The quantum criticality of metals in two dimensions: antiferromagnetism, d-wave superconductivity and bond order

Transcription

1 The quantum criticality of metals in two dimensions: antiferromagnetism, d-wave superconductivity and bond order ICTP, Trieste July 2, 2013 Subir Sachdev HARVARD

2 Max Metlitski Erez Berg Rolando La Placa

3 Quantum phase transitions with loss of antiferromagnetic order

4 Quantum phase transitions with loss of antiferromagnetic order 1. Dimerized insulators: Conformal field theory 2. Metals (at weaker coupling): The iron-based superconductors 3. Metals (at stronger coupling): The hole-doped cuprate superconductors

5 Quantum phase transitions with loss of antiferromagnetic order 1. Dimerized insulators: Conformal field theory 2. Metals (at weaker coupling): The iron-based superconductors 3. Metals (at stronger coupling): The hole-doped cuprate superconductors

6 Quantum phase transitions with loss of antiferromagnetic order 1. Dimerized insulators: Conformal field theory 2. Metals (at weaker coupling): The iron-based superconductors 3. Metals (at stronger coupling): The hole-doped cuprate superconductors

7 Quantum phase transitions with loss of antiferromagnetic order 1. Dimerized insulators: Conformal field theory 2. Metals (at weaker coupling): The iron-based superconductors 3. Metals (at stronger coupling): The hole-doped cuprate superconductors

8 Square lattice antiferromagnet H = J ij Si S j ij Ground state has long-range Néel order Order parameter is a single vector field = i = ±1 on two sublattices = 0 in Néel state. i Si

9 Square lattice antiferromagnet H = J ij Si S j ij J J/ Weaken some bonds to induce spin entanglement in a new quantum phase

10 Square lattice antiferromagnet H = J ij Si S j ij J J/ Ground state is a quantum paramagnet with spins locked in valence bond singlets = 1 2

11 = 1 2 c

12 = 1 2 c Quantum critical point with non-local entanglement in spin wavefunction

13 = 1 2 c CFT3 O(3) order parameter Z h(@ S = d 2 rd ~' ) 2 + c 2 (r r ~' ) 2 +( c)~' 2 + u ~' 2 2i

14 Quantum critical Classical spin waves Dilute triplon gas Neel order S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

15 CFT3 at T>0 Quantum critical Classical spin waves Dilute triplon gas Neel order S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

16 Quantum phase transitions with loss of antiferromagnetic order 1. Dimerized insulators: Conformal field theory 2. Metals (at weaker coupling): The iron-based superconductors 3. Metals (at stronger coupling): The hole-doped cuprate superconductors

17 Quantum phase transitions with loss of antiferromagnetic order 1. Dimerized insulators: Conformal field theory 2. Metals (at weaker coupling): The iron-based superconductors 3. Metals (at stronger coupling): The hole-doped cuprate superconductors

18 Iron pnictides: a new class of high temperature superconductors Ishida, Nakai, and Hosono arxiv: v1

19 Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW 1.0 Superconductivity 0 S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, (2010)

20 Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " Neel (AF) order 2.0 AF SDW 1.0 Superconductivity 0 S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, (2010)

21 Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW 1.0 Superconductivity Superconductor Bose condensate of pairs of electrons S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, (2010) 0

22 Resistivity 0 + AT BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW 1.0 Superconductivity 0 Ordinary metal S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, (2010) (Fermi liquid)

23 Resistivity 0 + AT Strange Metal no quasiparticles, Landau-Boltzmann theory does not apply BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW 1.0 Superconductivity 0 S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, (2010)

24 Fermi surface+antiferromagnetism Metal with large Fermi surface + The electron spin polarization obeys S(r, ) = (r, )e ik r where K is the ordering wavevector.

25 Fermi surface+antiferromagnetism H 0 = X i<j t ij c i c i X k " k c k c k + The electron spin polarization obeys S(r, ) = (r, )e ik r where K is the ordering wavevector.

26 Fermi surface+antiferromagnetism H 0 = X i<j t ij c i c i X k " k c k c k + H sdw = ~' X k,, c k, ~ c k+k, where ~ are the Pauli matrices.

