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

Lecture 14: Spin glasses

Outline:• the EA and SK models• heuristic theory• dynamics I: using random matrices• dynamics II: using MSR

Random Ising model

So far we dealt with “uniform systems” Jij was the same for all pairs of neighbours.

Random Ising model


What if every Jij is picked (independently) from some distribution?

Random Ising model



We want to know the average of physical quantities (thermodynamicfunctions, correlation functions, etc) over the distribution of Jij’s.

Random Ising model




Today: a simple model with <Jij> = 0

Random Ising model




Today: a simple model with <Jij> = 0: spin glass

Simple model (Edwards-Anderson)

€

Jij[ ]av

= 0, Jij2

[ ]av

=J 2

z(Jij = J ji)

Nearest-neighbour model with z neighbours


€

Jij[ ]av

= 0, Jij2

[ ]av

=J 2

z(Jij = J ji)


note averages over different “samples” (1 sample = 1 realization of choices of Jij’s for all pairs (ij) indicated by [ … ]av


€

Jij[ ]av

= 0, Jij2

[ ]av

=J 2

z(Jij = J ji)



€

E = − JijSi

ij

∑ S j − hi

i

∑ Si

= − 12 JijSi

ij

∑ S j − hi

i

∑ Si


€

Jij[ ]av

= 0, Jij2

[ ]av

=J 2

z(Jij = J ji)



€

E = − JijSi

ij

∑ S j − hi

i

∑ Si

= − 12 JijSi

ij

∑ S j − hi

i

∑ Si

non-uniform J: anticipate nonuniform magnetization

€

mi = Si

Sherrington-Kirkpatrick model

Every spin is a neighbour of every other one: z = (N – 1)



€

Jij2

[ ]av

=J 2

N −1≈

J 2

N



€

Jij2

[ ]av

=J 2

N −1≈

J 2

N

Mean field theory is exact for this model



€

Jij2

[ ]av

=J 2

N −1≈

J 2

N

Mean field theory is exact for this model (but it is not simple)

Heuristic mean field theoryreplace total field on Si,


€

H i = JijS j

j

∑


€

H i = JijS j

j

∑ (take hi = 0)


by its mean

€

H i = JijS j

j

∑ (take hi = 0)


by its mean

€

H i = JijS j

j

∑

€

Jijm j

j

∑(take hi = 0)


by its meanand calculate mi as the average S of a single spin in field H:

€

H i = JijS j

j

∑

€

Jijm j

j

∑(take hi = 0)



€

H i = JijS j

j

∑

€

Jijm j

j

∑

€

mi = tanh β Jijm j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

(take hi = 0)



€

H i = JijS j

j

∑

€

Jijm j

j

∑

€

mi = tanh β Jijm j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

no preference for mi > 0 or <0:

(take hi = 0)



€

H i = JijS j

j

∑

€

Jijm j

j

∑

€

mi = tanh β Jijm j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

no preference for mi > 0 or <0: [mij]av = 0

(take hi = 0)



€

H i = JijS j

j

∑

€

Jijm j

j

∑

€

mi = tanh β Jijm j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


if there are local spontaneous magnetizations mi ≠ 0,measure them by the order parameter (Edwards-Anderson)

(take hi = 0)



€

H i = JijS j

j

∑

€

Jijm j

j

∑

€

mi = tanh β Jijm j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


if there are local spontaneous magnetizations mi ≠ 0,measure them by the order parameter (Edwards-Anderson)

(take hi = 0)

€

q = mi2

[ ]av

self-consistent calculation of q:

To compute q: Hi is a sum of many (seemingly) independent terms


To compute q: Hi is a sum of many (seemingly) independent terms=> Hi is Gaussian


To compute q: Hi is a sum of many (seemingly) independent terms=> Hi is Gaussian with variance

€

Jijm j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥av

= JijJik

jk

∑ m jmk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥av

≈ [Jij2

j

∑ ]av[m j2]



