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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
So far we dealt with “uniform systems” Jij was the same for all pairs of neighbours.
What if every Jij is picked (independently) from some distribution?
Random Ising model
So far we dealt with “uniform systems” Jij was the same for all pairs of neighbours.
What if every Jij is picked (independently) from some distribution?
We want to know the average of physical quantities (thermodynamicfunctions, correlation functions, etc) over the distribution of Jij’s.
Random Ising model
So far we dealt with “uniform systems” Jij was the same for all pairs of neighbours.
What if every Jij is picked (independently) from some distribution?
We want to know the average of physical quantities (thermodynamicfunctions, correlation functions, etc) over the distribution of Jij’s.
Today: a simple model with <Jij> = 0
Random Ising model
So far we dealt with “uniform systems” Jij was the same for all pairs of neighbours.
What if every Jij is picked (independently) from some distribution?
We want to know the average of physical quantities (thermodynamicfunctions, correlation functions, etc) over the distribution of Jij’s.
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
Simple model (Edwards-Anderson)
€
Jij[ ]av
= 0, Jij2
[ ]av
=J 2
z(Jij = J ji)
Nearest-neighbour model with z neighbours
note averages over different “samples” (1 sample = 1 realization of choices of Jij’s for all pairs (ij) indicated by [ … ]av
Simple model (Edwards-Anderson)
€
Jij[ ]av
= 0, Jij2
[ ]av
=J 2
z(Jij = J ji)
Nearest-neighbour model with z neighbours
note averages over different “samples” (1 sample = 1 realization of choices of Jij’s for all pairs (ij) indicated by [ … ]av
€
E = − JijSi
ij
∑ S j − hi
i
∑ Si
= − 12 JijSi
ij
∑ S j − hi
i
∑ Si
Simple model (Edwards-Anderson)
€
Jij[ ]av
= 0, Jij2
[ ]av
=J 2
z(Jij = J ji)
Nearest-neighbour model with z neighbours
note averages over different “samples” (1 sample = 1 realization of choices of Jij’s for all pairs (ij) indicated by [ … ]av
€
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)
Sherrington-Kirkpatrick model
Every spin is a neighbour of every other one: z = (N – 1)
€
Jij2
[ ]av
=J 2
N −1≈
J 2
N
Sherrington-Kirkpatrick model
Every spin is a neighbour of every other one: z = (N – 1)
€
Jij2
[ ]av
=J 2
N −1≈
J 2
N
Mean field theory is exact for this model
Sherrington-Kirkpatrick model
Every spin is a neighbour of every other one: z = (N – 1)
€
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,
Heuristic mean field theoryreplace total field on Si,
€
H i = JijS j
j
∑
Heuristic mean field theoryreplace total field on Si,
€
H i = JijS j
j
∑ (take hi = 0)
Heuristic mean field theoryreplace total field on Si,
by its mean
€
H i = JijS j
j
∑ (take hi = 0)
Heuristic mean field theoryreplace total field on Si,
by its mean
€
H i = JijS j
j
∑
€
Jijm j
j
∑(take hi = 0)
Heuristic mean field theoryreplace total field on Si,
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)
Heuristic mean field theoryreplace total field on Si,
by its meanand calculate mi as the average S of a single spin in field H:
€
H i = JijS j
j
∑
€
Jijm j
j
∑
€
mi = tanh β Jijm j
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
(take hi = 0)
Heuristic mean field theoryreplace total field on Si,
by its meanand calculate mi as the average S of a single spin in field H:
€
H i = JijS j
j
∑
€
Jijm j
j
∑
€
mi = tanh β Jijm j
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
no preference for mi > 0 or <0:
(take hi = 0)
Heuristic mean field theoryreplace total field on Si,
by its meanand calculate mi as the average S of a single spin in field H:
€
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)
Heuristic mean field theoryreplace total field on Si,
by its meanand calculate mi as the average S of a single spin in field H:
€
H i = JijS j
j
∑
€
Jijm j
j
∑
€
mi = tanh β Jijm j
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
no preference for mi > 0 or <0: [mij]av = 0
if there are local spontaneous magnetizations mi ≠ 0,measure them by the order parameter (Edwards-Anderson)
(take hi = 0)
Heuristic mean field theoryreplace total field on Si,
by its meanand calculate mi as the average S of a single spin in field H:
€
H i = JijS j
j
∑
€
Jijm j
j
∑
€
mi = tanh β Jijm j
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
no preference for mi > 0 or <0: [mij]av = 0
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
self-consistent calculation of q:
To compute q: Hi is a sum of many (seemingly) independent terms=> Hi is Gaussian
self-consistent calculation of q:
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]
self-consistent calculation of q:
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]
= [Jij2
j
∑ ]av q ≡ J 2q
self-consistent calculation of q:
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]
= [Jij2
j
∑ ]av q ≡ J 2q
so
€
q =dH
2πJ 2q∫ tanh2 βH( )exp −
H 2
2J 2q
⎛
⎝ ⎜
⎞
⎠ ⎟
self-consistent calculation of q:
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]
= [Jij2
j
∑ ]av q ≡ J 2q
so
€
q =dH
2πJ 2q∫ tanh2 βH( )exp −
H 2
2J 2q
⎛
⎝ ⎜
⎞
⎠ ⎟ (solve for q)
spin glass transition:
€
q =dH
2πJ 2q∫ tanh2 βH( )exp −
H 2
2J 2q
⎛
⎝ ⎜
⎞
⎠ ⎟
spin glass transition:
€
q =dH
2πJ 2q∫ tanh2 βH( )exp −
H 2
2J 2q
⎛
⎝ ⎜
⎞
⎠ ⎟
expand in β:
spin glass transition:
€
q =dH
2πJ 2q∫ tanh2 βH( )exp −
H 2
2J 2q
⎛
⎝ ⎜
⎞
⎠ ⎟
expand in β:
€
q =dH
2πJ 2q∫ βH − 1
3 (βH)3 +L[ ]2exp −
H 2
2J 2q
⎛
⎝ ⎜
⎞
⎠ ⎟
spin glass transition:
€
q =dH
2πJ 2q∫ tanh2 βH( )exp −
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
⎛
⎝ ⎜
⎞
⎠ ⎟
spin glass transition:
€
q =dH
2πJ 2q∫ tanh2 βH( )exp −
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
spin glass transition:
€
q =dH
2πJ 2q∫ tanh2 βH( )exp −
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
spin glass transition:
€
q =dH
2πJ 2q∫ tanh2 βH( )exp −
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
below Tc:
€
β 2J 2 −1( )q ≈ 2q2 ⇒ q ≈ Tc − T
spin glass transition:
€
q =dH
2πJ 2q∫ tanh2 βH( )exp −
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
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
∑
the trouble below Tc
In the ferromagnet, it was safe to approximate
€
H i = JijS j
j
∑ ≈ Jijm j
j
∑
because the next term in a systematic expansion in β,
the trouble below Tc
In the ferromagnet, it was safe to approximate
€
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
∑ − βmi Jij2
j
∑ (1− m j2) +L
the trouble below Tc
In the ferromagnet, it was safe to approximate
€
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
∑ − βmi Jij2
j
∑ (1− m j2) +L
was O(1/z).
the trouble below Tc
In the ferromagnet, it was safe to approximate
€
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
∑ − 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.
the trouble below Tc
In the ferromagnet, it was safe to approximate
€
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
∑ − 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.
Thouless-Anderson-Palmer (TAP) equations):
€
mi = tanh β Jijm j
j
∑ − β 2mi Jij2
j
∑ (1− m j2) + βhi
⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
the trouble below Tc
In the ferromagnet, it was safe to approximate
€
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
∑ − 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.