27 Fermi surface+antiferromagnetism H 0 = X i<j t ij c i c i X k " k c k c k The electron dispersions obtained by diagonalizing H 0 + H sdw for ~' / (0, 0, 1) are E k± = " k + " k+k 2 ± H sdw = ~' X s "k c k, ~ " k+k 2 c k+k, + ' 2 We now fill the lowest energy states to obtain the new Fermi k,, surface + where ~ are the Pauli matrices.

28 Fermi surface+antiferromagnetism Metal with large Fermi surface

29 Fermi surface+antiferromagnetism K Fermi surfaces translated by K =(, ).

30 Fermi surface+antiferromagnetism K Hot spots

31 Fermi surface+antiferromagnetism Electron and hole pockets in antiferromagnetic phase with antiferromagnetic order parameter h~' i6=0

32 Fermi surface+antiferromagnetism ncreasing SDW h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface r

33 Pairing glue from antiferromagnetic fluctuations A A C A C D C B D B D B V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983) 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) S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, (2010)

34 D c k c k E = (cos k x cos k y ) Unconventional pairing at and near hot spots

35 Sign-problem-free Quantum Monte Carlo for antiferromagnetism in metals P ± (x max ) 10 x _ P _ + P _ r c L = 14 L = 10 L = r s/d pairing amplitudes P + /P as a function of the tuning parameter r E. Berg, M. Metlitski, and S. Sachdev, Science 338, 1606 (2012).

36 Fermi surface+antiferromagnetism ncreasing SDW h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface r

37 Fermi surface+antiferromagnetism T * Fluctuating Fermi pockets Quantum Strange Critical Metal Large Fermi surface ncreasing SDW Spin density wave (SDW) Underlying SDW ordering quantum critical point in metal at x = x m

38 Fermi surface+antiferromagnetism Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Quantum Strange Critical Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) QCP for the onset of SDW order is actually within a superconductor

39 Resistivity 0 + AT Strange Metal no quasiparticles, Landau-Boltzmann theory does not apply BaFe 2 (As 1-x P x ) 2 T 0 T SD T c α " 2.0 AF SDW 1.0 Superconductivity 0 S. Kasahara, T. Shibauchi, K. Hashimoto, K. Ikada, S. Tonegawa, R. Okazaki, H. Shishido, H. Ikeda, H. Takeya, K. Hirata, T. Terashima, and Y. Matsuda, Physical Review B 81, (2010)

40 Quantum phase transitions with loss of antiferromagnetic order 1. Dimerized insulators: Conformal field theory 2. Metals (at weaker coupling): The iron-based superconductors 3. Metals (at stronger coupling): The hole-doped cuprate superconductors

41 Quantum phase transitions with loss of antiferromagnetic order 1. Dimerized insulators: Conformal field theory 2. Metals (at weaker coupling): The iron-based superconductors 3. Metals (at stronger coupling): The hole-doped cuprate superconductors

42 High temperature superconductors YBa 2 Cu 3 O 6+x

43 Strange Metal

44 What about the pseudogap? Strange Metal Pseudogap

45 Fermi surface+antiferromagnetism Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Quantum Strange Critical Metal Large Fermi surface d-wave superconductor Spin density wave (SDW) QCP for the onset of SDW order is actually within a superconductor

46 Theory of quantum criticality in the cuprates Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Quantum Strange Critical Metal d-wave superconductor Large Fermi surface 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 Quantum Strange Critical Metal d-wave superconductor Large Fermi surface 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.

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

49 Theory of quantum criticality in the cuprates Fluctuating, Small Fermi paired pockets Fermi with pairing pockets fluctuations Quantum Strange Critical Metal Large Fermi surface M. Vojta and S. Sachdev, Phys. Rev. Lett. 83, 3916 (1999) Bond d-wave superconductor Order M.A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010) x s Spin density wave (SDW) The metal has an instability to both d-wave superconductivity and a d-wave charge density wave (bond order).