€

Jijm j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥av

= JijJik

jk

∑ m jmk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥av

≈ [Jij2

j

∑ ]av[m j2]

= [Jij2

j

∑ ]av q ≡ J 2q



€

Jijm j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥av

= JijJik

jk

∑ m jmk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥av

≈ [Jij2

j

∑ ]av[m j2]

= [Jij2

j

∑ ]av q ≡ J 2q

so

€

q =dH

2πJ 2q∫ tanh2 βH( )exp −

H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟



€

Jijm j

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

2 ⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥av

= JijJik

jk

∑ m jmk

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥av

≈ [Jij2

j

∑ ]av[m j2]

= [Jij2

j

∑ ]av q ≡ J 2q

so

€

q =dH


H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟ (solve for q)

spin glass transition:

€

q =dH


H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟


€

q =dH


H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

expand in β:


€

q =dH


H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

expand in β:

€

q =dH

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟


€

q =dH


H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

expand in β:

€

q =dH

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

=dH

2πJ 2q∫ (βH)2 − 2

3 (βH)4 +L[ ]exp −H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟


€

q =dH


H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

expand in β:

€

q =dH

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

=dH

2πJ 2q∫ (βH)2 − 2

3 (βH)4 +L[ ]exp −H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

= β 2J 2q − 23 ⋅3β 4J 4q2 +L


€

q =dH


H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

expand in β:

€

q =dH

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

=dH

2πJ 2q∫ (βH)2 − 2

3 (βH)4 +L[ ]exp −H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

= β 2J 2q − 23 ⋅3β 4J 4q2 +L

critical temperature: Tc = J


€

q =dH


H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

expand in β:

€

q =dH

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

=dH

2πJ 2q∫ (βH)2 − 2

3 (βH)4 +L[ ]exp −H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

= β 2J 2q − 23 ⋅3β 4J 4q2 +L


below Tc:

€

β 2J 2 −1( )q ≈ 2q2 ⇒ q ≈ Tc − T


€

q =dH


H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

expand in β:

€

q =dH

2πJ 2q∫ βH − 1

3 (βH)3 +L[ ]2exp −

H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

=dH

2πJ 2q∫ (βH)2 − 2

3 (βH)4 +L[ ]exp −H 2

2J 2q

⎛

⎝ ⎜

⎞

⎠ ⎟

= β 2J 2q − 23 ⋅3β 4J 4q2 +L


below Tc:

€

β 2J 2 −1( )q ≈ 2q2 ⇒ q ≈ Tc − T

This heuristic theory is right up to this point, but wrong below Tc.

the trouble below Tc

In the ferromagnet, it was safe to approximate

€

H i = JijS j

j

∑ ≈ Jijm j

j

∑



€

H i = JijS j

j

∑ ≈ Jijm j

j

∑

because the next term in a systematic expansion in β,



€

H i = JijS j

j

∑ ≈ Jijm j

j

∑


€

H i = JijS j

j

∑ = Jijm j

j

∑ − βmi Jij2

j

∑ (1− m j2) +L



€

H i = JijS j

j

∑ ≈ Jijm j

j

∑


€

H i = JijS j

j

∑ = Jijm j

j

∑ − βmi Jij2

j

∑ (1− m j2) +L

was O(1/z).



€

H i = JijS j

j

∑ ≈ Jijm j

j

∑


€

H i = JijS j

j

∑ = Jijm j

j

∑ − βmi Jij2

j

∑ (1− m j2) +L

was O(1/z). But here, the average of the 1st 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.



