1. Lecture 14: Spin glasses Outline: the EA and SK models heuristic theory dynamics I: using random matrices dynamics II: using MSR

2. Random Ising model So far we dealt with “uniform systems” J ij was the same for all pairs of neighbours.

3. Random Ising model So far we dealt with “uniform systems” J ij was the same for all pairs of neighbours. What if every J ij is picked (independently) from some distribution?

4. Random Ising model So far we dealt with “uniform systems” J ij was the same for all pairs of neighbours. What if every J ij is picked (independently) from some distribution? We want to know the average of physical quantities (thermodynamic functions, correlation functions, etc) over the distribution of J ij ’s.

5. Random Ising model So far we dealt with “uniform systems” J ij was the same for all pairs of neighbours. What if every J ij is picked (independently) from some distribution? We want to know the average of physical quantities (thermodynamic functions, correlation functions, etc) over the distribution of J ij ’s. Today: a simple model with = 0

6. Random Ising model So far we dealt with “uniform systems” J ij was the same for all pairs of neighbours. What if every J ij is picked (independently) from some distribution? We want to know the average of physical quantities (thermodynamic functions, correlation functions, etc) over the distribution of J ij ’s. Today: a simple model with = 0 : spin glass

7. Simple model (Edwards-Anderson) Nearest-neighbour model with z neighbours

8. Simple model (Edwards-Anderson) Nearest-neighbour model with z neighbours note averages over different “samples” (1 sample = 1 realization of choices of J ij ’s for all pairs ( ij ) indicated by [ … ] av

9. Simple model (Edwards-Anderson) Nearest-neighbour model with z neighbours note averages over different “samples” (1 sample = 1 realization of choices of J ij ’s for all pairs ( ij ) indicated by [ … ] av

10. Simple model (Edwards-Anderson) Nearest-neighbour model with z neighbours note averages over different “samples” (1 sample = 1 realization of choices of J ij ’s for all pairs ( ij ) indicated by [ … ] av non-uniform J: anticipate nonuniform magnetization

11. Sherrington-Kirkpatrick model Every spin is a neighbour of every other one: z = (N – 1)

12. Sherrington-Kirkpatrick model Every spin is a neighbour of every other one: z = (N – 1)

13. Sherrington-Kirkpatrick model Every spin is a neighbour of every other one: z = (N – 1) Mean field theory is exact for this model

14. Sherrington-Kirkpatrick model Every spin is a neighbour of every other one: z = (N – 1) Mean field theory is exact for this model (but it is not simple)

15. Heuristic mean field theory replace total field on S i,

16. Heuristic mean field theory replace total field on S i,

17. Heuristic mean field theory replace total field on S i, (take h i = 0 )

18. Heuristic mean field theory replace total field on S i, by its mean (take h i = 0 )

19. Heuristic mean field theory replace total field on S i, by its mean (take h i = 0 )

20. Heuristic mean field theory replace total field on S i, by its mean and calculate m i as the average S of a single spin in field H : (take h i = 0 )

21. Heuristic mean field theory replace total field on S i, by its mean and calculate m i as the average S of a single spin in field H : (take h i = 0 )

22. Heuristic mean field theory replace total field on S i, by its mean and calculate m i as the average S of a single spin in field H : no preference for m i > 0 or <0 : (take h i = 0 )

23. Heuristic mean field theory replace total field on S i, by its mean and calculate m i as the average S of a single spin in field H : no preference for m i > 0 or <0 : [m ij ] av = 0 (take h i = 0 )

24. Heuristic mean field theory replace total field on S i, by its mean and calculate m i as the average S of a single spin in field H : no preference for m i > 0 or <0 : [m ij ] av = 0 if there are local spontaneous magnetizations m i ≠ 0, measure them by the order parameter (Edwards-Anderson) (take h i = 0 )

25. Heuristic mean field theory replace total field on S i, by its mean and calculate m i as the average S of a single spin in field H : no preference for m i > 0 or <0 : [m ij ] av = 0 if there are local spontaneous magnetizations m i ≠ 0, measure them by the order parameter (Edwards-Anderson) (take h i = 0 )

26. self-consistent calculation of q : To compute q : H i is a sum of many (seemingly) independent terms

27. self-consistent calculation of q : To compute q : H i is a sum of many (seemingly) independent terms => H i is Gaussian