50 Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa 2 Cu 3 O y Tao Wu 1, Hadrien Mayaffre 1, Steffen Krämer 1, Mladen Horvatić 1, Claude Berthier 1, W. N. Hardy 2,3, Ruixing Liang 2,3, D. A. Bonn 2,3 & Marc-Henri Julien 1 8 S E P T E M B E R V O L N AT U R E erved Superconducting T (K) 40 Field-induced charge order Spin order p (hole/cu)

51 There is an approximate pseudospin symmetry in metals with antiferromagnetic spin correlations. The pseudospin partner of d-wave superconductivity is an incommensurate d-wave bond order These orders form a pseudospin doublet, which is responsible for the pseudogap phase. M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010) T. Holder and W. Metzner, Phys. Rev. B 85, (2012) C. Husemann and W. Metzner, Phys. Rev. B 86, (2012) M. Bejas, A. Greco, and H. Yamase, Phys. Rev. B 86, (2012) K. B. Efetov, H. Meier, and C. Pépin, Nature Physics 9, 442 (2013). S. Sachdev and R. La Placa, Phys. Rev. Lett. in press arxiv:

52 1. Pseudospin symmetry between d-wave superconductivity and bond order Continuum field theory with exact pseudospin symmetry 2. Approximate pseudospin symmetry on the lattice t-j model Pseudogap and bond order in the underdoped cuprates

53 1. Pseudospin symmetry between d-wave superconductivity and bond order Continuum field theory with exact pseudospin symmetry 2. Approximate pseudospin symmetry on the lattice t-j model Pseudogap and bond order in the underdoped cuprates

54 Pseudospin symmetry of the exchange interaction H J = X i<j J ij ~ Si ~S j with ~ S i = 1 2 c i ~ c i is the antiferromagnetic exchange interaction. Introduce the Nambu spinor i" = ci" c i#, i# = ci# c i" Then we can write X H J = 1 8 J ij i ~ i j ~ i i<j which is invariant under independent SU(2) pseudospin transformations on each site i! U i i This pseudospin (gauge) symmetry is important in classifying spin liquid ground states of H J. It is fully broken by the electron hopping t ij but does have remnant consequences in doped spin liquid states. I. A eck, Z. Zou, T. Hsu, and P. W. Anderson, Phys. Rev. B 38, 745 (1988) E. Dagotto, E. Fradkin, and A. Moreo, Phys. Rev. B 38, 2926 (1988) P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)

55 Pseudospin symmetry of the exchange interaction H tj = X i,j t H J = ij c i c j + X J ij Si ~ ~S j i<j with ~ S i = 1 2 c i ~ c i is the antiferromagnetic exchange interaction. Introduce the Nambu spinor i" = ci" c i#, i# = ci# c i" Then we can write X H J = 1 8 J ij i ~ i j ~ i i<j which is invariant under independent SU(2) pseudospin transformations on each site i! U i i This pseudospin (gauge) symmetry is important in classifying spin liquid ground states of H J. It is fully broken by the electron hopping t ij but does have remnant consequences in doped spin liquid states. I. A eck, Z. Zou, T. Hsu, and P. W. Anderson, Phys. Rev. B 38, 745 (1988) E. Dagotto, E. Fradkin, and A. Moreo, Phys. Rev. B 38, 2926 (1988) P. A. Lee, N. Nagaosa, and X.-G. Wen, Rev. Mod. Phys. 78, 17 (2006)

56 Pseudospin symmetry of the exchange interaction H tj = X i,j t H J = ij c i c j + X J ij Si ~ ~S j i<j with ~ S i = 1 2 c i ~ c i is the antiferromagnetic exchange interaction. Introduce the Nambu spinor i" = ci" c i#, i# = ci# c i" Then we can write X H J = 1 8 J ij i ~ i j ~ i i<j which is invariant under independent SU(2) pseudospin transformations on each site i! U i i This pseudospin (gauge) symmetry is important in classifying spin liquid ground states of H J. It is fully broken by the electron hopping t ij but does have remnant consequences in doped spin liquid states. We will find important consequences of the pseudospin symmetry in ordinary metals with antiferromagnetic correlations. M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010)

57 Fermi surface+antiferromagnetism ncreasing SDW h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface r

58 Fermi surface+antiferromagnetism ncreasing SDW K Focus on this region h~' i6=0 h~' i =0 Metal with electron and hole pockets Metal with large Fermi surface r

59 Fermi surface+antiferromagnetism K Hot spots

60 Fermi surface+antiferromagnetism K Low energy theory for critical point near hot spots

61 Fermi surface+antiferromagnetism K Low energy theory for critical point near hot spots

62 Theory has fermions 1,2 (with Fermi velocities v 1,2 ) and boson order parameter ~', interacting with coupling v 1 v 2 k y k x 1 fermions occupied Ar. Abanov and A. V. Chubukov, Phys. Rev. Lett. 84, 5608 (2000). 2 fermions occupied

