Lecture 11: Ising model
Outline:• equilibrium theory
• d = 1• mean field theory, phase transition• critical phenomena
• kinetics (Glauber model)• critical dynamics
• continuum description: Landau-Ginzburg model•Langevin dynamics
“spins”
binary variables Si = ±1 (or 0/1)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)representing up/down, firing/not firing, opinions, decisions, atom present/not present, …
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)representing up/down, firing/not firing, opinions, decisions, atom present/not present, … (always an idealization)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)representing up/down, firing/not firing, opinions, decisions, atom present/not present, … (always an idealization)Energy:
€
E = − JijSi
ij
∑ S j − hi
i
∑ Si
= − 12 JijSi
ij
∑ S j − hi
i
∑ Si
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)representing up/down, firing/not firing, opinions, decisions, atom present/not present, … (always an idealization)Energy:
€
E = − JijSi
ij
∑ S j − hi
i
∑ Si
= − 12 JijSi
ij
∑ S j − hi
i
∑ Si (low energy is “favorable”)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)representing up/down, firing/not firing, opinions, decisions, atom present/not present, … (always an idealization)Energy:
€
E = − JijSi
ij
∑ S j − hi
i
∑ Si
= − 12 JijSi
ij
∑ S j − hi
i
∑ Si (low energy is “favorable”)
role of geometry: i,j can label points on a lattice of dimensionality d
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)representing up/down, firing/not firing, opinions, decisions, atom present/not present, … (always an idealization)Energy:
€
E = − JijSi
ij
∑ S j − hi
i
∑ Si
= − 12 JijSi
ij
∑ S j − hi
i
∑ Si (low energy is “favorable”)
role of geometry: i,j can label points on a lattice of dimensionality dWe will consider especially connections between neighbors in d=1 (chain)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)representing up/down, firing/not firing, opinions, decisions, atom present/not present, … (always an idealization)Energy:
€
E = − JijSi
ij
∑ S j − hi
i
∑ Si
= − 12 JijSi
ij
∑ S j − hi
i
∑ Si (low energy is “favorable”)
role of geometry: i,j can label points on a lattice of dimensionality dWe will consider especially connections between neighbors in d=1 (chain) and d=∞ ( all-to-all connectivity)
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)representing up/down, firing/not firing, opinions, decisions, atom present/not present, … (always an idealization)Energy:
€
E = − JijSi
ij
∑ S j − hi
i
∑ Si
= − 12 JijSi
ij
∑ S j − hi
i
∑ Si (low energy is “favorable”)
role of geometry: i,j can label points on a lattice of dimensionality dWe will consider especially connections between neighbors in d=1 (chain) and d=∞ ( all-to-all connectivity)Jij > 0: favour Si = Sj:
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)representing up/down, firing/not firing, opinions, decisions, atom present/not present, … (always an idealization)Energy:
€
E = − JijSi
ij
∑ S j − hi
i
∑ Si
= − 12 JijSi
ij
∑ S j − hi
i
∑ Si (low energy is “favorable”)
role of geometry: i,j can label points on a lattice of dimensionality dWe will consider especially connections between neighbors in d=1 (chain) and d=∞ ( all-to-all connectivity)Jij > 0: favour Si = Sj: ferromagnetism
“spins”
binary variables Si = ±1 (or 0/1) (quantized but not quantum-mechanical)representing up/down, firing/not firing, opinions, decisions, atom present/not present, … (always an idealization)Energy:
€
E = − JijSi
ij
∑ S j − hi
i
∑ Si
= − 12 JijSi
ij
∑ S j − hi
i
∑ Si (low energy is “favorable”)
role of geometry: i,j can label points on a lattice of dimensionality dWe will consider especially connections between neighbors in d=1 (chain) and d=∞ ( all-to-all connectivity)Jij > 0: favour Si = Sj: ferromagnetism hi > 0: favour Si = +1.
equilibrium stat mech
€
P[S] =1
Zexp(−βE)Gibbs distribution
equilibrium stat mech