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:
Dynamics (I: simple way)
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 )
Dynamics (I: simple way)
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 )
€
τ 0
d Si(t)
dt= − Si(t) + tanh β JijS j (t)
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
recall we derived from this
Dynamics (I: simple way)
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 )
€
τ 0
d Si(t)
dt= − Si(t) + tanh β JijS j (t)
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
recall we derived from this
mean field:
Dynamics (I: simple way)
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 )
€
τ 0
d Si(t)
dt= − Si(t) + tanh β JijS j (t)
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
recall we derived from this
mean field:
€
τ 0
dmi
dt= −mi + tanh βH i[ ]
Dynamics I (continued)
€
τ 0
dmi
dt= −mi + tanh βH i[ ]
Dynamics I (continued)
€
τ 0
dmi
dt= −mi + tanh βH i[ ]
= −mi + βH ilinearize (above Tc):
Dynamics I (continued)
€
τ 0
dmi
dt= −mi + tanh βH i[ ]
= −mi + βH i
≈ −mi + β Jij
j
∑ m j − β 2mi Jij2(1− q
j
∑ ) + βhi
linearize (above Tc):
use TAP:
Dynamics I (continued)
€
τ 0
dmi
dt= −mi + tanh βH i[ ]
= −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 )
linearize (above Tc):
use TAP:
Dynamics I (continued)
€
τ 0
dmi
dt= −mi + tanh βH i[ ]
= −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 )
linearize (above Tc):
use TAP:
In basis where J is diagonal:
Dynamics I (continued)
€
τ 0
dmi
dt= −mi + tanh βH i[ ]
= −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 )
linearize (above Tc):
use TAP:
In basis where J is diagonal:
€
τ 0
dmλ
dt= −mλ 1+ β 2J 2 − βJλ[ ] + βhλ
Dynamics I (continued)
€
τ 0
dmi
dt= −mi + tanh βH i[ ]
= −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 )
linearize (above Tc):
use TAP:
In basis where J is diagonal:
€
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
Dynamics I (continued)
€
τ 0
dmi
dt= −mi + tanh βH i[ ]
= −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 )
linearize (above Tc):
use TAP:
In basis where J is diagonal:
€
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
Dynamics I (continued)
€
τ 0
dmi
dt= −mi + tanh βH i[ ]
= −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 )
linearize (above Tc):
use TAP:
In basis where J is diagonal:
€
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”:
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
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
€
(Jλ )max = 2 ⇒ β c =1so
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
€
(Jλ )max = 2 ⇒ β c =1so
local susceptibility
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
€
(Jλ )max = 2 ⇒ β c =1so
local susceptibility
€
χ(ω) =1
Nχ ii
i
∑ (ω) =1
Nχ λ
λ
∑ (ω) = dλρ∫ (λ )χ λ (ω)
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
€
(Jλ )max = 2 ⇒ β c =1so
local susceptibility
€
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
∫
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
€
(Jλ )max = 2 ⇒ β c =1so
local susceptibility
€
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
∫
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
€
(Jλ )max = 2 ⇒ β c =1so
local susceptibility
€
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 ⎧
⎨ ⎩
⎫ ⎬ ⎭
critical slowing down
€
⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )
2− 4T 2 ⎧
⎨ ⎩
⎫ ⎬ ⎭
(J = 1)
critical slowing down
€
⇒ χ(ω) = 12 β T 2(1− iωτ 0) +1− T 2(1− iωτ 0) +1( )
2− 4T 2 ⎧
⎨ ⎩
⎫ ⎬ ⎭
small ω:
(J = 1)
critical slowing down
€
⇒ 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)
critical slowing down
€
⇒ 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)
critical slowing down
€
⇒ 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
critical slowing down
(J = 1)
critical slowing down
€
⇒ 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
critical slowing down
(J = 1)
but note: for the softest mode (with eigenvalue 2J)
critical slowing down
€
⇒ 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
critical slowing down
(J = 1)
but note: for the softest mode (with eigenvalue 2J)
€
χλ =β
1− iωτ 0 + β 2J 2 − 2βJ
critical slowing down
€
⇒ 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
critical slowing down
(J = 1)
but note: for the softest mode (with eigenvalue 2J)
€
χλ =β
1− iωτ 0 + β 2J 2 − 2βJ
=β
(1− βJ)2 − iωτ 0
critical slowing down
€
⇒ 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
critical slowing down
(J = 1)
but note: for the softest mode (with eigenvalue 2J)
€
χλ =β
1− iωτ 0 + β 2J 2 − 2βJ
=β
(1− βJ)2 − iωτ 0
so its relaxation time diverges twice as strongly:
critical slowing down
€
⇒ 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
critical slowing down
(J = 1)
but note: for the softest mode (with eigenvalue 2J)
€
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:
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