63 S = Z apple d 2 rd 1 (@ iv 1 r r ) (@ iv 2 r r ) (r r ~' ) 2 + s 2 ~' 2 + u 4 ~' 4 ~' 1 ~ ~ 1 v 1 v 2 k y k x 1 fermions occupied Ar. Abanov and A. V. Chubukov, Phys. Rev. Lett. 84, 5608 (2000). 2 fermions occupied

64 S = Z apple d 2 rd 1 (@ iv 1 r r ) (@ iv 2 r r ) (r r ~' ) 2 + s 2 ~' 2 + u 4 ~' 4 ~' 1 ~ ~ 1 v 1 v 2 This low-energy theory is invariant under k y independent SU(2) pseudospin rotations on each pair of hot-spots. k x 1 fermions occupied M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010) 2 fermions occupied

65 Pairing glue from antiferromagnetic fluctuations A A C A C D C B D B D B V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983) 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) S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, (2010)

66 Same glue leads to particle-hole pairing A A C A C D C B D B D B M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010)

67 D c k c k E = " S (cos k x cos k y ) V. J. Emery, J. Phys. (Paris) Colloq. 44, C3-977 (1983) 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) S. Raghu, S.A. Kivelson, and D.J. Scalapino, Phys. Rev. B 81, (2010) K S S d-wave superconductor: particle-particle pairing at and near hot spots, with sign-changing pairing amplitude

68 D c k Q/2, c k+q/2, E = Q (cos k x cos k y ) After pseudospin Q is 2k F wavevector rotation on half the K Q Q hot-spots M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010) Incommensurate d-wave bond order: particle-hole pairing at and near hot spots, with sign-changing pairing amplitude

69 Incommensurate d-wave bond order Q M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010) Q D c k Q/2, c k+q/2, E = Q (cos k x cos k y )

70 Incommensurate d-wave bond order Q M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010) Q Q Q D c k Q/2, c k+q/2, E = Q (cos k x cos k y )

71 Incommensurate d-wave bond order Consider modulation in an o -site density like variable at sites r i and r j " # D E X c i c j Q(k)e ik (r i r j ) e iq (r i+r j )/2 k relative co-ord. average co-ord. The wavevector Q is associated with a modulation in the average coordinate (r i r j )/2: this determines the wavevector of the neutron/x-ray scattering peak. The interesting part is the dependence on the relative co-ordinate r i r j. Assuming time-reversal, the order parameter Q(k) can always be expanded as Q(k) =c s + c s 0(cos k x + cos k y )+c d (cos k x cos k y )+... The usual charge-density-wave has only c s 6= 0. The bond-ordered state we find has c d c s,c s 0,...

72 Incommensurate d-wave bond order +1 Bond density measures amplitude for electrons to be in spin-singlet valence bond. -1 M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010) X X c r c s = Q k e iq (r+s)/2 e ik (r s) D c k Q/2, c k+q/2, E where Q extends over Q =(±Q 0, ±Q 0 )withq 0 =2 /(7.3) and D E c k Q/2, c k+q/2, = Q (cos k x cos k y ) Note c r c s is non-zero only when r, s are nearest neighbors.

73 Order parameter for the pseudogap 4-component order parameter d-wave pairing Re[ S], Im[ S] d-wave bond order Re[ Q], Im[ Q] O(4) symmetry is exact in the continuum hot-spot theory for the onset of antiferromagnetism M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010)

74 Order parameter for the pseudogap 4-component order parameter d-wave pairing Re[ S], Im[ S] d-wave bond order Re[ Q], Im[ Q] Symmetry is anomalous because it requires independent rotation of left- and right-movers: it is broken by any lattice regularization of the theory. M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010)

75 Order parameter for the pseudogap 4-component order parameter d-wave pairing Re[ S], Im[ S] d-wave bond order Re[ Q], Im[ Q] Symmetry is broken by Fermi surface curvature (which prefers S) and density-density Coulomb repulsion (which prefers Q). M. A. Metlitski and S. Sachdev, Phys. Rev. B 85, (2010)