€
P[S] =1
Zexp(−βE) =
1
Zexp 1
2 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟Gibbs distribution
equilibrium stat mech
€
P[S] =1
Zexp(−βE) =
1
Zexp 1
2 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟Gibbs distribution
€
Z = exp 12 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
equilibrium stat mech
€
P[S] =1
Zexp(−βE) =
1
Zexp 1
2 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟Gibbs distribution
partition function
€
Z = exp 12 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
equilibrium stat mech
€
P[S] =1
Zexp(−βE) =
1
Zexp 1
2 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟Gibbs distribution
partition function
€
Z = exp 12 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
€
F = −T log Zfree energy:
equilibrium stat mech
€
P[S] =1
Zexp(−βE) =
1
Zexp 1
2 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟Gibbs distribution
partition function
the original Ising model: nearest-neighbor interactions, J > 0, d = 1, hi = 0€
Z = exp 12 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
€
F = −T log Zfree energy:
equilibrium stat mech
€
P[S] =1
Zexp(−βE) =
1
Zexp 1
2 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟Gibbs distribution
partition function
the original Ising model: nearest-neighbor interactions, J > 0, d = 1, hi = 0
€
Z = exp βJ Si
i
∑ Si+1
⎛
⎝ ⎜
⎞
⎠ ⎟
{S}
∑€
Z = exp 12 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
€
F = −T log Zfree energy:
equilibrium stat mech
€
P[S] =1
Zexp(−βE) =
1
Zexp 1
2 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟Gibbs distribution
partition function
the original Ising model: nearest-neighbor interactions, J > 0, d = 1, hi = 0
€
Z = exp βJ Si
i
∑ Si+1
⎛
⎝ ⎜
⎞
⎠ ⎟
{S}
∑€
Z = exp 12 β JijSi
ij
∑ S j + β hi
i
∑ Si
⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
can also have “3-spin interactions”, etc:
€
E = − 16 K ijkSiS jSk
ijk
∑ − 12 JijSi
ij
∑ S j − hi
i
∑ Si
€
F = −T log Zfree energy:
solving 1-d Ising model by decimation
€
Z = exp βJ Si
i
∑ Si+1
⎛
⎝ ⎜
⎞
⎠ ⎟
{S}
∑ = eβJS1S2 eβJS2S3 eβJS3S4
S4
∑S3
∑ eβJS4 S5 LS5
∑S1S2
∑
€
solving 1-d Ising model by decimation
€
Z = exp βJ Si
i
∑ Si+1
⎛
⎝ ⎜
⎞
⎠ ⎟
{S}
∑ = eβJS1S2 eβJS2S3 eβJS3S4
S4
∑S3
∑ eβJS4 S5 LS5
∑S1S2
∑
€
eβJS i Si+1 =1+ βJSiSi+1 + 12 (βJ)2(SiSi+1)
2 + 13! (βJ)3(SiSi+1)
3
= cosh(βJ) + SiSi+1 sinh(βJ)
= cosh(βJ) 1+ SiSi+1 tanh(βJ)[ ]
€
solving 1-d Ising model by decimation
€
Z = exp βJ Si
i
∑ Si+1
⎛
⎝ ⎜
⎞
⎠ ⎟
{S}
∑ = eβJS1S2 eβJS2S3 eβJS3S4
S4
∑S3
∑ eβJS4 S5 LS5
∑S1S2
∑
€
Z = coshN βJ 1+ S1S2 tanh(βJ)( ) 1+ S2S3 tanh(βJ)( ) 1+ S3S4 tanh(βJ)( )S4
∑S3
∑ LS1S2
∑€
eβJS i Si+1 =1+ βJSiSi+1 + 12 (βJ)2(SiSi+1)
2 + 13! (βJ)3(SiSi+1)
3
= cosh(βJ) + SiSi+1 sinh(βJ)
= cosh(βJ) 1+ SiSi+1 tanh(βJ)[ ]
€
solving 1-d Ising model by decimation
€
Z = exp βJ Si
i
∑ Si+1
⎛
⎝ ⎜
⎞
⎠ ⎟
{S}
∑ = eβJS1S2 eβJS2S3 eβJS3S4
S4
∑S3
∑ eβJS4 S5 LS5
∑S1S2
∑
sum on every other spin:
€
Z = coshN βJ 1+ S1S2 tanh(βJ)( ) 1+ S2S3 tanh(βJ)( ) 1+ S3S4 tanh(βJ)( )S4
∑S3
∑ LS1S2
∑€
eβJS i Si+1 =1+ βJSiSi+1 + 12 (βJ)2(SiSi+1)
2 + 13! (βJ)3(SiSi+1)
3
= cosh(βJ) + SiSi+1 sinh(βJ)
= cosh(βJ) 1+ SiSi+1 tanh(βJ)[ ]
€ €
(1+ Si−1Si tanhβJ)(1+ SiSi+1 tanhβJ)Si
∑ = 2(1+ Si−1Si+1 tanh2 βJ)
solving 1-d Ising model by decimation
€
Z = exp βJ Si
i
∑ Si+1
⎛
⎝ ⎜
⎞
⎠ ⎟
{S}
∑ = eβJS1S2 eβJS2S3 eβJS3S4
S4
∑S3
∑ eβJS4 S5 LS5
∑S1S2
∑
sum on every other spin:
€
Z = coshN βJ 1+ S1S2 tanh(βJ)( ) 1+ S2S3 tanh(βJ)( ) 1+ S3S4 tanh(βJ)( )S4
∑S3
∑ LS1S2
∑€
eβJS i Si+1 =1+ βJSiSi+1 + 12 (βJ)2(SiSi+1)
2 + 13! (βJ)3(SiSi+1)
3
= cosh(βJ) + SiSi+1 sinh(βJ)
= cosh(βJ) 1+ SiSi+1 tanh(βJ)[ ]
€
But this is an interaction J’ between Si-1 and Si+1 with
€
tanhβ ′ J = tanh2 βJ
€
(1+ Si−1Si tanhβJ)(1+ SiSi+1 tanhβJ)Si
∑ = 2(1+ Si−1Si+1 tanh2 βJ)
correlation function
Repeat (“renormalization group”):
correlation function
Repeat (“renormalization group”):after m steps,
€
tanhβJm = (tanhβJ)2m
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins.
