1. N ON - EQUILIBRIUM DYNAMIC CRITICAL SCALING OF THE QUANTUM I SING CHAIN Michael Kolodrubetz Princeton University In collaboration with: Bryan Clark, David Huse David Pekker Krishnendu Sengupta

2. Q UANTUM STATE OF TRANSVERSE - FIELD I SING MODEL DURING SLOW RAMP IS … Universal Non-equilibrium Experimentally viable Non-thermal Dephasing resistant

3. C LASSICAL P HASE T RANSITIONS “Magnetization” Landau-Ginzburg functional

4. C LASSICAL P HASE T RANSITIONS “Magnetization”

5.  Thermal fluctuations C LASSICAL P HASE T RANSITIONS

6. Q UANTUM P HASE T RANSITIONS One-dimensional transverse-field Ising chain

7. Q UANTUM P HASE T RANSITIONS One-dimensional transverse-field Ising chain Paramagnet (PM) Ferromagnet (FM)

8. Q UANTUM P HASE T RANSITIONS One-dimensional transverse-field Ising chain Paramagnet (PM) Ferromagnet (FM)  Quantum fluctuations

9. C RITICAL S CALING [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]

10. C RITICAL S CALING [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]

11. C RITICAL S CALING Correlation length critical exponent Dynamic critical exponent, [Smirnov, php.math.unifi.it/users/paf/ LaPietra/files/Chelkak01.ppt]

12. C RITICAL S CALING, Ising: Correlation length critical exponent Dynamic critical exponent

13. C RITICAL S CALING, Ising: Correlation length critical exponent Dynamic critical exponent Order parameter critical exponent

14. C RITICAL S CALING, Ising: Correlation length critical exponent Dynamic critical exponent Order parameter critical exponent

15. K IBBLE -Z UREK RAMPS Ramp rate

16. K IBBLE -Z UREK RAMPS Ramp rate

17. K IBBLE -Z UREK RAMPS Adiabatic Ramp rate

18. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate

19. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate

20. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate

21. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate

22. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate

23. K IBBLE -Z UREK RAMPS METHODIDEAWHEN IT WORKS “Old-school” Kibble-Zurek [Kibble 1976, Zurek 1985] and set the “interesting” time and length scales -Ramp to the QCP -Ramp to deep in the FM phase

24. K IBBLE -Z UREK RAMPS METHODIDEAWHEN IT WORKS “Old-school” Kibble-Zurek [Kibble 1976, Zurek 1985] and set the “interesting” time and length scales -Ramp to the QCP -Ramp to deep in the FM phase Kibble-Zurek scaling [Deng et. al. 2008, Erez et. al., in prep., Polkovnikov, …] Most quantities show scaling collapse when scaled by and Throughout the ramp

25. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate

26. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate

27. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate

28. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate

29. T RANSVERSE - FIELD I SING CHAIN Sachdev: “Quantum Phase Transitions”

30. T RANSVERSE - FIELD I SING CHAIN Wigner fermionize Sachdev: “Quantum Phase Transitions”  phase

31. T RANSVERSE - FIELD I SING CHAIN Wigner fermionize Sachdev: “Quantum Phase Transitions”  phase

32. T RANSVERSE - FIELD I SING CHAIN Wigner fermionize Quadratic  Integrable Sachdev: “Quantum Phase Transitions”  phase

33. T RANSVERSE - FIELD I SING CHAIN Wigner fermionize Quadratic  Integrable Hamiltonian conserves parity for each mode k Sachdev: “Quantum Phase Transitions”  phase

34. T RANSVERSE - FIELD I SING CHAIN Wigner fermionize Quadratic  Integrable Hamiltonian conserves parity for each mode k Work in subspace where parity is even Sachdev: “Quantum Phase Transitions”  phase

35. T RANSVERSE - FIELD I SING CHAIN Wigner fermionize Quadratic  Integrable Hamiltonian conserves parity for each mode k Work in subspace where parity is even Sachdev: “Quantum Phase Transitions”  phase

36. T RANSVERSE - FIELD I SING CHAIN

42. Low energy, long wavelength theory

43. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate Low energy, long wavelength theory?

44. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate Low energy, long wavelength theory

45. K IBBLE -Z UREK RAMPS Adiabatic Impulse Ramp rate Low energy, long wavelength theory

46. K IBBLE -Z UREK SCALING LIMIT Schrödinger Equation OR Observable Fixed

47. K IBBLE -Z UREK SCALING LIMIT

51. K IBBLE -Z UREK OBSERVABLES Excess heat Spin-spin correlation function

52. K IBBLE -Z UREK OBSERVABLES Excess heat Spin-spin correlation function

53. K IBBLE -Z UREK OBSERVABLES

60. F INITE - SIZE SCALING

62. Finite size effects can be ignored

63. F INITE - SIZE SCALING

65. E QUILIBRIUM VIA DYNAMICS KZ scaling function Equilibrium scaling function If dynamic scaling functions exist, they must have the equilibrium critical exponents

66. F INITE - SIZE SCALING

68. L ANDAU -Z ENER DYNAMICS

71. F INITE - SIZE SCALING

72. L ANDAU -Z ENER DYNAMICS

73. A THERMAL PROPERTIES

77. Inverted

78. A THERMAL PROPERTIES Kibble-Zurek

79. A THERMAL PROPERTIES Kibble-Zurek Thermal

80. D EPHASING Protocol Ramp to create excitations Freeze the Hamiltonian Wait

81. D EPHASING Protocol Ramp to create excitations Freeze the Hamiltonian Wait … …

82. D EPHASING … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait

83. D EPHASING … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait

84. D EPHASING … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait

85. D EPHASING … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait

86. D EPHASING Dephasing in integrable model: Generalized Gibb’s ensemble (GGE) … … Protocol Ramp to create excitations Freeze the Hamiltonian Wait

87. D EPHASING Dephasing in integrable model: Generalized Gibb’s ensemble (GGE) … … Does dephasing occur during the Kibble- Zurek ramp?

88. D EPHASING Dephasing in integrable model: Generalized Gibb’s ensemble (GGE) … …

89. D EPHASING Dephasing in integrable model: Generalized Gibb’s ensemble (GGE) as … …

90. D EPHASING Dephasing in integrable model: Generalized Gibb’s ensemble (GGE) as … …

91. D EPHASING Cubic ramp: … …

92. D EPHASING Cubic ramp: as … …

93. D EPHASING Cubic ramp: as … …

94. U NIVERSALITY

96. Additional terms change (renormalize) the non- universal aspects of the critical point They do not change critical scaling Critical exponents Scaling functions Debated for non-integrable system dynamics

97. U NIVERSALITY

99. Paramagnet Antiferromagnet

100. U NIVERSALITY Paramagnet Antiferromagnet Ramp the tilt ( ) linearly in time

101. U NIVERSALITY

105. C ONCLUSIONS Solved dynamic critical scaling behavior of the TFI chain Athermal  negative correlations Phase-locked high order ramps Strong numerical evidence for universality Tilted boson model has same scaling functions Experimentally accessible Athermal features robust against open boundary conditions Open b.c. simplifies measurement Time scales already available [Simon et. al., 2007]

106. D EPHASING VIA QUASIPARTICLES

108. O PEN BOUNDARY CONDITIONS

111. U NIVERSALITY Remove spin ups on neighboring sites