Quantum Phase Transitions Subir Sachdev Talks online at sachdev.physics.harvard

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 Quantum phase transitions • Spin ordering in “Han purple” • 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. • Entanglement of valence bonds • Conclusions

6. Outline Quantum phase transitions • Spin ordering in “Han purple” • 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. • Entanglement of valence bonds • Conclusions

7. Han Purple – BaCuSi2O6 Chinese Terracotta warriors (479-221 BC) M. Jaime et al.,Phys. Rev. Lett.93, 087203 (2004)

8. Han Purple – BaCuSi2O6 Each Cu2+ has a single free electron spin M. Jaime et al.,Phys. Rev. Lett.93, 087203 (2004)

9. Weak magnetic field Han Purple – BaCuSi2O6 M. Jaime et al.,Phys. Rev. Lett.93, 087203 (2004)

10. Strong magnetic field Han Purple – BaCuSi2O6 Magnetic field, H M. Jaime et al.,Phys. Rev. Lett.93, 087203 (2004)

11. Han Purple – BaCuSi2O6 SPM XY-AFM QPM M. Jaime et al.,Phys. Rev. Lett.93, 087203 (2004)

12. Outline Quantum phase transitions • Spin ordering in “Han purple” • 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. • Entanglement of valence bonds • Conclusions

13. Outline Quantum phase transitions • Spin ordering in “Han purple” • 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. • Entanglement of valence bonds • Conclusions

14. Outline Quantum phase transitions • Spin ordering in “Han purple” • 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. • Entanglement of valence bonds • Conclusions

15. Han Purple – BaCuSi2O6 Each Cu2+ has a single free electron spin. Vary the ratio J/J’ M. Jaime et al.,Phys. Rev. Lett.93, 087203 (2004)

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

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

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

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

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

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

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

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

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

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

26. Temperature, T 0 Quantum critical transport Spin diffusion constant where Q is a universal number Spin gap Neel J/J’ Vary the ratio J/J’ A.V. Chubukov, S. Sachdev and J. Ye, Phys. Rev. B49, 11919 (1994).

27. Outline Quantum phase transitions • Spin ordering in “Han purple” • 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. • Entanglement of valence bonds • Conclusions

28. Outline Quantum phase transitions • Spin ordering in “Han purple” • 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. • Entanglement of valence bonds • Conclusions

29. Trap for ultracold 87Rb atoms

30. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature415, 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. Breaking up the Bose-Einstein condensate Lowest energy state for many atoms 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”

33. Velocity distribution of 87Rb atoms Superfliud M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature415, 39 (2002).

34. Velocity distribution of 87Rb atoms Insulator M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature415, 39 (2002).

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

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

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

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

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

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

41. Non-zero temperature phase diagram Collisionless-to hydrodynamic crossover of a conformal field theory (CFT) Insulator Superfluid Depth of periodic potential K. Damle and S. Sachdev, Phys. Rev. B56, 8714 (1997).

42. Non-zero temperature phase diagram Needed: Cold atom experiments in this regime Collisionless-to hydrodynamic crossover of a conformal field theory (CFT) Insulator Superfluid Depth of periodic potential K. Damle and S. Sachdev, Phys. Rev. B56, 8714 (1997).

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. 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. Holographic representation of black hole physics in a 2+1 dimensional CFT at a temperature equal to the Hawking temperature of the black hole. 3+1 dimensional AdS space Black hole

45. Hydrodynamics of a conformal field theory (CFT) Hydrodynamics of a CFT Waves of gauge fields in a curved background

46. Charge diffusion constant Conductivity 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 kBT K. Damle and S. Sachdev, Phys. Rev. B56, 8714 (1997).

47. Spin diffusion constant Spin conductivity Hydrodynamics of a conformal field theory (CFT) 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: P. Kovtun, C. Herzog, S. Sachdev, and D.T. Son, hep-th/0701036

48. Outline Quantum phase transitions • Spin ordering in “Han purple” • 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. • Entanglement of valence bonds • Conclusions

49. Outline Quantum phase transitions • Spin ordering in “Han purple” • 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. • Entanglement of valence bonds • Conclusions

50. Outline Quantum phase transitions • Spin ordering in “Han purple” • 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. • Entanglement of valence bonds • Conclusions