€
tanhβJm = (tanhβJ)2m
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:
€
tanhβJm = (tanhβJ)2m
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:
€
tanhβJm = (tanhβJ)2m
€
Z ∝ exp βJM S1SN( )
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:
€
tanhβJm = (tanhβJ)2m
€
Z ∝ exp βJM S1SN( )
€
S1SN =2exp(βJM ) − 2exp(−βJM )
2exp(βJM ) + 2exp(−βJM )
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:
€
tanhβJm = (tanhβJ)2m
€
Z ∝ exp βJM S1SN( )
€
S1SN =2exp(βJM ) − 2exp(−βJM )
2exp(βJM ) + 2exp(−βJM )= tanhβJM
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:
€
tanhβJm = (tanhβJ)2m
€
Z ∝ exp βJM S1SN( )
€
S1SN =2exp(βJM ) − 2exp(−βJM )
2exp(βJM ) + 2exp(−βJM )= tanhβJM = tanhβJ( )
N
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:
€
tanhβJm = (tanhβJ)2m
€
Z ∝ exp βJM S1SN( )
€
S1SN =2exp(βJM ) − 2exp(−βJM )
2exp(βJM ) + 2exp(−βJM )= tanhβJM = tanhβJ( )
N
€
SiSi+n = exp −n /ξ( )
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:
€
tanhβJm = (tanhβJ)2m
€
Z ∝ exp βJM S1SN( )
€
S1SN =2exp(βJM ) − 2exp(−βJM )
2exp(βJM ) + 2exp(−βJM )= tanhβJM = tanhβJ( )
N
=> correlation length
€
SiSi+n = exp −n /ξ( )
€
ξ =−1
logtanhβJ
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:
€
tanhβJm = (tanhβJ)2m
€
Z ∝ exp βJM S1SN( )
€
S1SN =2exp(βJM ) − 2exp(−βJM )
2exp(βJM ) + 2exp(−βJM )= tanhβJM = tanhβJ( )
N
=> correlation length
€
SiSi+n = exp −n /ξ( )
€
tanhβJ →1− 2exp(−2βJ)low T:
€
ξ =−1
logtanhβJ
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:
€
tanhβJm = (tanhβJ)2m
€
Z ∝ exp βJM S1SN( )
€
S1SN =2exp(βJM ) − 2exp(−βJM )
2exp(βJM ) + 2exp(−βJM )= tanhβJM = tanhβJ( )
N
=> correlation length
€
SiSi+n = exp −n /ξ( )
€
tanhβJ →1− 2exp(−2βJ) ⇒ ξ → 12 exp(2βJ)low T:
€
ξ =−1
logtanhβJ
correlation function
Repeat (“renormalization group”):after m steps,Suppose we started with N = 2M spins. After M decimation steps there are just 2 spins left:
€
tanhβJm = (tanhβJ)2m
€
Z ∝ exp βJM S1SN( )
€
S1SN =2exp(βJM ) − 2exp(−βJM )
2exp(βJM ) + 2exp(−βJM )= tanhβJM = tanhβJ( )
N
=> correlation length
€
SiSi+n = exp −n /ξ( )
€
tanhβJ →1− 2exp(−2βJ) ⇒ ξ → 12 exp(2βJ)low T:
€
ξ =−1
logtanhβJ
Correlation length grows toward ∞ at low T, but no ordering
infinite-range modelThe opposite limit: every Jij = J/Nalso soluble:
infinite-range modelThe opposite limit: every Jij = J/Nalso soluble:
d = ∞
infinite-range modelThe opposite limit: every Jij = J/Nalso soluble:
€
Z = expβJ
2NSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
d = ∞
infinite-range modelThe opposite limit: every Jij = J/Nalso soluble:
€
Z = expβJ
2NSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑ = expβJ
2NSi
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
+ βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
d = ∞
infinite-range modelThe opposite limit: every Jij = J/Nalso soluble:
€
Z = expβJ
2NSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑ = expβJ
2NSi
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
+ βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
€
Z = dmδ Nm − Si
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟∫ exp 1
2 NβJm2 + Nβhm( ){S}
∑
d = ∞
infinite-range modelThe opposite limit: every Jij = J/Nalso soluble:
€
Z = expβJ
2NSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑ = expβJ
2NSi
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
+ βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
€
Z = dmδ Nm − Si
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟∫ exp 1
2 NβJm2 + Nβhm( ){S}
∑
€
Z = dmdy
2πi∫ exp −y Nm − Si
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥∫ exp 1
2 NβJm2 + Nβhm( ){S}
∑
d = ∞
infinite-range modelThe opposite limit: every Jij = J/Nalso soluble:
€
Z = expβJ
2NSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑ = expβJ
2NSi
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
+ βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
€
Z = dmδ Nm − Si
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟∫ exp 1
2 NβJm2 + Nβhm( ){S}
∑
€
Z = dmdy
2πi∫ exp −y Nm − Si
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥∫ exp 1
2 NβJm2 + Nβhm( ){S}
∑
€
Z = dmdy
2πi∫∫ exp N −ym + log(2cosh y) + 1
2 βJm2 + βhm( )[ ]
d = ∞
infinite-range modelThe opposite limit: every Jij = J/Nalso soluble:
€
Z = expβJ
2NSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑ = expβJ
2NSi
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
2
+ βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
{S}
∑
€
Z = dmδ Nm − Si
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟∫ exp 1
2 NβJm2 + Nβhm( ){S}
∑
€
Z = dmdy
2πi∫ exp −y Nm − Si
i
∑ ⎛
⎝ ⎜
⎞
⎠ ⎟
⎡
⎣ ⎢
⎤
⎦ ⎥∫ exp 1
2 NβJm2 + Nβhm( ){S}
∑
€
Z = dmdy
