1. Quantum Phase Transitions Subir Sachdev Talks online at http://sachdev.physics.harvard.edu

2. What is a “phase transition” ? A change in the collective properties of a macroscopic number of atoms

3. What is a “quantum phase transition” ? Change in the nature of entanglement in a macroscopic quantum system.

4. Entanglement Hydrogen atom: Hydrogen molecule: = _ Superposition of two electron states leads to non-local correlations between spins

5. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

6. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

7. Chinese Terracotta warriors (479-221 BC) M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Han Purple – BaCuSi 2 O 6

8. M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Each Cu 2+ has a single free electron spin Han Purple – BaCuSi 2 O 6

9. M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Weak magnetic field Han Purple – BaCuSi 2 O 6

10. M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Strong magnetic field Magnetic field, H Han Purple – BaCuSi 2 O 6

11. QPM XY-AFM SPM M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Han Purple – BaCuSi 2 O 6

12. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

13. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

14. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

15. M. Jaime et al., Phys. Rev. Lett. 93, 087203 (2004) Each Cu 2+ has a single free electron spin. Han Purple – BaCuSi 2 O 6 Vary the ratio J/J’

16. J/J’ Vary the ratio J/J’ Neel Spin gap

17. J/J’ Vary the ratio J/J’ Temperature, T 0 Neel Spin gap

18. J/J’ Vary the ratio J/J’ Temperature, T 0 Spin wave

19. J/J’ Vary the ratio J/J’ Temperature, T 0 Neel Spin gap

20. J/J’ Vary the ratio J/J’ Temperature, T 0 “Triplet magnon” Neel Spin gap

21. J/J’ Vary the ratio J/J’ Temperature, T 0 Neel Spin gap

22. J/J’ Vary the ratio J/J’ Temperature, T 0 S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989) Neel Spin gap

23. J/J’ Vary the ratio J/J’ Temperature, T 0 Quantum Criticality S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992).

24. J/J’ Vary the ratio J/J’ Temperature, T 0 S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). Thermal excitations interact via a universal S matrix. Quantum Criticality Neel Spin gap

25. J/J’ Vary the ratio J/J’ Temperature, T 0 Decoherence time S. Sachdev and J. Ye, Phys. Rev. Lett. 69, 2411 (1992). Neel Spin gap

26. J/J’ Vary the ratio J/J’ Temperature, T 0 A.V. Chubukov, S. Sachdev and J. Ye, Phys. Rev. B 49, 11919 (1994). Quantum critical transport Spin diffusion constant where  is a universal number Neel Spin gap

27. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

28. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

29. Trap for ultracold 87 Rb atoms

30. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

31. The Bose-Einstein condensate in a periodic potential Lowest energy state for many atoms Large fluctuations in number of atoms in each potential well – superfluidity (atoms can “flow” without dissipation)

32. By tuning repulsive interactions between the atoms, states with multiple atoms in a potential well can be suppressed. The lowest energy state is then a Mott insulator – it has negligible number fluctuations, and atoms cannot “flow” Breaking up the Bose-Einstein condensate Lowest energy state for many atoms

33. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Velocity distribution of 87 Rb atoms Superfliud

34. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002). Velocity distribution of 87 Rb atoms Insulator

35. Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential

36. Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential Dynamics of the classical Gross-Pitaevski equation

37. Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential Dilute Boltzmann gas of particle and holes

38. Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential No wave or quasiparticle description

39. D. B. Haviland, Y. Liu, and A. M. Goldman, Phys. Rev. Lett. 62, 2180 (1989) Resistivity of Bi films M. P. A. Fisher, Phys. Rev. Lett. 65, 923 (1990)

40. Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential

41. Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential Collisionless-to hydrodynamic crossover of a conformal field theory (CFT) K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

42. Superfluid Insulator Non-zero temperature phase diagram Depth of periodic potential Collisionless-to hydrodynamic crossover of a conformal field theory (CFT) K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997). Needed: Cold atom experiments in this regime

43. Hydrodynamics of a conformal field theory (CFT) Maldacena’s AdS/CFT correspondence relates the hydrodynamics of CFTs to the quantum gravity theory of the horizon of a black hole in Anti-de Sitter space.

44. Holographic representation of black hole physics in a 2+1 dimensional CFT at a temperature equal to the Hawking temperature of the black hole. Black hole 3+1 dimensional AdS space Hydrodynamics of a conformal field theory (CFT)

45. Hydrodynamics of a CFT Waves of gauge fields in a curved background

46. Hydrodynamics of a conformal field theory (CFT) The scattering cross-section of the thermal excitations is universal and so transport co- efficients are universally determined by k B T Charge diffusion constant Conductivity K. Damle and S. Sachdev, Phys. Rev. B 56, 8714 (1997).