76 Order parameter for the pseudogap 4-component order parameter d-wave pairing Re[ S], Im[ S] d-wave bond order Re[ Q], Im[ Q] Thermal fluctuations restore symmetry, and the pseudogap is a regime of a fluctuating O(4) order parameter (which cannot have a phase transition with O(4) symmetry). K. B. Efetov, H. Meier, and C. Pepin, Nature Physics 9, 442 (2013)

77 1. Pseudospin symmetry between d-wave superconductivity and bond order Continuum field theory with exact pseudospin symmetry 2. Approximate pseudospin symmetry on the lattice t-j model Pseudogap and bond order in the underdoped cuprates

78 1. Pseudospin symmetry between d-wave superconductivity and bond order Continuum field theory with exact pseudospin symmetry 2. Approximate pseudospin symmetry on the lattice t-j model Pseudogap and bond order in the underdoped cuprates

79 H tj = X i,j t ij c i c j + X i<j J ij ~ Si ~S j Optimize the free energy w.r.t. a mean field Hamiltonian which allows for spin-singlet charge order ( Q(k)): H MF = X i,j t ij c i c j + X k,q Q(k)c k+q/2, c k Q/2, Expanding the free energy in powers of the order parameters we obtain F = F 0 + X k,q Q(k)M Q (k, k 0 ) Q(k 0 ) We compute the eigenvalues, 1 + Q(k) of the kernel M Q (k, k 0 ) Q, and eigenfunctions,

80 H tj = X i,j t ij c i c j + X i<j J ij ~ Si ~S j Optimize the free energy w.r.t. a mean field Hamiltonian which allows for spin-singlet charge order ( Q(k)): H MF = X i,j t ij c i c j + X k,q Q(k)c k+q/2, c k Q/2, Expanding the free energy in powers of the order parameters we obtain F = F 0 + X k,q Q(k)M Q (k, k 0 ) Q(k 0 ) We compute the eigenvalues, 1 + Q(k) of the kernel M Q (k, k 0 ) Q, and eigenfunctions,

81 Charge-ordering eigenvalue Q/J 0. S. Sachdev and R. La Placa, Phys. Rev. Lett. in press, arxiv:

82 Minimum at Q =(Q m,q m )with Q(k) = 0.993(cos k x cos k y ) 0.069(cos(2k x ) cos(2k y )) 0.009(cos k x cos k y ) p 8 sin k x sin k y Incommensurate d-wave bond order Charge-ordering eigenvalue Q/J 0. S. Sachdev and R. La Placa, Phys. Rev. Lett. in press, arxiv:

83 K 1. (±Q 0,Q 0 ) Remarkable agreement between the value of Q m from Hartree-Fock in a metal with short-range incommensurate spin correlations, and the value of Q 0 from hot spots of commensurate antiferromagnetism. Q Q m

84 Minimum at Q =(Q m,q m )with Q(k) = 0.993(cos k x cos k y ) 0.069(cos(2k x ) cos(2k y )) 0.009(cos k x cos k y ) p 8 sin k x sin k y Incommensurate d-wave bond order Charge-ordering eigenvalue Q/J 0. S. Sachdev and R. La Placa, Phys. Rev. Lett. in press, arxiv:

85 Charge-ordering eigenvalue Q/J 0. S. Sachdev and R. La Placa, Phys. Rev. Lett. in press, arxiv:

86 Q =(, ) with Q(k) =i(sin k x sin k y ) Orbital currents Charge-ordering eigenvalue Q/J 0. S. Sachdev and R. La Placa, Phys. Rev. Lett. in press, arxiv:

87 Charge-ordering eigenvalue Q/J 0. S. Sachdev and R. La Placa, Phys. Rev. Lett. in press, arxiv:

88 Q =(0, 0) with Q(k) = cos k x cos k y Ising-nematic order Charge-ordering eigenvalue Q/J 0. S. Sachdev and R. La Placa, Phys. Rev. Lett. in press, arxiv:

89 Charge-ordering eigenvalue Q/J 0. S. Sachdev and R. La Placa, Phys. Rev. Lett. in press, arxiv:

90 Q =(Q m, 0) with Q(k) = 0.963(cos k x cos k y ) d +s 0.067(cos(2k x ) cos(2k y )) 0.044(cos k x + cos k y ) 0.046(cos(2k x ) + cos(2k y )) Incommensurate -wave bond order Charge-ordering eigenvalue Q/J 0. S. Sachdev and R. La Placa, Phys. Rev. Lett. in press, arxiv:

91 K 1. (0,Q 0 ) Remarkable agreement between the value of Q m from Hartree-Fock in a metal with short-range incommensurate spin correlations, and the value of Q 0 from hot spots of commensurate antiferromagnetism. Q Q m

92 Incommensurate d-wave bond order Observed low T ordering. K Our computations show that the charge order is predominantly d-wave also at this Q. This Q is preferred in computations of bond order within the superconducting phase. S. Sachdev and R. La Placa, Physical Review Letters in press; arxiv: M. Vojta and S. Sachdev, Physical Review Letters 83, 3916 (1999) M. Vojta and O. Rosch, Physical Review B 77, (2008)

93 An Intrinsic Bond-Centered Electronic Glass with Unidirectional Domains in Underdoped Cuprates Y. Kohsaka, 1 C. Taylor, 1 K. Fujita, 1,2 A. Schmidt, 1 C. Lupien, 3 T. Hanaguri, 4 M. Azuma, 5 M. Takano, 5 H. Eisaki, 6 H. Takagi, 2,4 S. Uchida, 2,7 J. C. Davis 1,8 * 9 MARCH 2007 VOL 315 SCIENCE PHYSICAL REVIEW B 77, Superconducting d-wave stripes in cuprates: Valence bond order coexisting with nodal quasiparticles Matthias Vojta and Oliver Rösch Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, Köln, Germany We point out that unidirectional bond-centered charge-density-wave states in cuprates involve electronic order in both s- and d-wave channels, with nonlocal Coulomb repulsion suppressing the s-wave component. The resulting bond-charge-density wave, coexisting with superconductivity, is compatible with recent photo-

94 Direct observation of competition between superconductivity and charge density wave order in YBa 2 Cu 3 O 6.67 NATURE PHYSICS VOL 8 DECEMBER 2012 J. Chang 1,2 *, E. Blackburn 3, A. T. Holmes 3, N. B. Christensen 4, J. Larsen 4,5, J. Mesot 1,2, Ruixing Liang 6,7, D. A. Bonn 6,7, W. N. Hardy 6,7, A. Watenphul 8, M. v. Zimmermann 8, E. M. Forgan 3 and S. M. Hayden 9 a T C 250 T N T YBCO Intensity (cnts s ) T 15 T 7.5 T 0 T Q = (1.695, 0, 0.5) Q = (0, 3.691, 0.5) (x4) T CDW Temperature (K) T SDW T H T NMR T CDW a T (K) b I (cnts s 1 ) T c 0 T T = 2 K AF Doping, p (holes/cu) Cut 1 17 T c Cut 1 17 T c Cut T 0 T I (cnts s 1 ) T = 66 K T = 150 K h in (h, 0, 0.5) (r.l.u.) h in (h, 0, 0.5) (r.l.u.) h in (h, 0, 0.5) (r.l.u.) c I (cnts s 1 ) SC 0.20