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
€
Jij2
[ ]av
=J 2
N
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
€
Jij2
[ ]av
=J 2
N
Langevin dynamics:
€
∂φi
∂t= −
∂ E[φ]( )∂φi
+ η i(t) = −r0φi − u0φi3 + Jijφ j
j
∑ + hi + η i(t)
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
€
Jij2
[ ]av
=J 2
N
Langevin dynamics:
€
∂φi
∂t= −
∂ E[φ]( )∂φi
+ η i(t) = −r0φi − u0φi3 + Jijφ j
j
∑ + hi + η i(t)
Generating functional:
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
€
Jij2
[ ]av
=J 2
N
Langevin dynamics:
€
∂φi
∂t= −
∂ E[φ]( )∂φi
+ η i(t) = −r0φi − u0φi3 + Jijφ j
j
∑ + hi + η i(t)
Generating functional:
€
Z[J,h,θ] = DφD ˆ φ exp −S[φ, ˆ φ ,J,h,θ]( )∫
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
€
Jij2
[ ]av
=J 2
N
Langevin dynamics:
€
∂φi
∂t= −
∂ E[φ]( )∂φi
+ η i(t) = −r0φi − u0φi3 + Jijφ j
j
∑ + hi + η i(t)
Generating functional:
€
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
∑ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
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
∑ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
The exponent contains
€
C(t − ′ t ) =1
Nφi(t)φi(t '), R(t − ′ t )
i
∑ =i
Nφi(t) ˆ φ i(t'),
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
∑ ⎡
⎣ ⎢ ⎢
⎤
⎦ ⎥ ⎥
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 i dt( ˆ φ iφ j + ˆ φ jφi)Jij∫( )ij
∏
= 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)
€
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
saddle-point equations:
(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
saddle-point equations:
€
wrt ˆ C : C(t − t') = φ(t)φ( ′ t )loc
(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
saddle-point equations:
€
wrt ˆ C : C(t − t') = φ(t)φ( ′ t )loc
wrt C : ˆ C (t − t') = − 12 J 2 ˆ φ (t) ˆ φ ( ′ t )
loc= 0
(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
saddle-point equations:
€
wrt ˆ C : C(t − t') = φ(t)φ( ′ t )loc
wrt C : ˆ C (t − t') = − 12 J 2 ˆ φ (t) ˆ φ ( ′ t )
loc= 0
wrt ˆ R : R(t − t') = i φ(t) ˆ φ ( ′ t )loc
(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
saddle-point equations:
€
wrt ˆ C : C(t − t') = φ(t)φ( ′ t )loc
wrt C : ˆ C (t − t') = − 12 J 2 ˆ φ (t) ˆ φ ( ′ t )
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
effective 1-spin problem:
The average correlation and response functions are equal to those of a self-consistent single-spin problem with action
€
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 ˆ φ (t)C(t − ′ t ) ˆ φ ( ′ t )∫
−J 2 dt d ′ t ∫ i ˆ φ (t)R(t − ′ t )φ( ′ t )[ ]
effective 1-spin problem:
The average correlation and response functions are equal to those of a self-consistent single-spin problem with action
€
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 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
effective 1-spin problem:
The average correlation and response functions are equal to those of a self-consistent single-spin problem with action
€
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 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’)
effective 1-spin problem:
The average correlation and response functions are equal to those of a self-consistent single-spin problem with action
€
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 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
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)
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)
ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(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)
ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )
Fourier transform (u0 = 0)
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)
ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )
Fourier transform (u0 = 0)
€
−iωS(ω) = −r0S(ω) + J 2R0(ω)S(ω) + h(ω) + ξ (ω)
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)
ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )
Fourier transform (u0 = 0)
€
−iωS(ω) = −r0S(ω) + J 2R0(ω)S(ω) + h(ω) + ξ (ω)
response function (susceptibility)
€
R0(ω) =∂ S(ω)
∂h(ω)=
1
r0 − J 2R0(ω)
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)
ξ (t)ξ ( ′ t ) = 2Tδ(t − ′ t ) + J 2C(t − ′ t )
Fourier transform (u0 = 0)
€
−iωS(ω) = −r0S(ω) + J 2R0(ω)S(ω) + h(ω) + ξ (ω)
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
critical slowing downat small ω, R0
-1(ω) ~ 1 - iωτ
€
R0−1(ω) = r0 − J 2R0(ω)from
critical slowing downat small ω, R0
-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)
⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥
critical slowing downat small ω, R0
-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
critical slowing downat small ω, R0
-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 downat small ω, R0
-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
critical slowing downat small ω, R0
-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)