28. self-consistent calculation of q : To compute q : H i is a sum of many (seemingly) independent terms => H i is Gaussian with variance

29. self-consistent calculation of q : To compute q : H i is a sum of many (seemingly) independent terms => H i is Gaussian with variance

30. self-consistent calculation of q : To compute q : H i is a sum of many (seemingly) independent terms => H i is Gaussian with variance so

31. self-consistent calculation of q : To compute q : H i is a sum of many (seemingly) independent terms => H i is Gaussian with variance so (solve for q )

32. spin glass transition:

33. expand in β :

34. spin glass transition: expand in β :

35. spin glass transition: expand in β :

36. spin glass transition: expand in β :

37. spin glass transition: expand in β : critical temperature: T c = J

38. spin glass transition: expand in β : critical temperature: T c = J below T c :

39. spin glass transition: expand in β : critical temperature: T c = J below T c : This heuristic theory is right up to this point, but wrong below T c.

40. the trouble below T c In the ferromagnet, it was safe to approximate

41. the trouble below T c In the ferromagnet, it was safe to approximate because the next term in a systematic expansion in β,

42. the trouble below T c In the ferromagnet, it was safe to approximate because the next term in a systematic expansion in β,

43. the trouble below T c In the ferromagnet, it was safe to approximate because the next term in a systematic expansion in β, was O(1/z).

44. the trouble below T c In the ferromagnet, it was safe to approximate because the next term in a systematic expansion in β, was O(1/z). But here, the average of the 1 st term is zero and you have to keep the second order term, the mean of which is of the order of the rms value of the first term.

45. the trouble below T c In the ferromagnet, it was safe to approximate because the next term in a systematic expansion in β, was O(1/z). But here, the average of the 1 st term is zero and you have to keep the second order term, the mean of which is of the order of the rms value of the first term.  Thouless-Anderson-Palmer (TAP) equations):

46. the trouble below T c In the ferromagnet, it was safe to approximate because the next term in a systematic expansion in β, was O(1/z). But here, the average of the 1 st term is zero and you have to keep the second order term, the mean of which is of the order of the rms value of the first term.  Thouless-Anderson-Palmer (TAP) equations): ______________ Onsager correction to mean field

47. Dynamics (I: simple way) Glauber dynamics:

48. Dynamics (I: simple way) Glauber dynamics:

49. Dynamics (I: simple way) Glauber dynamics: recall we derived from this

50. Dynamics (I: simple way) Glauber dynamics: recall we derived from this mean field:

51. Dynamics (I: simple way) Glauber dynamics: recall we derived from this mean field:

52. Dynamics I (continued)

53. linearize (above T c ):

54. Dynamics I (continued) linearize (above T c ): use TAP:

55. Dynamics I (continued) linearize (above T c ): use TAP:

56. Dynamics I (continued) linearize (above T c ): use TAP: In basis where J is diagonal:

57. Dynamics I (continued) linearize (above T c ): use TAP: In basis where J is diagonal:

58. Dynamics I (continued) linearize (above T c ): use TAP: In basis where J is diagonal: susceptibility:

59. Dynamics I (continued) linearize (above T c ): use TAP: In basis where J is diagonal: instability (transition) reached when maximum eigenvalue susceptibility:

60. Dynamics I (continued) linearize (above T c ): use TAP: In basis where J is diagonal: instability (transition) reached when maximum eigenvalue susceptibility:

61. eigenvalue spectrum of a random matrix For a dense random matrix with mean square element value J 2 /N, the eigenvalue density is “semicircular”:

62. eigenvalue spectrum of a random matrix For a dense random matrix with mean square element value J 2 /N, the eigenvalue density is “semicircular”:

63. eigenvalue spectrum of a random matrix For a dense random matrix with mean square element value J 2 /N, the eigenvalue density is “semicircular”: so

64. eigenvalue spectrum of a random matrix For a dense random matrix with mean square element value J 2 /N, the eigenvalue density is “semicircular”: so local susceptibility

65. eigenvalue spectrum of a random matrix For a dense random matrix with mean square element value J 2 /N, the eigenvalue density is “semicircular”: so local susceptibility

66. eigenvalue spectrum of a random matrix For a dense random matrix with mean square element value J 2 /N, the eigenvalue density is “semicircular”: so local susceptibility

67. eigenvalue spectrum of a random matrix For a dense random matrix with mean square element value J 2 /N, the eigenvalue density is “semicircular”: so local susceptibility use