€

H i = JijS j

j

∑ ≈ Jijm j

j

∑


€

H i = JijS j

j

∑ = Jijm j

j

∑ − βmi Jij2

j

∑ (1− m j2) +L


Thouless-Anderson-Palmer (TAP) equations):

€

mi = tanh β Jijm j

j

∑ − β 2mi Jij2

j

∑ (1− m j2) + βhi

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥



€

H i = JijS j

j

∑ ≈ Jijm j

j

∑


€

H i = JijS j

j

∑ = Jijm j

j

∑ − βmi Jij2

j

∑ (1− m j2) +L


Thouless-Anderson-Palmer (TAP) equations):

€

mi = tanh β Jijm j

j

∑ − β 2mi Jij2

j

∑ (1− m j2) + βhi

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥______________

Onsager correction to mean field

Dynamics (I: simple way)

Glauber dynamics:


Glauber dynamics:

€

τ 0

dP({S},t)

dt= 1

2 1+ Si tanh βhi(t)( )[ ]i

∑ P(S1L − SiL SN )

− 12 1− Si tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )


Glauber dynamics:

€

τ 0

dP({S},t)

dt= 1



− 12 1− Si tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )

€

τ 0

d Si(t)

dt= − Si(t) + tanh β JijS j (t)

j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

recall we derived from this


Glauber dynamics:

€

τ 0

dP({S},t)

dt= 1



− 12 1− Si tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )

€

τ 0

d Si(t)


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


mean field:


Glauber dynamics:

€

τ 0

dP({S},t)

dt= 1



− 12 1− Si tanh βhi(t)( )[ ]

i

∑ P(S1L SiL SN )

€

τ 0

d Si(t)


j

∑ ⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟


mean field:

€

τ 0

dmi

dt= −mi + tanh βH i[ ]

Dynamics I (continued)

€

τ 0

dmi



€

τ 0

dmi


= −mi + βH ilinearize (above Tc):


€

τ 0

dmi


= −mi + βH i

≈ −mi + β Jij

j

∑ m j − β 2mi Jij2(1− q

j

∑ ) + βhi

linearize (above Tc):

use TAP:


€

τ 0

dmi


= −mi + βH i

≈ −mi + β Jij

j

∑ m j − β 2mi Jij2(1− q

j

∑ ) + βhi

= −mi + β Jij

j

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )


use TAP:


€

τ 0

dmi


= −mi + βH i

≈ −mi + β Jij

j

∑ m j − β 2mi Jij2(1− q

j

∑ ) + βhi

= −mi + β Jij

j

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )


use TAP:

In basis where J is diagonal:


€

τ 0

dmi


= −mi + βH i

≈ −mi + β Jij

j

∑ m j − β 2mi Jij2(1− q

j

∑ ) + βhi

= −mi + β Jij

j

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )


use TAP:


€

τ 0

dmλ

dt= −mλ 1+ β 2J 2 − βJλ[ ] + βhλ


€

τ 0

dmi


= −mi + βH i

≈ −mi + β Jij

j

∑ m j − β 2mi Jij2(1− q

j

∑ ) + βhi

= −mi + β Jij

j

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )


use TAP:


€

τ 0

dmλ

dt= −mλ 1+ β 2J 2 − βJλ[ ] + βhλ

susceptibility:

€

χλ =∂mλ (ω)

∂hλ (ω)=

β

1− iωτ 0 + β 2J 2 − βJλ


€

τ 0

dmi


= −mi + βH i

≈ −mi + β Jij

j

∑ m j − β 2mi Jij2(1− q

j

∑ ) + βhi

= −mi + β Jij

j

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )


use TAP:


€

τ 0

dmλ

dt= −mλ 1+ β 2J 2 − βJλ[ ] + βhλ

instability (transition) reached when maximum eigenvalue

susceptibility:

€

χλ =∂mλ (ω)

∂hλ (ω)=

β

1− iωτ 0 + β 2J 2 − βJλ


€

τ 0

dmi


= −mi + βH i

≈ −mi + β Jij

j

∑ m j − β 2mi Jij2(1− q

j

∑ ) + βhi

= −mi + β Jij

j

∑ m j − miβ2J 2 + βhi (q = 0, T > Tc )


use TAP:


€

τ 0

dmλ

dt= −mλ 1+ β 2J 2 − βJλ[ ] + βhλ

instability (transition) reached when maximum eigenvalue

€

(Jλ )max =1+ β 2J 2

β

susceptibility:

€

χλ =∂mλ (ω)

∂hλ (ω)=

β

1− iωτ 0 + β 2J 2 − βJλ

eigenvalue spectrum of a random matrix

For a dense random matrix with mean square element value J2/N,the eigenvalue density is “semicircular”:



€

ρ(Jλ ) =1

2πJ4J 2 − Jλ

2



€

ρ(Jλ ) =1

2πJ4J 2 − Jλ

2

€

(Jλ )max = 2 ⇒ β c =1so



€

ρ(Jλ ) =1

2πJ4J 2 − Jλ

2

€

(Jλ )max = 2 ⇒ β c =1so

local susceptibility



€

ρ(Jλ ) =1

2πJ4J 2 − Jλ

2

€

(Jλ )max = 2 ⇒ β c =1so


€

χ(ω) =1

Nχ ii

i

∑ (ω) =1

Nχ λ

λ

∑ (ω) = dλρ∫ (λ )χ λ (ω)



€

ρ(Jλ ) =1

2πJ4J 2 − Jλ

2

€

(Jλ )max = 2 ⇒ β c =1so


€

χ(ω) =1

Nχ ii

i

∑ (ω) =1

Nχ λ

λ

∑ (ω) = dλρ∫ (λ )χ λ (ω)

€

=β

2πJdλ

4J 2 − λ2

1− iωτ 0 + β 2J 2 − βλ−2J

2J

∫



€

ρ(Jλ ) =1

2πJ4J 2 − Jλ

2

€

(Jλ )max = 2 ⇒ β c =1so


€

χ(ω) =1

Nχ ii

i

∑ (ω) =1

Nχ λ

λ

∑ (ω) = dλρ∫ (λ )χ λ (ω)

use

€

1

πdx

4J 2 − x 2

y − x−2J

2J

∫ = y − y 2 − 4J 2[ ]

€

=β

2πJdλ

4J 2 − λ2

1− iωτ 0 + β 2J 2 − βλ−2J

2J

∫



€

ρ(Jλ ) =1

2πJ4J 2 − Jλ

2

€

(Jλ )max = 2 ⇒ β c =1so


€

χ(ω) =1

Nχ ii

i

∑ (ω) =1

Nχ λ

λ

∑ (ω) = dλρ∫ (λ )χ λ (ω)

use

€

1

πdx

4J 2 − x 2

y − x−2J

2J

∫ = y − y 2 − 4J 2[ ]

€

y =1− iωτ 0 + β 2J 2, x = βJλwith€

=β

2πJdλ

4J 2 − λ2

1− iωτ 0 + β 2J 2 − βλ−2J

2J

∫

critical slowing down

€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭


€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

(J = 1)


€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

(J = 1)


€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

€

χ−1(ω) ≈ T(1− iωτ )

(J = 1)


€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

€

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc

(J = 1)


€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

€

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc


(J = 1)


€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

€

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc


(J = 1)

but note: for the softest mode (with eigenvalue 2J)


€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

€

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc


(J = 1)


€

χλ =β

1− iωτ 0 + β 2J 2 − 2βJ


€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

€

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc


(J = 1)


€

χλ =β

1− iωτ 0 + β 2J 2 − 2βJ

=β

(1− βJ)2 − iωτ 0


€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

€

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc


(J = 1)


€

χλ =β

1− iωτ 0 + β 2J 2 − 2βJ

=β

(1− βJ)2 − iωτ 0

so its relaxation time diverges twice as strongly:


€

⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )

2− 4T 2 ⎧

⎨ ⎩

⎫ ⎬ ⎭

small ω:

€

χ−1(ω) ≈ T(1− iωτ ), τ ∝1

T − Tc


(J = 1)


€

χλ =β

1− iωτ 0 + β 2J 2 − 2βJ

=β

(1− βJ)2 − iωτ 0

so its relaxation time diverges twice as strongly:

€

τ ∝ 1

(T − Tc )2

Dynamics II: using MSRUse a “soft-spin” SK model:


€

E[φ] = 12 r0φi

2 + 14 u0φi

4( )

i

∑ − 12 Jijφi

ij

∑ φ j − hi

i

∑ φi


€

E[φ] = 12 r0φi

2 + 14 u0φi

4( )

i

∑ − 12 Jijφi

ij

∑ φ j − hi

i

∑ φi

€

Jij2

[ ]av

=J 2

N


€

E[φ] = 12 r0φi

2 + 14 u0φi

4( )

i

∑ − 12 Jijφi

ij

∑ φ j − hi

i

∑ φi

€

Jij2

[ ]av

=J 2

N

Langevin dynamics:

€

∂φi

∂t= −

∂ E[φ]( )∂φi

+ η i(t) = −r0φi − u0φi3 + Jijφ j

j

∑ + hi + η i(t)


€

E[φ] = 12 r0φi

2 + 14 u0φi

4( )

i

∑ − 12 Jijφi

ij

∑ φ j − hi

i

∑ φi

€

Jij2

[ ]av

=J 2

N

Langevin dynamics:

€

∂φi

∂t= −

∂ E[φ]( )∂φi


j

∑ + hi + η i(t)

Generating functional:


€

E[φ] = 12 r0φi

2 + 14 u0φi

4( )

i

∑ − 12 Jijφi

ij

∑ φ j − hi

i

∑ φi

€

Jij2

[ ]av

=J 2

N

Langevin dynamics:

€

∂φi

∂t= −

∂ E[φ]( )∂φi


j

∑ + hi + η i(t)


€

Z[J,h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,J,h,θ]( )∫


€

E[φ] = 12 r0φi

2 + 14 u0φi

4( )

i

∑ − 12 Jijφi

ij

∑ φ j − hi

i

∑ φi

€

Jij2

[ ]av

=J 2

N

Langevin dynamics:

€

∂φi

∂t= −

∂ E[φ]( )∂φi


j

∑ + hi + η i(t)


€

Z[J,h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,J,h,θ]( )∫

€

S[φ, ˆ φ ,J,h,θ] = d∫ t T ˆ φ i2 + i ˆ φ i

∂φi

∂t+ r0φi + u0φi

3 − Jijφ j

j

∑ − hi

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟− iθ iφi

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥i

∑

averaging over the Jij

€

exp i dt( ˆ φ iφ j + ˆ φ jφi)Jij∫( )ij

∏

= exp −J 2

2Ndt d ′ t ˆ φ i(t) ˆ φ i(t')φ j (t)φ j (t') + ˆ φ i(t)φi( ′ t )φ j (t) ˆ φ j ( ′ t )( )∫

ij

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥


€


∏

= exp −J 2


ij

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

The exponent contains

€

C(t − ′ t ) =1

Nφi(t)φi(t '), R(t − ′ t )

i

∑ =i

Nφi(t) ˆ φ i(t'),

i

∑


€


∏

= exp −J 2


ij

∑ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

The exponent contains

€

C(t − ′ t ) =1

Nφi(t)φi(t '), R(t − ′ t )

i

∑ =i

Nφi(t) ˆ φ i(t'),

i

∑

so replace them in the exponent

€


∏

= exp − 12 J 2 dt d ′ t ˆ φ i(t) ˆ φ i(t')C(t − ′ t ) − i ˆ φ i(t)φi(t')R(t − ′ t )( )∫

i

∑ ⎡

⎣ ⎢

⎤

⎦ ⎥

decoupling sitesand introduce delta functions

€

1= DCD ˆ C ∫ exp − dtd ′ t ˆ C (t − ′ t ) NC(t − ′ t ) − φi(t)φi(t ')i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟∫

⎡

⎣ ⎢

⎤

⎦ ⎥

1= DRDCR∫ exp − dtd ′ t ˆ R (t − ′ t ) NR(t − ′ t ) − i φi(t) ˆ φ i(t ')i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟∫

⎡

⎣ ⎢

⎤

⎦ ⎥

decoupling sitesand introduce delta functions

€

1= DCD ˆ C ∫ exp − dtd ′ t ˆ C (t − ′ t ) NC(t − ′ t ) − φi(t)φi(t ')i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟∫

⎡

⎣ ⎢

⎤

⎦ ⎥

1= DRDCR∫ exp − dtd ′ t ˆ R (t − ′ t ) NR(t − ′ t ) − i φi(t) ˆ φ i(t ')i

∑ ⎛

⎝ ⎜

⎞

⎠ ⎟∫

⎡

⎣ ⎢

⎤

⎦ ⎥

We are left with

€

W ≡ Z[0,0,J][ ]av

= DC D ˆ C DR D ˆ R exp −N dt d ′ t ˆ C (t − ′ t )C(t − ′ t ) − ˆ R (t − ′ t )R(t − ′ t )[ ]∫( )∫

×exp N log DφD ˆ φ exp∫ −Sloc[φ, ˆ φ ,C, ˆ C ,R, ˆ R ]( )( )

(almost there)

€

Sloc[φ, ˆ φ ,C, ˆ C ,R, ˆ R ] = d∫ t T ˆ φ 2 + i ˆ φ ˙ φ + r0φ + u0φ3

( )[ ]

+ 12 J 2 dt d ′ t C(t − ′ t ) ˆ φ (t) ˆ φ ( ′ t ) − iR(t − ′ t ) ˆ φ (t)φ( ′ t )[ ]∫

+ dt d ′ t ˆ C (t − ′ t )φ(t)φ( ′ t ) − i ˆ R (t − ′ t )φ(t) ˆ φ ( ′ t )[ ]∫

where

(almost there)

€


( )[ ]



where

saddle-point equations:

(almost there)

€


( )[ ]



where


€

wrt ˆ C : C(t − t') = φ(t)φ( ′ t )loc

(almost there)

€


( )[ ]



where


€


wrt C : ˆ C (t − t') = − 12 J 2 ˆ φ (t) ˆ φ ( ′ t )

loc= 0

(almost there)

€


( )[ ]



where


€



loc= 0

wrt ˆ R : R(t − t') = i φ(t) ˆ φ ( ′ t )loc

(almost there)

€


( )[ ]



where


€



loc= 0

wrt ˆ R : R(t − t') = i φ(t) ˆ φ ( ′ t )loc

wrt R : ˆ R (t − ′ t ) = 12 iJ 2 ˆ φ (t)φ( ′ t )

loc= 1

2 J 2R( ′ t − t)

effective 1-spin problem:

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



€


( )[ ]

+ 12 J 2 dt d ′ t ˆ φ (t)C(t − ′ t ) ˆ φ ( ′ t )∫

−J 2 dt d ′ t ∫ i ˆ φ (t)R(t − ′ t )φ( ′ t )[ ]



€


( )[ ]

+ 12 J 2 dt d ′ t ˆ φ (t)C(t − ′ t ) ˆ φ ( ′ t )∫

−J 2 dt d ′ t ∫ i ˆ φ (t)R(t − ′ t )φ( ′ t )[ ]

describing a single spin



€


( )[ ]

+ 12 J 2 dt d ′ t ˆ φ (t)C(t − ′ t ) ˆ φ ( ′ t )∫

−J 2 dt d ′ t ∫ i ˆ φ (t)R(t − ′ t )φ( ′ t )[ ]

describing a single spin subject tonoise with correlation function 2Tδ(t – t’) +J2C(t - t’)



€


( )[ ]

+ 12 J 2 dt d ′ t ˆ φ (t)C(t − ′ t ) ˆ φ ( ′ t )∫

−J 2 dt d ′ t ∫ i ˆ φ (t)R(t − ′ t )φ( ′ t )[ ]

describing a single spin subject tonoise with correlation function 2Tδ(t – t’) +J2C(t - t’)and retarded self-interaction J2R(t - t’)

local response function

single effective spin obeys



€

dS

dt= −r0S − u0S

3 + J 2 d ′ t −∞

t

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)