47. Hydrodynamics of a conformal field theory (CFT) P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, hep-th/0701036 Spin diffusion constant Spin conductivity For the (unique) CFT with a SU(N) gauge field and 16 supercharges, we know the exact diffusion constant associated with a global SO(8) symmetry:

48. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

49. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

50. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

51. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

52. Resonance in benzene leads to a symmetric configuration of valence bonds (F. Kekulé, L. Pauling) Valence bonds in benzene

53. Resonance in benzene leads to a symmetric configuration of valence bonds (F. Kekulé, L. Pauling) Valence bonds in benzene

54. Resonance in benzene leads to a symmetric configuration of valence bonds (F. Kekulé, L. Pauling) Valence bonds in benzene

55. Temperature-doping phase diagram of the cuprate superconductors

56. Antiferromagnetic (Neel) order in the insulator

57. Induce formation of valence bonds by e.g. ring-exchange interactions A. W. Sandvik, cond-mat/0611343

58. = As in H 2 and benzene, each electron wants to pair up with another electron and form a valence bond

59. =

60. =

61. =

62. =

63. = Entangled liquid of valence bonds (Resonating valence bonds – RVB) P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974).

64. = N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS)

65. = N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS)

66. = N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

67. = N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

68. = N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

69. = N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

70. = N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

71. = N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

72. = N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989). R. Moessner and S. L. Sondhi, Phys. Rev. B 63, 224401 (2001). Valence bond solid (VBS) More possibilities for entanglement with nearby states

73. = Excitations of the RVB liquid

74. =

75. =

76. =

77. = Electron fractionalization: Excitations carry spin S=1/2 but no charge

78. = Excitations of the VBS

79. =

80. =

81. =

82. = Free spins are unable to move apart: no fractionalization, but confinement

83. Phase diagram of square lattice antiferromagnet A. W. Sandvik, cond-mat/0611343

84. Phase diagram of square lattice antiferromagnet VBS order K/J Neel order T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

85. Phase diagram of square lattice antiferromagnet VBS order K/J Neel order T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004). RVB physics appears at the quantum critical point which has fractionalized excitations: “deconfined criticality”

86. Phase diagram of square lattice antiferromagnet VBS order K/J Neel order T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).

87. Temperature, T 0 Quantum criticality of fractionalized excitations K/J

88. Phases of nuclear matter

89. X[Pd(dmit) 2 ] 2 Pd C S One free electron spin on each vertex of a triangular lattice M. Tamura et al., J. Phys. Soc. Jpn. 75, 093701 (2006) Observation of a valence bond solid (VBS)

90. Pressure-temperature phase diagram of ETMe 3 P[Pd(dmit) 2 ] 2 Y. Shimizu et al. cond-mat/0612545 RVB (?) Observation of a valence bond solid (VBS)

91. Temperature-doping phase diagram of the cuprate superconductors

92. VBS order K/J Neel order Deconfined quantum critical point (DQCP)

93. Temperature-doping phase diagram of the cuprate superconductors Hole concentration Neel order + d-wave supercon- ductivity “Supercon- ducting algebraic holon liquid” d-wave supercon- ductivity R.K. Kaul, Y.-B. Kim, S. Sachdev and T. Senthil, to appear DQCP

94. Temperature-doping phase diagram of the cuprate superconductors Hole concentration Neel order + d-wave supercon- ductivity “Supercon- ducting algebraic holon liquid” d-wave supercon- ductivity R.K. Kaul, Y.-B. Kim, S. Sachdev and T. Senthil, to appear DQCP Quantum critical phases with enhanced VBS correlations

95. Temperature-doping phase diagram of the cuprate superconductors STM in zero field

96. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

97. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

98. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

99. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

100. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007)

101. Y. Kohsaka, C. Taylor, K. Fujita, A. Schmidt, C. Lupien, T. Hanaguri, M. Azuma, M. Takano, H. Eisaki, H. Takagi, S. Uchida, and J. C. Davis, Science 315, 1380 (2007) “Glassy” Valence Bond Solid (VBS)

102. Temperature-doping phase diagram of the cuprate superconductors “Glassy” Valence Bond Solid (VBS)

103. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

104. Outline 1.Spin ordering in “Han purple” 2.Entanglement at the critical point: physical consequences at non-zero temperatures (a) Double-layer antiferromagnet (b) Superfluid-insulator transition (c) Hydrodynamics via mapping to quantum theory of black holes. 3.Entanglement of valence bonds 4.Conclusions Quantum phase transitions

105. Studies of new materials and trapped ultracold atoms are yielding new quantum phases, with novel forms of quantum entanglement. Some materials are of technological importance: e.g. high temperature superconductors. Real-world studies on the entanglement of large numbers of qubits: insights may be important for quantum cryptography and quantum computing. Tabletop “laboratories for the entire universe”: quantum mechanics of black holes, quark-gluon plasma, neutrons stars, and big-bang physics. Conclusions