2πi∫∫ exp N −ym + log(2cosh y) + 1
2 βJm2 + βhm( )[ ]
exponent prop to N => can use saddle point to evaluate Z
d = ∞
saddle point equations
€
Z = dmdy
2πi∫∫ exp N −ym + log(2cosh y) + 1
2 βJm2 + βhm( )[ ]
saddle point equations
€
Z = dmdy
2πi∫∫ exp N −ym + log(2cosh y) + 1
2 βJm2 + βhm( )[ ]
€
∂∂y
−ym + log(2cosh y) + 12 βJm2 + βhm( ) = 0 ⇒ m = tanh y
saddle point equations
€
Z = dmdy
2πi∫∫ exp N −ym + log(2cosh y) + 1
2 βJm2 + βhm( )[ ]
€
∂∂y
−ym + log(2cosh y) + 12 βJm2 + βhm( ) = 0 ⇒ m = tanh y
∂
∂m−ym + log(2cosh y) + 1
2 βJm2 + βhm( ) = 0 ⇒ y = β (Jm + h)
saddle point equations
€
Z = dmdy
2πi∫∫ exp N −ym + log(2cosh y) + 1
2 βJm2 + βhm( )[ ]
€
∂∂y
−ym + log(2cosh y) + 12 βJm2 + βhm( ) = 0 ⇒ m = tanh y
∂
∂m−ym + log(2cosh y) + 1
2 βJm2 + βhm( ) = 0 ⇒ y = β (Jm + h)
put them together:
€
m = tanh β (Jm + h)[ ]
saddle point equations
€
Z = dmdy
2πi∫∫ exp N −ym + log(2cosh y) + 1
2 βJm2 + βhm( )[ ]
€
∂∂y
−ym + log(2cosh y) + 12 βJm2 + βhm( ) = 0 ⇒ m = tanh y
∂
∂m−ym + log(2cosh y) + 1
2 βJm2 + βhm( ) = 0 ⇒ y = β (Jm + h)
put them together:
€
m = tanh β (Jm + h)[ ]
h = 0:
€
m = tanhβJm
saddle point equations
€
Z = dmdy
2πi∫∫ exp N −ym + log(2cosh y) + 1
2 βJm2 + βhm( )[ ]
€
∂∂y
−ym + log(2cosh y) + 12 βJm2 + βhm( ) = 0 ⇒ m = tanh y
∂
∂m−ym + log(2cosh y) + 1
2 βJm2 + βhm( ) = 0 ⇒ y = β (Jm + h)
put them together:
€
m = tanh β (Jm + h)[ ]
h = 0:
€
m = tanhβJm has 2 solutions (i.e 2 saddle points) for βJ > 1, i.e., T < J
saddle point equations
€
Z = dmdy
2πi∫∫ exp N −ym + log(2cosh y) + 1
2 βJm2 + βhm( )[ ]
€
∂∂y
−ym + log(2cosh y) + 12 βJm2 + βhm( ) = 0 ⇒ m = tanh y
∂
∂m−ym + log(2cosh y) + 1
2 βJm2 + βhm( ) = 0 ⇒ y = β (Jm + h)
put them together:
€
m = tanh β (Jm + h)[ ]
h = 0:
€
m = tanhβJm has 2 solutions (i.e 2 saddle points) for βJ > 1, i.e., T < J
solution m: spontaneous magnetization
intuition: heuristic approach
€
P[S] =1
Zexp 1
2 β JijSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟from
intuition: heuristic approach
€
P[S] =1
Zexp 1
2 β JijSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟from
total field on Si is
€
H i = h + JijS j
j
∑
intuition: heuristic approach
€
P[S] =1
Zexp 1
2 β JijSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟from
total field on Si is
replace it by its mean:
€
H i = h + JijS j
j
∑
€
H = h + Jij S j
j
∑ = h + Jm
intuition: heuristic approach
€
P[S] =1
Zexp 1
2 β JijSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟from
total field on Si is
replace it by its mean:
and calculate m as the average S of a single spin in field H: €
H i = h + JijS j
j
∑
€
H = h + Jij S j
j
∑ = h + Jm
intuition: heuristic approach
€
P[S] =1
Zexp 1
2 β JijSi
ij
∑ S j + βh Si
i
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟from
total field on Si is
replace it by its mean:
and calculate m as the average S of a single spin in field H: €
H i = h + JijS j
j
∑
€
H = h + Jij S j
j
∑ = h + Jm
€
m = S =eβH − eβH
eβH + eβH= tanhβ (h + Jm)
broken symmetry
T > TcT = TcT < Tc
Near Tc
expand
€
m = tanh(βJm + βh) = βJm + βh − 13 (βJm)3
Near Tc
expand
€
m = tanh(βJm + βh) = βJm + βh − 13 (βJm)3
€
(βJ −1)m = 13 (βJm)3h = 0:
Near Tc
expand
€
m = tanh(βJm + βh) = βJm + βh − 13 (βJm)3
€
(βJ −1)m = 13 (βJm)3 ⇒ m =
3(Tc − T)
Tc
, Tc = Jh = 0:
Near Tc
expand
€
m = tanh(βJm + βh) = βJm + βh − 13 (βJm)3
€
(βJ −1)m = 13 (βJm)3 ⇒ m =
3(Tc − T)
Tc
, Tc = Jh = 0:
1st order in h:
€
m = βJm + βh
Near Tc
expand
€
m = tanh(βJm + βh) = βJm + βh − 13 (βJm)3
€
(βJ −1)m = 13 (βJm)3 ⇒ m =
3(Tc − T)
Tc
, Tc = Jh = 0:
1st order in h:
€
m = βJm + βh ⇒ m =h
T − Tc
Near Tc
expand
€
m = tanh(βJm + βh) = βJm + βh − 13 (βJm)3
€
(βJ −1)m = 13 (βJm)3 ⇒ m =
3(Tc − T)
Tc
, Tc = Jh = 0:
1st order in h:
€
m = βJm + βh ⇒ m =h
T − Tc
, i.e., χ =1
T − Tc
Near Tc
expand
€
m = tanh(βJm + βh) = βJm + βh − 13 (βJm)3
€
(βJ −1)m = 13 (βJm)3 ⇒ m =
3(Tc − T)
Tc
, Tc = Jh = 0:
T = Tc:
€
βh = 13 (βJm)3 ⇒ m∝ h
13
1st order in h:
€
m = βJm + βh ⇒ m =h
T − Tc
, i.e., χ =1
T − Tc
Near Tc
expand
€
m = tanh(βJm + βh) = βJm + βh − 13 (βJm)3
€
(βJ −1)m = 13 (βJm)3 ⇒ m =
3(Tc − T)
Tc
, Tc = Jh = 0:
T = Tc:
€
βh = 13 (βJm)3 ⇒ m∝ h
13
1st order in h:
€
m = βJm + βh ⇒ m =h
T − Tc
, i.e., χ =1
T − Tc
mean field critical behaviour
critical behaviour:MFT describes the phase transition qualitatively correctly(right exponents) for d > 4.
critical behaviour:MFT describes the phase transition qualitatively correctly(right exponents) for d > 4.For 1 < d < 4, there is a phase transition, but the critical exponentsare different.
critical behaviour:MFT describes the phase transition qualitatively correctly(right exponents) for d > 4.For 1 < d < 4, there is a phase transition, but the critical exponentsare different.