€

dS

dt= −r0S − u0S

3 + J 2 d ′ t −∞

t

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )



€

dS

dt= −r0S − u0S

3 + J 2 d ′ t −∞

t

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )

Fourier transform (u0 = 0)



€

dS

dt= −r0S − u0S

3 + J 2 d ′ t −∞

t

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )


€

−iωS(ω) = −r0S(ω) + J 2R0(ω)S(ω) + h(ω) + ξ (ω)



€

dS

dt= −r0S − u0S

3 + J 2 d ′ t −∞

t

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )


€


response function (susceptibility)

€

R0(ω) =∂ S(ω)

∂h(ω)=

1

r0 − J 2R0(ω)



€

dS

dt= −r0S − u0S

3 + J 2 d ′ t −∞

t

∫ R(t − ′ t )S( ′ t ) + h(t) + ξ (t)

ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )


€


response function (susceptibility)

€

R0(ω) =∂ S(ω)

∂h(ω)=

1

r0 − J 2R0(ω)

(Can solve quadratic equation for R0 to find it explicitly)

critical slowing downat small ω, R0

-1(ω) ~ 1 - iωτ


-1(ω) ~ 1 - iωτ

€

R0−1(ω) = r0 − J 2R0(ω)from


-1(ω) ~ 1 - iωτ

€

R0−1(ω) = r0 − J 2R0(ω)from

compute

€

τ =limω →0

∂R0−1(ω)

∂(−iω)

⎛

⎝ ⎜

⎞

⎠ ⎟=1+ J 2R0

2(0) limω →0

∂R0−1(ω)

∂(−iω)

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥


-1(ω) ~ 1 - iωτ

€

R0−1(ω) = r0 − J 2R0(ω)from

compute

€

τ =limω →0

∂R0−1(ω)

∂(−iω)

⎛

⎝ ⎜

⎞

⎠ ⎟=1+ J 2R0

2(0) limω →0

∂R0−1(ω)

∂(−iω)

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

τ =1+ J 2R02(0)τ


-1(ω) ~ 1 - iωτ

€

R0−1(ω) = r0 − J 2R0(ω)from

compute

€

τ =limω →0

∂R0−1(ω)

∂(−iω)

⎛

⎝ ⎜

⎞

⎠ ⎟=1+ J 2R0

2(0) limω →0

∂R0−1(ω)

∂(−iω)

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

τ =1+ J 2R02(0)τ

€

⇒ τ =1

1− J 2R02(0)

R(0) =1

T

⎛

⎝ ⎜

⎞

⎠ ⎟


-1(ω) ~ 1 - iωτ

€

R0−1(ω) = r0 − J 2R0(ω)from

compute

€

τ =limω →0

∂R0−1(ω)

∂(−iω)

⎛

⎝ ⎜

⎞

⎠ ⎟=1+ J 2R0

2(0) limω →0

∂R0−1(ω)

∂(−iω)

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

τ =1+ J 2R02(0)τ

€

⇒ τ =1

1− J 2R02(0)

R(0) =1

T

⎛

⎝ ⎜

⎞

⎠ ⎟ critical slowing down

at Tc = J


-1(ω) ~ 1 - iωτ

€

R0−1(ω) = r0 − J 2R0(ω)from

compute

€

τ =limω →0

∂R0−1(ω)

∂(−iω)

⎛

⎝ ⎜

⎞

⎠ ⎟=1+ J 2R0

2(0) limω →0

∂R0−1(ω)

∂(−iω)

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

τ =1+ J 2R02(0)τ

€

⇒ τ =1

1− J 2R02(0)

R(0) =1

T

⎛

⎝ ⎜

⎞

⎠ ⎟ critical slowing down

at Tc = J

(u0 > 0: perturbation theory does not change this qualitatively)

Documents

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