95 Distinct Charge Orders in the Planes and Chains of Ortho-III-Ordered YBa 2 Cu 3 O 6þ Superconductors Identified by Resonant Elastic X-ray Scattering hkar, 1 R. Sutarto, 2,3 X. Mao, 1 F. He, 3 A. Frano, 4,5 S. Blanco-Canosa, 4 M. Le Tacon, 4 G. Ghiringhe raicovich, 6 M. Minola, 6 M. Moretti Sala, 7 C. Mazzoli, 6 Ruixing Liang, 2 D. A. Bonn, 2 W. N. Hardy, 2 B. Keimer, 4 G. A. Sawatzky, 2 and D. G. Hawthorn 1, * A. J. Achkar, 1 R. Sutarto, 2,3 X. Mao, 1 F. He, 3 A. Frano, 4,5 S. Blanco-Canosa, 4 M. Le Tacon, 4 G. Ghiringhelli, 6 L. Braicovich, 6 M. Minola, 6 M. Moretti Sala, 7 C. Mazzoli, 6 Ruixing Liang, 2 D. A. Bonn, 2 W. N. Hardy, 2 B. Keimer, 4 G. A. Sawatzky, 2 and D. G. Hawthorn 1, * 1 Department of Physics and Astronomy, University of Waterloo, Waterloo, Ontario, Canada N2L 3G (2013) P H Y S I C A L R E V I E W L E T T E R S tment of Physics and 1 Astronomy, University of British Columbia, Vancouver, British Columbia, Canada V6T 1 PRL 109, (2012) 3 Canadian Light Source, University of Saskatchewan, Saskatoon, PRL 109, (2012) PHYSICSaskatchewan, AL REVIECanada W LES7N TT0X4 ERS 4 Max-Planck-Institut für Festkörperforschung, Heisenbergstraße 1, D Stuttgart, Germany elmholtz-zentrum nt X-Ray Scattering Berlin für Materialien Measurements und Energie, of Albert-Einstein-Straße a Spatial Modulation 15, D of the Berlin, CuGermany 3d and N, CNISM and Energies Dipartimentoindi Stripe-Ordered Fisica, Politecnico di Milano, Cuprate Piazza Superconductors Leonardo da Vinci 32, I Milano 7 European Synchrotron Radiation Facility, BP 220, F Grenoble Cedex, France (Received 16 July 2012; published 17 October 2012) A. J. Achkar, 1 F. He, 2 R. Sutarto, 3 J. Geck, 4 H. Zhang, 5 Y.-J. Kim, 5 and D. G. Hawthorn 1 Recently, 1 Department chargeof density Physics wave and(cdw) Astronomy, orderuniversity in the CuOof 2 Waterloo, planes of Waterloo underdoped N2L YBa 3G1, 2 CuCanada 3 O 6þ was detected 2 Canadian usinglight resonant Source, soft x-ray University scattering. of Saskatchewan, An important question Saskatoon, remains: Saskatchewan is the chain S7N layer 0X4, responsible Canada for 3 Department this chargeofordering? Physics and Here, Astronomy, we explore University the energy of British and polarization Columbia, dependence Vancouver V6T of the 1Z4, resonant Canada zscattering Institute for intensity Solid in State a detwinned and Materials sampleresearch of YBa 2 IFW Cu 3 ODresden, 6:75 with ortho-iii Helmholtzstraße oxygen ordering 20, in Dresden, the chain Germ layer. We show 5 Department that the ortho-iii of Physics, CDW order University in the of chains Toronto, is distinct Toronto fromm5s the1a7, CDWCanada order in the planes. The ortho-iii structure gives (Received rise to 12 a March commensurate 2012; published superlattice 2 January reflection 2013) at Q ¼½0:33 0 LŠ whose energy A prevailing and polarization description dependence of the stripe agrees phase withinexpectations underdoped for cuprate oxygen superconductors ordering and aisspatial that the modulation of(holes) the Cu phase valencesegregate in the chains. on a Incommensurate microscopic scale peaks intoat hole-rich [ L] and and hole-poor [ L] regions. from the WeCDW report charge carriers resonant order in the elastic planes x-ray arescattering shown tomeasurements be distinct in Qof as stripe-ordered well as theirla temperature, 1:475 Nd 0:4 Srenergy, 0:125 CuOand 4 atpolarization the Cu L and Odependence, K absorption andedges are FIG. thus that 2 unrelated (color identify online). an to the additional structure The [Hfeature 0of L] the [(a) of chain and stripe layer. (b)] order. and Moreover, [0 Analysis KL] the ofenergy the energy dependence dependence of of the the CDW scattering order intensity [(c) the and planes reveals (d)] isnormalized shown that the todominant scattering result fromsignature intensity, a spatial Iof sc =I modulation the 0, in stripe arbitrary order of energies is a spatial of the modulation Cu 2p toin the 3d x units. The scattering intensity was measured with [(a) and (c)] 2 energies y 2 transition, of Cusimilar 3d andtoo stripe-ordered 2p states rather 214than cuprates. the large modulation of the charge density (valence) and [(b) and (d)] incident photon polarization at T ¼ 60 K. envisioned in the common stripe paradigm. These energy shifts are interpreted as a spatial modulation of DOI: /PhysRevLett r.l.u., reciprocal lattice units. PACS numbers: Gh, cp, Lr, Dm the electronic structure and may point to a valence-bond-solid interpretation of the stripe phase. we 4 JAN

