Lecture 14: Spin glasses

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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

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Random Ising model
So far we dealt with uniform systems Jij was
the same for all pairs of neighbours.
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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?
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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 (thermodynamic functions,
correlation functions, etc) over the distribution
of Jijs.
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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 (thermodynamic functions,
correlation functions, etc) over the distribution
of Jijs. Today a simple model with ltJijgt 0
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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 (thermodynamic functions,
correlation functions, etc) over the distribution
of Jijs. Today a simple model with ltJijgt 0
spin glass
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Simple model (Edwards-Anderson)
Nearest-neighbour model with z neighbours
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Simple model (Edwards-Anderson)
Nearest-neighbour model with z neighbours
note averages over different samples (1 sample
1 realization of choices of Jijs for all
pairs (ij) indicated by av
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Simple model (Edwards-Anderson)
Nearest-neighbour model with z neighbours
note averages over different samples (1 sample
1 realization of choices of Jijs for all
pairs (ij) indicated by av
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Simple model (Edwards-Anderson)
Nearest-neighbour model with z neighbours
note averages over different samples (1 sample
1 realization of choices of Jijs for all
pairs (ij) indicated by av
non-uniform J anticipate nonuniform
magnetization
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Sherrington-Kirkpatrick model
Every spin is a neighbour of every other one z
(N 1)
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Sherrington-Kirkpatrick model
Every spin is a neighbour of every other one z
(N 1)
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Sherrington-Kirkpatrick model
Every spin is a neighbour of every other one z
(N 1)
Mean field theory is exact for this model
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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)
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Heuristic mean field theory
replace total field on Si,
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Heuristic mean field theory
replace total field on Si,
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Heuristic mean field theory
(take hi 0)
replace total field on Si,
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Heuristic mean field theory
(take hi 0)
replace total field on Si, by its mean
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Heuristic mean field theory
(take hi 0)
replace total field on Si, by its mean
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Heuristic mean field theory
(take hi 0)
replace total field on Si, by its mean and
calculate mi as the average S of a single spin in
field H
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Heuristic mean field theory
(take hi 0)
replace total field on Si, by its mean and
calculate mi as the average S of a single spin in
field H
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Heuristic mean field theory
(take hi 0)
replace total field on Si, by its mean and
calculate mi as the average S of a single spin in
field H
no preference for mi gt 0 or lt0
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Heuristic mean field theory
(take hi 0)
replace total field on Si, by its mean and
calculate mi as the average S of a single spin in
field H
no preference for mi gt 0 or lt0 mijav 0
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Heuristic mean field theory
(take hi 0)
replace total field on Si, by its mean and
calculate mi as the average S of a single spin in
field H
no preference for mi gt 0 or lt0 mijav 0 if
there are local spontaneous magnetizations mi ?
0, measure them by the order parameter
(Edwards-Anderson)
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Heuristic mean field theory
(take hi 0)
replace total field on Si, by its mean and
calculate mi as the average S of a single spin in
field H
no preference for mi gt 0 or lt0 mijav 0 if
there are local spontaneous magnetizations mi ?
0, measure them by the order parameter
(Edwards-Anderson)
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self-consistent calculation of q
To compute q Hi is a sum of many (seemingly)
independent terms
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self-consistent calculation of q
To compute q Hi is a sum of many (seemingly)
independent terms gt Hi is Gaussian
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self-consistent calculation of q
To compute q Hi is a sum of many (seemingly)
independent terms gt Hi is Gaussian with variance
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self-consistent calculation of q
To compute q Hi is a sum of many (seemingly)
independent terms gt Hi is Gaussian with variance
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self-consistent calculation of q
To compute q Hi is a sum of many (seemingly)
independent terms gt Hi is Gaussian with variance
so
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self-consistent calculation of q
To compute q Hi is a sum of many (seemingly)
independent terms gt Hi is Gaussian with variance
so
(solve for q)
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spin glass transition
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spin glass transition
expand in ß
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spin glass transition
expand in ß
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spin glass transition
expand in ß
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spin glass transition
expand in ß
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spin glass transition
expand in ß
critical temperature Tc J
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spin glass transition
expand in ß
critical temperature Tc J
below Tc
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spin glass transition
expand in ß
critical temperature Tc J
below Tc
This heuristic theory is right up to this point,
but wrong below Tc.
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the trouble below Tc
In the ferromagnet, it was safe to approximate
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the trouble below Tc
In the ferromagnet, it was safe to approximate
because the next term in a systematic expansion
in ß,
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the trouble below Tc
In the ferromagnet, it was safe to approximate
because the next term in a systematic expansion
in ß,
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the trouble below Tc
In the ferromagnet, it was safe to approximate
because the next term in a systematic expansion
in ß,
was O(1/z).
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the trouble below Tc
In the ferromagnet, it was safe to approximate
because the next term in a systematic expansion
in ß,
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.
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the trouble below Tc
In the ferromagnet, it was safe to approximate
because the next term in a systematic expansion
in ß,

• 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)

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the trouble below Tc
In the ferromagnet, it was safe to approximate
because the next term in a systematic expansion
in ß,

• 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)