68. eigenvalue spectrum of a random matrix For a dense random matrix with mean square element value J 2 /N, the eigenvalue density is “semicircular”: so local susceptibility use with

69. critical slowing down

70. ( J = 1 )

71. critical slowing down small ω: ( J = 1 )

72. critical slowing down small ω: ( J = 1 )

73. critical slowing down small ω: ( J = 1 )

74. critical slowing down small ω:critical slowing down ( J = 1 )

75. critical slowing down small ω:critical slowing down ( J = 1 ) but note: for the softest mode (with eigenvalue 2J )

76. critical slowing down small ω:critical slowing down ( J = 1 ) but note: for the softest mode (with eigenvalue 2J )

77. critical slowing down small ω:critical slowing down ( J = 1 ) but note: for the softest mode (with eigenvalue 2J )

78. critical slowing down small ω:critical slowing down ( J = 1 ) but note: for the softest mode (with eigenvalue 2J ) so its relaxation time diverges twice as strongly:

79. critical slowing down small ω:critical slowing down ( J = 1 ) but note: for the softest mode (with eigenvalue 2J ) so its relaxation time diverges twice as strongly:

80. Dynamics II: using MSR Use a “soft-spin” SK model:

81. Dynamics II: using MSR Use a “soft-spin” SK model:

82. Dynamics II: using MSR Use a “soft-spin” SK model:

83. Dynamics II: using MSR Use a “soft-spin” SK model: Langevin dynamics:

84. Dynamics II: using MSR Use a “soft-spin” SK model: Langevin dynamics: Generating functional:

85. Dynamics II: using MSR Use a “soft-spin” SK model: Langevin dynamics: Generating functional:

86. Dynamics II: using MSR Use a “soft-spin” SK model: Langevin dynamics: Generating functional:

87. averaging over the J ij

88. The exponent contains

89. averaging over the J ij The exponent contains so replace them in the exponent

90. decoupling sites and introduce delta functions

91. decoupling sites and introduce delta functions We are left with

92. (almost there) where

93. (almost there) where saddle-point equations:

94. (almost there) where saddle-point equations:

95. (almost there) where saddle-point equations:

96. (almost there) where saddle-point equations:

97. (almost there) where saddle-point equations:

98. effective 1-spin problem: The average correlation and response functions are equal to those of a self-consistent single-spin problem with action

99. effective 1-spin problem: The average correlation and response functions are equal to those of a self-consistent single-spin problem with action

100. effective 1-spin problem: The average correlation and response functions are equal to those of a self-consistent single-spin problem with action describing a single spin

101. effective 1-spin problem: The average correlation and response functions are equal to those of a self-consistent single-spin problem with action describing a single spin subject to noise with correlation function 2Tδ(t – t’) +J 2 C(t - t’)

102. effective 1-spin problem: The average correlation and response functions are equal to those of a self-consistent single-spin problem with action describing a single spin subject to noise with correlation function 2Tδ(t – t’) +J 2 C(t - t’) and retarded self-interaction J 2 R(t - t’)

103. local response function single effective spin obeys

104. local response function single effective spin obeys

105. local response function single effective spin obeys

106. local response function single effective spin obeys Fourier transform ( u 0 = 0 )

107. local response function single effective spin obeys Fourier transform ( u 0 = 0 )

108. local response function single effective spin obeys Fourier transform ( u 0 = 0 ) response function (susceptibility)

109. local response function single effective spin obeys Fourier transform ( u 0 = 0 ) response function (susceptibility) (Can solve quadratic equation for R 0 to find it explicitly)

110. critical slowing down at small ω, R 0 -1 (ω) ~ 1 - iωτ

111. critical slowing down at small ω, R 0 -1 (ω) ~ 1 - iωτ from

112. critical slowing down at small ω, R 0 -1 (ω) ~ 1 - iωτ from compute

113. critical slowing down at small ω, R 0 -1 (ω) ~ 1 - iωτ from compute

114. critical slowing down at small ω, R 0 -1 (ω) ~ 1 - iωτ from compute

115. critical slowing down at small ω, R 0 -1 (ω) ~ 1 - iωτ from compute critical slowing down at T c = J

116. critical slowing down at small ω, R 0 -1 (ω) ~ 1 - iωτ from compute critical slowing down at T c = J ( u 0 > 0 : perturbation theory does not change this qualitatively)