€
m∝ (T − Tc )β , χ ∝ (T − Tc )−γ , m∝ h1δ
β < 12 , γ >1, δ > 3
critical behaviour:MFT describes the phase transition qualitatively correctly(right exponents) for d > 4.For 1 < d < 4, there is a phase transition, but the critical exponentsare different.
€
m∝ (T − Tc )β , χ ∝ (T − Tc )−γ , m∝ h1δ
β < 12 , γ >1, δ > 3
d = 2: Onsager exact solution:
critical behaviour:MFT describes the phase transition qualitatively correctly(right exponents) for d > 4.For 1 < d < 4, there is a phase transition, but the critical exponentsare different.
€
m∝ (T − Tc )β , χ ∝ (T − Tc )−γ , m∝ h1δ
β < 12 , γ >1, δ > 3
d = 2: Onsager exact solution:
€
Tc = 2.269J, β = 18 , γ = 7
4 , δ =15
Dynamics: Glauber model
74
Ising model: Binary “spins” Si(t) = ±1
Dynamics: Glauber model
75
Ising model: Binary “spins” Si(t) = ±1Dynamics: at every time step,
Dynamics: Glauber model
76
Ising model: Binary “spins” Si(t) = ±1Dynamics: at every time step,
(1) choose a spin (i) at random
Dynamics: Glauber model
77
Ising model: Binary “spins” Si(t) = ±1Dynamics: at every time step,
(1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t)
Dynamics: Glauber model
78
Ising model: Binary “spins” Si(t) = ±1Dynamics: at every time step,
(1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t) Jij = Jji
Dynamics: Glauber model
79
Ising model: Binary “spins” Si(t) = ±1Dynamics: at every time step,
(1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t) Jij = Jji (3) Si(t + Δt) = ±1 with probability
€
P(hi) =exp ±βhi( )
exp βhi( ) + exp −βhi( )=
1
1+ exp(m2βhi)
Dynamics: Glauber model
80
Ising model: Binary “spins” Si(t) = ±1Dynamics: at every time step,
(1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t) Jij = Jji (3) Si(t + Δt) = ±1 with probability
€
P(hi) =exp ±βhi( )
exp βhi( ) + exp −βhi( )=
1
1+ exp(m2βhi)= 1
2 1± tanh(βhi)( )
Dynamics: Glauber model
81
Ising model: Binary “spins” Si(t) = ±1Dynamics: at every time step,
(1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t) Jij = Jji (3) Si(t + Δt) = ±1 with probability
(equilibration of Si, given current values of other S’s)
€
P(hi) =exp ±βhi( )
exp βhi( ) + exp −βhi( )=
1
1+ exp(m2βhi)= 1
2 1± tanh(βhi)( )
Dynamics: Glauber model
82
Ising model: Binary “spins” Si(t) = ±1Dynamics: at every time step,
(1) choose a spin (i) at random(2) compute “field” of neighbors hi(t) = ΣjJijSj(t) Jij = Jji (3) Si(t + Δt) = ±1 with probability
(equilibration of Si, given current values of other S’s)
€
P(hi) =exp ±βhi( )
exp βhi( ) + exp −βhi( )=
1
1+ exp(m2βhi)= 1
2 1± tanh(βhi)( )
master equation for Glauber model
€
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 )
master equation for Glauber model
€
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 )
magnetization
€
Si(t) = Si
{S}
∑ P({S}, t)
master equation for Glauber model
€
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 )
magnetization
€
Si(t) = Si
{S}
∑ P({S}, t)
€
τ 0
d Si
dt= τ 0 Si
{S}
∑ dP({S},t)
dttime evolution:
master equation for Glauber model
€
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 )
magnetization
€
Si(t) = Si
{S}
∑ P({S}, t)
€
τ 0
d Si
dt= τ 0 Si
{S}
∑ dP({S},t)
dt
= 12 Si + tanh βhi(t)( )[ ]
i
∑ P(S1L − SiL SN )
− 12 Si − tanh βhi(t)( )[ ]
i
∑ P(S1L SiL SN )
time evolution:
master equation for Glauber model
€
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 )
magnetization
€
Si(t) = Si
{S}
∑ P({S}, t)
€
τ 0
d Si
dt= τ 0 Si
{S}
∑ dP({S},t)
dt
= 12 Si + tanh βhi(t)( )[ ]
i
∑ P(S1L − SiL SN )
− 12 Si − tanh βhi(t)( )[ ]
i
∑ P(S1L SiL SN )
= − Si + tanh βhi(t)( )
time evolution:
mean field dynamics
€
τ 0
d Si
dt= − Si(t) + tanh β JijS j (t) + βh
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
mean field dynamics
€
τ 0
d Si
dt= − Si(t) + tanh β JijS j (t) + βh
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
mean field approximation:
mean field dynamics
€
τ 0
d Si
dt= − Si(t) + tanh β JijS j (t) + βh
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
mean field approximation:
€
S j (t) → S j (t) = m(t)
mean field dynamics
€
τ 0
d Si
dt= − Si(t) + tanh β JijS j (t) + βh
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
mean field approximation:
€
S j (t) → S j (t) = m(t)
€
⇒ τ 0
dm
dt= −m + tanhβ (Jm + h)
mean field dynamics
€
τ 0
d Si
dt= − Si(t) + tanh β JijS j (t) + βh
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
mean field approximation:
€
S j (t) → S j (t) = m(t)
€
⇒ τ 0
dm
dt= −m + tanhβ (Jm + h)
t -> ∞: recover equilibrium result
mean field dynamics
€
τ 0
d Si
dt= − Si(t) + tanh β JijS j (t) + βh
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
mean field approximation:
€
S j (t) → S j (t) = m(t)
€
⇒ τ 0
dm
dt= −m + tanhβ (Jm + h)
t -> ∞: recover equilibrium result
T > Tc, h = 0:
€
τ 0
dm
dt= −m(1− βJ)
mean field dynamics
€
τ 0
d Si
dt= − Si(t) + tanh β JijS j (t) + βh
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
mean field approximation:
€
S j (t) → S j (t) = m(t)
€
⇒ τ 0
dm
dt= −m + tanhβ (Jm + h)
t -> ∞: recover equilibrium result
T > Tc, h = 0:
€
τ 0
dm
dt= −m(1− βJ) ⇒ m = m(0)exp −
T − Tc
Tτ 0
⎛
⎝ ⎜
⎞
⎠ ⎟t
⎡
⎣ ⎢
⎤
⎦ ⎥
mean field dynamics
€
τ 0
d Si
dt= − Si(t) + tanh β JijS j (t) + βh
j
∑ ⎛
⎝ ⎜ ⎜
⎞
⎠ ⎟ ⎟
mean field approximation:
€
S j (t) → S j (t) = m(t)
€
⇒ τ 0
dm
dt= −m + tanhβ (Jm + h)
t -> ∞: recover equilibrium result
T > Tc, h = 0:
€
τ 0
dm
dt= −m(1− βJ) ⇒ m = m(0)exp −
T − Tc
Tτ 0
⎛
⎝ ⎜
⎞
⎠ ⎟t
⎡
⎣ ⎢
⎤
⎦ ⎥
“critical slowing down”
€
τ =τ 0
T
T − Tc
⎛
⎝ ⎜
⎞
⎠ ⎟
below Tc
€
τ 0
dm
dt= −m + tanhβ (Jm + h)
below Tc
€
τ 0
dm
dt= −m + tanhβ (Jm + h)
€
m0 = tanh(βJm0)expand around
below Tc
€
τ 0
dm
dt= −m + tanhβ (Jm + h)
€
m0 = tanh(βJm0)
τ 0
dδm
dt= −δm + sech2(βJm0) ⋅βJδm
expand around
below Tc
€
τ 0
dm
dt= −m + tanhβ (Jm + h)
€
m0 = tanh(βJm0)
τ 0
dδm
dt= −δm + sech2(βJm0) ⋅βJδm
= −δm + (1− m02)βJδm
expand around
below Tc
€
τ 0
dm
dt= −m + tanhβ (Jm + h)
€
m0 = tanh(βJm0)
τ 0
dδm
dt= −δm + sech2(βJm0) ⋅βJδm
= −δm + (1− m02)βJδm
expand around
€
τ 0
dδm
dt= − 1− (βJ)(1− m0
2)( )δmjust below Tc:
below Tc
€
τ 0
dm
dt= −m + tanhβ (Jm + h)
€
m0 = tanh(βJm0)
τ 0
dδm
dt= −δm + sech2(βJm0) ⋅βJδm
= −δm + (1− m02)βJδm
expand around
€
τ 0
dδm
dt= − 1− (βJ)(1− m0
2)( )δm
= − 1− (βJ)(1− 3(Tc − T) /Tc )( )δm
just below Tc:
below Tc
€
τ 0
dm
dt= −m + tanhβ (Jm + h)
€
m0 = tanh(βJm0)
τ 0
dδm
dt= −δm + sech2(βJm0) ⋅βJδm
= −δm + (1− m02)βJδm
expand around
€
τ 0
dδm
dt= − 1− (βJ)(1− m0
2)( )δm
= − 1− (βJ)(1− 3(Tc − T) /Tc )( )δm
= −2(Tc − T)δm
just below Tc:
below Tc
€
τ 0
dm
dt= −m + tanhβ (Jm + h)
€
m0 = tanh(βJm0)
τ 0
dδm
dt= −δm + sech2(βJm0) ⋅βJδm
= −δm + (1− m02)βJδm
expand around
€
τ 0
dδm
dt= − 1− (βJ)(1− m0
2)( )δm
= − 1− (βJ)(1− 3(Tc − T) /Tc )( )δm
= −2(Tc − T)δm
€
τ =12 τ 0
Tc
Tc − T
⎛
⎝ ⎜
⎞
⎠ ⎟
just below Tc:
correlation time
beyond MFT
still have critical slowing down, different critical exponents
Landau-Ginzburg model
continuous-valued spins in continuous space:
Landau-Ginzburg model
continuous-valued spins in continuous space:
€
Si → φ(x)
Landau-Ginzburg model
continuous-valued spins in continuous space:
€
Si → φ(x)
€
Z = exp −βE({S})( ){S}
∑
Landau-Ginzburg model
continuous-valued spins in continuous space:
€
Si → φ(x)
€
Z = exp −βE({S})( ){S}
∑
→ Z = Dφexp∫ −βE[φ]( )
Landau-Ginzburg model
continuous-valued spins in continuous space:
€
Si → φ(x)
€
Z = exp −βE({S})( ){S}
∑
→ Z = Dφexp∫ −βE[φ]( ) functional integral:
Landau-Ginzburg model
continuous-valued spins in continuous space:
€
Si → φ(x)
€
Z = exp −βE({S})( ){S}
∑
→ Z = Dφexp∫ −βE[φ]( ) functional integral:
€
Dφ ≡ limxi+1 −xi →0
∫ dφi
i
∏
Landau-Ginzburg model
continuous-valued spins in continuous space:
€
Si → φ(x)
€
Z = exp −βE({S})( ){S}
∑
→ Z = Dφexp∫ −βE[φ]( ) functional integral:
€
Dφ ≡ limxi+1 −xi →0
∫ dφi
i
∏
€
E[φ] = 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2[ ]
(free) energyfunctional:
Landau-Ginzburg model
continuous-valued spins in continuous space:
€
Si → φ(x)
€
Z = exp −βE({S})( ){S}
∑
→ Z = Dφexp∫ −βE[φ]( ) functional integral:
€
Dφ ≡ limxi+1 −xi →0
∫ dφi
i
∏
€
E[φ] = 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2[ ]
(free) energyfunctional:
“potential”
€
V (φ) = 12 r0φ
2 + 14 u0φ
4
Landau-Ginzburg model
continuous-valued spins in continuous space:
€
Si → φ(x)
€
Z = exp −βE({S})( ){S}
∑
→ Z = Dφexp∫ −βE[φ]( ) functional integral:
€
Dφ ≡ limxi+1 −xi →0
∫ dφi
i
∏
€
E[φ] = 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2[ ]
(free) energyfunctional:
“potential”
€
V (φ) = 12 r0φ
2 + 14 u0φ
4
Landau-Ginzburg model
continuous-valued spins in continuous space:
€
Si → φ(x)
€
Z = exp −βE({S})( ){S}
∑
→ Z = Dφexp∫ −βE[φ]( ) functional integral:
€
Dφ ≡ limxi+1 −xi →0
∫ dφi
i
∏
€
E[φ] = 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2[ ]
(free) energyfunctional:
“potential” “bending energy”
€
∇φ(x)( )2
€
V (φ) = 12 r0φ
2 + 14 u0φ
4
Landau-Ginzburg model
continuous-valued spins in continuous space:
€
Si → φ(x)
€
Z = exp −βE({S})( ){S}
∑
→ Z = Dφexp∫ −βE[φ]( ) functional integral:
€
Dφ ≡ limxi+1 −xi →0
∫ dφi
i
∏
€
E[φ] = 12 dd x∫ r0φ
2(x) + 12 u0φ
4 (x) + (∇φ(x))2[ ]
(free) energyfunctional:
“potential” “bending energy”
€
∇φ(x)( )2
€
V (φ) = 12 r0φ
2 + 14 u0φ
4
€
− Si+1Si −1( )i
∑ = 12 Si+1 − Si( )
i
∑2
continuum limit of
mean field theory
no fluctuations: ϕ uniform
mean field theory
no fluctuations: ϕ uniform
€
dV
dφ= 0
mean field theory
no fluctuations: ϕ uniform
€
dV
dφ= 0 ⇒ r0φ0 + u0φ0
3 = 0
mean field theory
no fluctuations: ϕ uniform
€
dV
dφ= 0 ⇒ r0φ0 + u0φ0
3 = 0
€
φ0 = 0
φ0 = 0, ± −r0 u0
solutions:
mean field theory
no fluctuations: ϕ uniform
€
dV
dφ= 0 ⇒ r0φ0 + u0φ0
3 = 0
€
φ0 = 0
φ0 = 0, ± −r0 u0
solutions:
take r0 prop to T - Tc
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
Fourier transform:
€
E[φ] = 12
dd p
(2π )d∫ r0 φ( p)2
+ p2 φ( p)2
[ ]
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
Fourier transform:
€
E[φ] = 12
dd p
(2π )d∫ r0 φ( p)2
+ p2 φ( p)2
[ ] ≡ 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
Fourier transform:
€
E[φ] = 12
dd p
(2π )d∫ r0 φ( p)2
+ p2 φ( p)2
[ ] ≡ 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫
€
⇒ P[φ] =1
Zexp −βE[φ]( )
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
Fourier transform:
€
E[φ] = 12
dd p
(2π )d∫ r0 φ( p)2
+ p2 φ( p)2
[ ] ≡ 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫
€
⇒ P[φ] =1
Zexp −βE[φ]( ) =
1
Zexp − 1
2 (r0 + p2)φ(p)2
[ ]p
∏
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
Fourier transform:
€
E[φ] = 12
dd p
(2π )d∫ r0 φ( p)2
+ p2 φ( p)2
[ ] ≡ 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫
€
⇒ P[φ] =1
Zexp −βE[φ]( ) =
1
Zexp − 1
2 (r0 + p2)φ(p)2
[ ]p
∏ (T = 1)
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
Fourier transform:
€
E[φ] = 12
dd p
(2π )d∫ r0 φ( p)2
+ p2 φ( p)2
[ ] ≡ 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫
€
⇒ P[φ] =1
Zexp −βE[φ]( ) =
1
Zexp − 1
2 (r0 + p2)φ(p)2
[ ]p
∏
product of independent Gaussians, one for each pair (p,-p)
(T = 1)
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
Fourier transform:
€
E[φ] = 12
dd p
(2π )d∫ r0 φ( p)2
+ p2 φ( p)2
[ ] ≡ 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫
€
⇒ P[φ] =1
Zexp −βE[φ]( ) =
1
Zexp − 1
2 (r0 + p2)φ(p)2
[ ]p
∏
product of independent Gaussians, one for each pair (p,-p)
€
φ( p)2
=1
r0 + p2
(T = 1)
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
Fourier transform:
€
E[φ] = 12
dd p
(2π )d∫ r0 φ( p)2
+ p2 φ( p)2
[ ] ≡ 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫
€
⇒ P[φ] =1
Zexp −βE[φ]( ) =
1
Zexp − 1
2 (r0 + p2)φ(p)2
[ ]p
∏
product of independent Gaussians, one for each pair (p,-p)
