 4 Introduce a 4-component Majorana fermion ζ i, i =1...4 and a SO(4) matrix R, and decompose: χ = R ζ SO(4) = SU(2) pseudospin SU(2) spin

55 Unified spin liquid theory Decompose electron operator into real fermions, χ: c = χ 1 + iχ 2 ; c = χ 3 + iχ 4 Introduce a 4-component Majorana fermion ζ i, i =1...4 and a SO(4) matrix R, and decompose: χ = R ζ O O T SO(4) gauge = SU(2) p;gauge SU(2) s;gauge

56 Unified spin liquid theory Decompose electron operator into real fermions, χ: c = χ 1 + iχ 2 ; c = χ 3 + iχ 4 Introduce a 4-component Majorana fermion ζ i, i =1...4 and a SO(4) matrix R, and decompose: χ = R ζ By breaking SO(4)gauge with different Higgs fields, we can reproduce essentially all earlier theories of spin liquids. We also find many new spin liquid phases, some with Majorana fermion excitations which carry neither spin nor charge

57 Outline 1. Quantum disordering magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases 2. Unified theory of spin liquids Majorana liquids 3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order

58 Outline 1. Quantum disordering magnetic order in two-dimensional antiferromagnets Topological defects and their Berry phases 2. Unified theory of spin liquids Majorana liquids 3. Loss of magnetic order in a metal d-wave pairing and (modulated) Ising-nematic order

59 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.

60 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).

61 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).

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 v 1 v 2 Hot spot Cold Fermi surfaces

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 2 ( τ ϕ ) 2 + s 2 ϕ 2 + u 4 ϕ 4

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

68 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)

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 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).

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 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.

71 =3 =2 =4 =1 Low energy fermions ψ 1α, ψ 2α =1,...,4

72 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

73 + 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

74 2 4 3 Γ k y Q =(π, 0) k x Similar theory applies to the pnictides, and leads to s ± pairing.

75 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.

76 + 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

77 + 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

78 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

79 ψ 3 2 ψ 3 1

80 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 )

81 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 )

82 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 )

83 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

84 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

85 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

86 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

87 Conclusions Quantum disordering magnetic order leads to valence bond solids and Z2 spin liquids Unified theory of spin liquids using Majorana fermions: also includes states obtained by projecting free fermion determinants