______________ Onsager correction to mean field
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Dynamics (I simple way)
Glauber dynamics
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Dynamics (I simple way)
Glauber dynamics
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Dynamics (I simple way)
Glauber dynamics
recall we derived from this
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Dynamics (I simple way)
Glauber dynamics
recall we derived from this
mean field
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Dynamics (I simple way)
Glauber dynamics
recall we derived from this
mean field
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Dynamics I (continued)
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Dynamics I (continued)
linearize (above Tc)
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Dynamics I (continued)
linearize (above Tc)
use TAP
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Dynamics I (continued)
linearize (above Tc)
use TAP
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Dynamics I (continued)
linearize (above Tc)
use TAP
In basis where J is diagonal
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Dynamics I (continued)
linearize (above Tc)
use TAP
In basis where J is diagonal
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Dynamics I (continued)
linearize (above Tc)
use TAP
In basis where J is diagonal
susceptibility
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Dynamics I (continued)
linearize (above Tc)
use TAP
In basis where J is diagonal
susceptibility
instability (transition) reached when maximum
eigenvalue
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Dynamics I (continued)
linearize (above Tc)
use TAP
In basis where J is diagonal
susceptibility
instability (transition) reached when maximum
eigenvalue
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eigenvalue spectrum of a random matrix
For a dense random matrix with mean square
element value J2/N, the eigenvalue density is
semicircular
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eigenvalue spectrum of a random matrix
For a dense random matrix with mean square
element value J2/N, the eigenvalue density is
semicircular
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eigenvalue spectrum of a random matrix
For a dense random matrix with mean square
element value J2/N, the eigenvalue density is
semicircular
so
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eigenvalue spectrum of a random matrix
For a dense random matrix with mean square
element value J2/N, the eigenvalue density is
semicircular
so
local susceptibility
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eigenvalue spectrum of a random matrix
For a dense random matrix with mean square
element value J2/N, the eigenvalue density is
semicircular
so
local susceptibility
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eigenvalue spectrum of a random matrix
For a dense random matrix with mean square
element value J2/N, the eigenvalue density is
semicircular
so
local susceptibility
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eigenvalue spectrum of a random matrix
For a dense random matrix with mean square
element value J2/N, the eigenvalue density is
semicircular
so
local susceptibility
use
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eigenvalue spectrum of a random matrix
For a dense random matrix with mean square
element value J2/N, the eigenvalue density is
semicircular
so
local susceptibility
use
with
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critical slowing down
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critical slowing down
(J 1)
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critical slowing down
(J 1)
small ?
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critical slowing down
(J 1)
small ?
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critical slowing down
(J 1)
small ?
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critical slowing down
(J 1)
small ?
critical slowing down
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critical slowing down
(J 1)
small ?
critical slowing down
but note for the softest mode (with eigenvalue
2J)
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critical slowing down
(J 1)
small ?
critical slowing down
but note for the softest mode (with eigenvalue
2J)
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critical slowing down
(J 1)
small ?
critical slowing down
but note for the softest mode (with eigenvalue
2J)
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critical slowing down
(J 1)
small ?
critical slowing down
but note for the softest mode (with eigenvalue
2J)
so its relaxation time diverges twice as strongly
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critical slowing down
(J 1)
small ?
critical slowing down
but note for the softest mode (with eigenvalue
2J)
so its relaxation time diverges twice as strongly
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Dynamics II using MSR
Use a soft-spin SK model
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Dynamics II using MSR
Use a soft-spin SK model
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Dynamics II using MSR
Use a soft-spin SK model
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Dynamics II using MSR
Use a soft-spin SK model
Langevin dynamics
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Dynamics II using MSR
Use a soft-spin SK model
Langevin dynamics
Generating functional
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Dynamics II using MSR
Use a soft-spin SK model
Langevin dynamics
Generating functional
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Dynamics II using MSR
Use a soft-spin SK model
Langevin dynamics
Generating functional
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averaging over the Jij
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averaging over the Jij
The exponent contains
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averaging over the Jij
The exponent contains
so replace them in the exponent
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decoupling sites
and introduce delta functions
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decoupling sites
and introduce delta functions
We are left with
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(almost there)
where
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(almost there)
where
saddle-point equations
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(almost there)
where
saddle-point equations
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(almost there)
where
saddle-point equations
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(almost there)
where
saddle-point equations
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(almost there)
where
saddle-point equations
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effective 1-spin problem
The average correlation and response functions
are equal to those of a self-consistent
single-spin problem with action
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effective 1-spin problem
The average correlation and response functions
are equal to those of a self-consistent
single-spin problem with action
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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
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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 2Td(t t) J2C(t - t)
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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 2Td(t t) J2C(t -
t) and retarded self-interaction J2R(t - t)
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local response function
single effective spin obeys
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local response function
single effective spin obeys
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local response function
single effective spin obeys
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local response function
single effective spin obeys
Fourier transform (u0 0)
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local response function
single effective spin obeys
Fourier transform (u0 0)
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local response function
single effective spin obeys
Fourier transform (u0 0)
response function (susceptibility)
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local response function
single effective spin obeys
Fourier transform (u0 0)
response function (susceptibility)
(Can solve quadratic equation for R0 to find it
explicitly)
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critical slowing down
at small ?, R0-1(?) 1 - i?t
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critical slowing down
at small ?, R0-1(?) 1 - i?t
from
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critical slowing down
at small ?, R0-1(?) 1 - i?t
from
compute
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critical slowing down
at small ?, R0-1(?) 1 - i?t
from
compute
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critical slowing down
at small ?, R0-1(?) 1 - i?t
from
compute
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critical slowing down
at small ?, R0-1(?) 1 - i?t
from
compute
critical slowing down at Tc J
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critical slowing down
at small ?, R0-1(?) 1 - i?t
from
compute
critical slowing down at Tc J
(u0 gt 0 perturbation theory does not change this
qualitatively)