96 Incommensurate d-wave bond order Observed low T ordering. K Our computations show that the charge order is predominantly d-wave also at this Q. This Q is preferred in computations of bond order within the superconducting phase. S. Sachdev and R. La Placa, Physical Review Letters in press; arxiv: M. Vojta and S. Sachdev, Physical Review Letters 83, 3916 (1999) M. Vojta and O. Rosch, Physical Review B 77, (2008)

97 Incommensurate d-wave bond order K High T pseudogap: Fluctuating O(4) order parameter of nearly degenerate d-wave pairing and incommensurate d-wave bond order. D c k Q/2, c k+q/2, E = Q (cos k x cos k y )

98 the et charg [10] J. Chang al., cq, (k) the single-electron Green s function, ImGk,k (! (9) + i ), and the Q (k) = arc featu [11] A. J. Achkar et results are shown in Fig. 4. The stability of the Fermi arc in argued by [12] D. LeBoeuf, can b D. A. arcs Bonn, and ere labels various orthonormal basis functions, and cq, [13] N. Doiron-Leyr In our c numerical coefficients that we determine. Thus we have [14] N. Harrison and ordering noteextended that Fig. 2s also has a broad local minimum s basis function s (k) Finally, = 1, the function (2011). blue val near Q = (, ). Here (k) is found to have the odd-parity Q (k) = cos k x + cos ky, the d function d (k) = cos k x cos ky [15] S. E. Sebastian, (±Q p form in Table I. Condensation of this mode will break 0, 0), x,y d so on, as shown in Table I. Depending upon the symmetry Phys. 75, due to an time-reversal symmetry, and lead to the state with sponta[16] B. Vignolle, D Q (in particular, the little group of the wavevector Q) and strong-co neous orbital currents proposed by Chakravarty C. et al. [25]. R. Physique he eigenvector, some of the cq, may be exactly zero. But We now study the electronic spectral function ina. the Q (0,Metlitsk 1) = dire [17] M. a generic Q, only time-reversal constrains the values of (Q0, 0) charge-ordered state. We will work with (2010). the state with c present, and we are allowed tobi-directional have an admixture of many charge order [13]; basis in our theory charge and or [18]the Ar.degeneracy Abanov ctions. Nevertheless, webetween will seethe that only a smalland number uni-directional bi-directional charge ordered (2000). as was the basis functions have appreciable coefficients, so quartic Eq. (9)in the[19]q,t.and state is broken only byand terms we have Holder and W be interes ImG(k,!accounted + i ) ImG(k,!+ i ) resents a useful notspectral for these Choosing the[20] largest 2 extension compo- an C. Husemann FIG. 4:expansion. Electron density in log thehere. phase with bidirectional [21] K. B. Efetov, H Table we theqorder parameter charge order at nents Q = from (Q0, 0) and I,(0, Q0have ) with 0 = 4 /11. The For th ( i ) at! = 0 and = 0.02; the[22] S. Chakravarty, left panel show ImG (! + right k,k D E ha cos ky ), Q = (±Q0, model 0) s + d (cos k x B 63 (k) = + 0i ) panel shows log same parameters, as a wayphys. Rev. (9) ck Q/2, ck+q/2, / ImG (cosfork xthe cos k ) Q(k) k,k (! Q y cos ky ), Q = (0, ±Qsociated w s d (cos k x 0) of enhancing the low intensities. The dashed line is the underlying by Eq. (9 Fermi surface ofwith the metal without charge order. The charge order is We computed the imaginary part of s/ d = to d as in Eq. (9) with = 0.3, and othergreen s parameters as in Fig. the dsingle-electron function, ImG2.k,k (! + i ), point and the S. Sachdev and R. La Placa, arxiv: the stran results are shown in Fig. 4. The stability of the Fermi arc in Electron spectral function

99 Summary Antiferromagnetism in metals and the high temperature superconductors Antiferromagnetic quantum criticality leads to d-wave superconductivity (supported by sign-problemfree Monte Carlo simulations)

100 Summary Antiferromagnetism in metals and the high temperature superconductors Antiferromagnetic quantum criticality leads to d-wave superconductivity (supported by sign-problemfree Monte Carlo simulations) Metals with antiferromagnetic spin correlations have nearly degenerate instabilities: to d-wave superconductivity, and to a charge density wave with a d- wave form factor. This is a promising explanation of the pseudogap regime.