€
φ( p)2
=1
r0 + p2
€
⇒ φ(x)φ(y) =1
4π x − yexp − r0 x − y( )
(T = 1)
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
Fourier transform:
€
E[φ] = 12
dd p
(2π )d∫ r0 φ( p)2
+ p2 φ( p)2
[ ] ≡ 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫
€
⇒ P[φ] =1
Zexp −βE[φ]( ) =
1
Zexp − 1
2 (r0 + p2)φ(p)2
[ ]p
∏
product of independent Gaussians, one for each pair (p,-p)
€
φ( p)2
=1
r0 + p2
€
⇒ φ(x)φ(y) =1
4π x − yexp − r0 x − y( )
correlation length
€
ξ =1
r0
∝1
T − Tc
(T = 1)
fluctuations around MFT
r0 > 0: ignore ϕ4 term:
€
E[φ] = 12 dd x∫ r0φ
2(x) + (∇φ(x))2[ ]
Fourier transform:
€
E[φ] = 12
dd p
(2π )d∫ r0 φ( p)2
+ p2 φ( p)2
[ ] ≡ 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫
€
⇒ P[φ] =1
Zexp −βE[φ]( ) =
1
Zexp − 1
2 (r0 + p2)φ(p)2
[ ]p
∏
product of independent Gaussians, one for each pair (p,-p)
€
φ( p)2
=1
r0 + p2
€
⇒ φ(x)φ(y) =1
4π x − yexp − r0 x − y( )
correlation length
€
ξ =1
r0
∝1
T − Tc
diverges at Tc
(T = 1)
Langevin dynamics
€
∂φ(x)
∂t= −
δE
δφ(x)+ η (x, t), η (x, t)η ( ′ x , ′ t ) = 2Tδ d (x − ′ x )δ(t − ′ t )
Langevin dynamics
€
∂φ(x)
∂t= −
δE
δφ(x)+ η (x, t), η (x, t)η ( ′ x , ′ t ) = 2Tδ d (x − ′ x )δ(t − ′ t )
€
∂φ( p)
∂t= −
∂E
∂φ*( p)+ η (p, t), η (p, t)η (−p, ′ t ) = 2Tδ(t − ′ t )
Fourier transform :
Langevin dynamics
€
∂φ(x)
∂t= −
δE
δφ(x)+ η (x, t), η (x, t)η ( ′ x , ′ t ) = 2Tδ d (x − ′ x )δ(t − ′ t )
€
∂φ( p)
∂t= −
∂E
∂φ*( p)+ η (p, t), η (p, t)η (−p, ′ t ) = 2Tδ(t − ′ t )
Fourier transform :
€
E[φ] = 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫ + 14 u0 φ(p)φ( ′ p )φ( ′ ′ p )φ(−p − ′ p − ′ ′ p )
p, ′ p , ′ ′ p
∫
Langevin dynamics
€
∂φ(x)
∂t= −
δE
δφ(x)+ η (x, t), η (x, t)η ( ′ x , ′ t ) = 2Tδ d (x − ′ x )δ(t − ′ t )
€
∂φ( p)
∂t= −
∂E
∂φ*( p)+ η (p, t), η (p, t)η (−p, ′ t ) = 2Tδ(t − ′ t )
Fourier transform :
€
E[φ] = 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫ + 14 u0 φ(p)φ( ′ p )φ( ′ ′ p )φ(−p − ′ p − ′ ′ p )
p, ′ p , ′ ′ p
∫
€
∂φ( p)
∂t= − r0 + p2
( )φ(p) − u0 φ( ′ p )φ( ′ ′ p )φ(p − ′ p − ′ ′ p )′ p , ′ ′ p
∫ + η ( p, t)
Langevin dynamics
€
∂φ(x)
∂t= −
δE
δφ(x)+ η (x, t), η (x, t)η ( ′ x , ′ t ) = 2Tδ d (x − ′ x )δ(t − ′ t )
€
∂φ( p)
∂t= −
∂E
∂φ*( p)+ η (p, t), η (p, t)η (−p, ′ t ) = 2Tδ(t − ′ t )
Fourier transform :
€
E[φ] = 12 r0 φ( p)
2+ p2 φ( p)
2
[ ]p
∫ + 14 u0 φ(p)φ( ′ p )φ( ′ ′ p )φ(−p − ′ p − ′ ′ p )
p, ′ p , ′ ′ p
∫
€
∂φ( p)
∂t= − r0 + p2
( )φ(p) − u0 φ( ′ p )φ( ′ ′ p )φ(p − ′ p − ′ ′ p )′ p , ′ ′ p
∫ + η ( p, t)
or, back in real space,
€
∂φ(x)
∂t= − r0 −∇ 2
( )φ(x) − u0φ3(x) + η (x, t)
small fluctuations above Tc
€
∂φ( p)
∂t= − r0 + p2
( )φ(p) + η ( p, t)ignoring u0:
small fluctuations above Tc
€
∂φ( p)
∂t= − r0 + p2
( )φ(p) + η ( p, t)ignoring u0:
We have solved this before:
small fluctuations above Tc
€
∂φ( p)
∂t= − r0 + p2
( )φ(p) + η ( p, t)ignoring u0:
We have solved this before:
€
φ( p, t) = exp −(r0 + p2)(t − ′ t )[ ]−∞
t
∫ η (p, ′ t )d ′ t
small fluctuations above Tc
€
∂φ( p)
∂t= − r0 + p2
( )φ(p) + η ( p, t)ignoring u0:
We have solved this before:
€
φ( p, t) = exp −(r0 + p2)(t − ′ t )[ ]−∞
t
∫ η (p, ′ t )d ′ t
average over noise (we did this before, too)
small fluctuations above Tc
€
∂φ( p)
∂t= − r0 + p2
( )φ(p) + η ( p, t)ignoring u0:
We have solved this before:
€
φ( p, t) = exp −(r0 + p2)(t − ′ t )[ ]−∞
t
∫ η (p, ′ t )d ′ t
average over noise (we did this before, too)
€
φ( p, t)φ(p, ′ t ) =T
r0 + p2exp −(r0 + p2) t − ′ t [ ]
small fluctuations above Tc
€
∂φ( p)
∂t= − r0 + p2
( )φ(p) + η ( p, t)ignoring u0:
We have solved this before:
€
φ( p, t) = exp −(r0 + p2)(t − ′ t )[ ]−∞
t
∫ η (p, ′ t )d ′ t
average over noise (we did this before, too)
€
φ( p, t)φ(p, ′ t ) =T
r0 + p2exp −(r0 + p2) t − ′ t [ ]
critical slowing down (long correlation time) of fluctuations near Tc (small r0) for small p (long wavelengths)
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