1. Quantum critical phenomena Talk online: sachdev.physics.harvard.edu Talk online: sachdev.physics.harvard.edu Quantum critical phenomena Talk online: sachdev.physics.harvard.edu Talk online: sachdev.physics.harvard.edu

2. 1. Coupled dimer antiferromagnets Landau-Ginzburg quantum criticality 2. Spin density waves in metals Paramagnon quantum criticality 3. Spin liquids and valence bond solids Schwinger-boson mean-field theory and U(1) gauge theory Outline

3. References Exotic phases and quantum phase transitions: model systems and experiments, Rapporteur talk at the 24th Solvay Conference on Physics, "Quantum Theory of Condensed Matter", arXiv:0901.4103 Quantum magnetism and criticality, Nature Physics 4, 173 (2008), arXiv:0711.3015 Quantum phases and phase transitions of Mott insulators, arXiv:cond-mat/0401041

4. 1. Coupled dimer antiferromagnets Landau-Ginzburg quantum criticality 2. Spin density waves in metals Paramagnon quantum criticality 3. Spin liquids and valence bond solids Schwinger-boson mean-field theory and U(1) gauge theory Outline

5. TlCuCl 3

6. An insulator whose spin susceptibility vanishes exponentially as the temperature T tends to zero.

7. N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001). TlCuCl 3 at ambient pressure

8. N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001). Sharp spin 1 particle excitation above an energy gap (spin gap) TlCuCl 3 at ambient pressure

9. Ground state has long-range Néel order Square lattice antiferromagnet

10. J J/ Weaken some bonds to induce spin entanglement in a new quantum phase

11. Square lattice antiferromagnet J J/ Ground state is a “quantum paramagnet” with spins locked in valence bond singlets

12. Pressure in TlCuCl 3

13. Quantum critical point with non-local entanglement in spin wavefunction

19. N. Cavadini, G. Heigold, W. Henggeler, A. Furrer, H.-U. Güdel, K. Krämer and H. Mutka, Phys. Rev. B 63 172414 (2001). Sharp spin 1 particle excitation above an energy gap (spin gap) TlCuCl 3 at ambient pressure

21. Spin waves

24. Discussion of quantum rotor model

25. CFT3

32. Spin waves

36. Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans–Ulrich Gudel, Severian Gvasaliya, Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, 205701 (2008) TlCuCl 3 with varying pressure

37. Prediction of quantum field theory Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans–Ulrich Gudel, Severian Gvasaliya, Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, 205701 (2008)

38. CFT3

39. S. Wenzel and W. Janke, arXiv:0808.1418 M. Troyer, M. Imada, and K. Ueda, J. Phys. Soc. Japan (1997) Quantum Monte Carlo - critical exponents

40. Field-theoretic RG of CFT3 E. Vicari et al. S. Wenzel and W. Janke, arXiv:0808.1418 M. Troyer, M. Imada, and K. Ueda, J. Phys. Soc. Japan (1997)

41. 1. Coupled dimer antiferromagnets Landau-Ginzburg quantum criticality 2. Spin density waves in metals Paramagnon quantum criticality 3. Spin liquids and valence bond solids Schwinger-boson mean-field theory and U(1) gauge theory Outline

42. 1. Coupled dimer antiferromagnets Landau-Ginzburg quantum criticality 2. Spin density waves in metals Paramagnon quantum criticality 3. Spin liquids and valence bond solids Schwinger-boson mean-field theory and U(1) gauge theory Outline

43. Fermi surfaces in electron- and hole-doped cuprates Hole states occupied Electron states occupied

44. Spin density wave theory

46. Spin density wave theory in electron-doped cuprates S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

47. Spin density wave theory in electron-doped cuprates S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

48. Spin density wave theory in electron-doped cuprates S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Hole pockets Electron pockets

49. Spin density wave theory in electron-doped cuprates S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Electron pockets

50. N. P. Armitage et al., Phys. Rev. Lett. 88, 257001 (2002). Photoemission in NCCO

51. Spin density wave theory in hole-doped cuprates S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

52. Spin density wave theory in hole-doped cuprates S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997).

53. Spin density wave theory in hole-doped cuprates S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Electron pockets Hole pockets

54. Spin density wave theory in hole-doped cuprates S. Sachdev, A. V. Chubukov, and A. Sokol, Phys. Rev. B 51, 14874 (1995). A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997). Hole pockets

55. Spin density wave theory

57. 1. Coupled dimer antiferromagnets Landau-Ginzburg quantum criticality 2. Spin density waves in metals Paramagnon quantum criticality 3. Spin liquids and valence bond solids Schwinger-boson mean-field theory and U(1) gauge theory Outline

58. 1. Coupled dimer antiferromagnets Landau-Ginzburg quantum criticality 2. Spin density waves in metals Paramagnon quantum criticality 3. Spin liquids and valence bond solids Schwinger-boson mean-field theory and U(1) gauge theory Outline

59. Half-filled band  Mott insulator with spin S = 1/2 Triangular lattice of [Pd(dmit) 2 ] 2  frustrated quantum spin system X[Pd(dmit) 2 ] 2 Pd SC X Pd(dmit) 2 t’ t t Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007)

62. Anisotropic triangular lattice antiferromagnet Neel ground state for small J’/J Broken spin rotation symmetry

63. Anisotropic triangular lattice antiferromagnet

64. Magnetic Criticality T N (K) Neel order Me 4 P Me 4 As EtMe 3 As Et 2 Me 2 As Me 4 Sb Et 2 Me 2 P EtMe 3 Sb Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007) X[Pd(dmit) 2 ] 2 Et 2 Me 2 Sb (CO)

65. Anisotropic triangular lattice antiferromagnet Possible ground state for intermediate J’/J N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)

66. Anisotropic triangular lattice antiferromagnet Possible ground state for intermediate J’/J Valence bond solid (VBS) Broken lattice space group symmetry N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)

67. Anisotropic triangular lattice antiferromagnet Broken lattice space group symmetry Possible ground state for intermediate J’/J Valence bond solid (VBS) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)

68. Anisotropic triangular lattice antiferromagnet Broken lattice space group symmetry Possible ground state for intermediate J’/J Valence bond solid (VBS) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)

69. Anisotropic triangular lattice antiferromagnet Broken lattice space group symmetry Possible ground state for intermediate J’/J Valence bond solid (VBS) N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)

70. Anisotropic triangular lattice antiferromagnet

71. = Triangular lattice antiferromagnet Z 2 spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

72. = Triangular lattice antiferromagnet Z 2 spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

73. = Triangular lattice antiferromagnet Z 2 spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

74. = Triangular lattice antiferromagnet Z 2 spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

75. = Triangular lattice antiferromagnet Z 2 spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

76. = Triangular lattice antiferromagnet Z 2 spin liquid N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991) X.-G. Wen, Phys. Rev. B 44, 2664 (1991)

77. Excitations of the Z 2 Spin liquid = A spinon

78. Excitations of the Z 2 Spin liquid = A spinon

79. Excitations of the Z 2 Spin liquid = A spinon

80. Excitations of the Z 2 Spin liquid = A spinon

81. Anisotropic triangular lattice antiferromagnet

82. Magnetic Criticality T N (K) Neel order Me 4 P Me 4 As EtMe 3 As Et 2 Me 2 As Me 4 Sb Et 2 Me 2 P EtMe 3 Sb Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007) X[Pd(dmit) 2 ] 2 Et 2 Me 2 Sb (CO)

83. Magnetic Criticality T N (K) Neel order Me 4 P Me 4 As EtMe 3 As Et 2 Me 2 As Me 4 Sb Et 2 Me 2 P EtMe 3 Sb EtMe 3 P Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007) X[Pd(dmit) 2 ] 2 Et 2 Me 2 Sb (CO) Spin gap Spin gap

84. Magnetic Criticality T N (K) Neel order Me 4 P Me 4 As EtMe 3 As Et 2 Me 2 As Me 4 Sb Et 2 Me 2 P EtMe 3 Sb EtMe 3 P Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007) X[Pd(dmit) 2 ] 2 Et 2 Me 2 Sb (CO) VBS order Spin gap Spin gap

85. M. Tamura, A. Nakao and R. Kato, J. Phys. Soc. Japan 75, 093701 (2006) Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, Phys. Rev. Lett. 99, 256403 (2007) Observation of a valence bond solid (VBS) in ETMe 3 P[Pd(dmit) 2 ] 2 Spin gap ~ 40 K J ~ 250 K X-ray scattering

86. Magnetic Criticality T N (K) Neel order Me 4 P Me 4 As EtMe 3 As Et 2 Me 2 As Me 4 Sb Et 2 Me 2 P EtMe 3 Sb EtMe 3 P Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, J. Phys.: Condens. Matter 19, 145240 (2007) X[Pd(dmit) 2 ] 2 Et 2 Me 2 Sb (CO) VBS order Spin gap Spin gap

87. Discussion of Schwinger bosons on the square lattice and U(1) gauge theory http://qpt.physics.harvard.edu/leshouches/schwinger_bosons.pdf

88. Schwinger boson mean field theory on the square lattice and perturbative fluctuations Origin of gauge invariance

89. Schwinger boson mean field theory on the square lattice and perturbative fluctuations