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Computational Statistical Physics
FS 2017
402-0812-00L
Friday 10.45 – 12.30 in HIT H51 Exercices: Friday 8.45- 10.30 in HIT F21
Oral exam
www.ifb.ethz.ch/education/msc-courses/ msc-computational-statphys.html
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Studiengänge
• Mathematics, Computer Science (Bachelor) • Mathematics, Computer Science (Master) • Physics (Wahlfach) • Material Science (Spezialvorlesung, Master) • Civil Engineering (Spezialvorlesung)
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Where do you find us? Computational Physics for Building
Materials (IfB)
Mirko Lukovic
Miller Mendoza
Malte Henkel
HIF E 28.2 HIT G 23.5 HIF E 23.1
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Computational Physics II Computational Quantum Physics 402-0810 G. Carleo Tuesday morning: V Di 10-12, U Di 13-15
Computational Statistical Physics 402-0812 M. Lukovic, M. Mendoza Jimenez and M. Henkel Friday morning: V Fr 11-13, U Fr 9-11
Molecular and Materials Modelling 327-5102 D. Passerone and C. Pignedoli Friday afternoon: V Fr 14-16, U Fr 16-18
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Computational Quantum Physics
Giuseppe Carleo
Tuesday morning: V Di 10-12, U Di 13-15
One particle quantum mechanics: scattering problem, time evolution shooting technique Numerov algorithm
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Computational Quantum Physics
Many particle systems: Fock space, etc (≈ 2 weeks theory) Hartree-Fock approximation density functional theory and electron structure (He & H2) strongly correlated electrons Hubbard and T-J models
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Computational Quantum Physics
Lanczos method Path integral Monte Carlo Bosonic world lines Variational Monte Carlo Density-Matrix Renorm. Group Fermions, QFT
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Molecular & Materials Modelling
Daniele Passerone and Carlo Pignedoli Friday afternoon; V Fr 14-16, U Fr 16-18
Empirical potentials and transition rates Bio-force fields, charges, peptides Embedded atom models, Wilff‘s theorem Pair-correlation function with MD for neutron scattering
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Melting temperature from phase coexistence MO-theory, basic SCF, chemical reactions Density functional theory, pseudopotentials DFT on realistic systems, hybrids Linear scaling, GPW Electronic spectroscopies, STM Bandstructure, graphene, free energies
Molecular & Materials Modelling
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Plan of this course
• 24.02. Statistical Physics, recapitulation MC • 03.03. Multi-spin-coding, dynamical scaling • 10.03. Glauber and Kawasaki dynamics • 17.03. Microcanonical simulations, Binder
…….............................…...cumulants, 1st oder transitions (Potts)
• 24.04. Cluster algorithm, histogram methods • 31.04. MC Renormalization Group,
…..................................…..parallelization and vectorization
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Plan of this course
• 07.04. Molecular Dynamics, Verlet scheme • 14.04 and 21.04 ETH vacations • 28.04. Linked cell, Ewald sums, particle-mesh
• 05.05. Reaction field, Lagrange multipliers, ……………….………..…rigid bodies, quaternions
• 12.05. Nosé-Hoover thermostat, stochastic ……………….………..…method, constant pressure ensemble
• 19.05. Event driven, inelastic collisions, friction • 26.05. Contact dynamics • 02.06. ab initio MD, Car – Parinello
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Prerequisites
• Introduction to Computational Physics • Ability to work with UNIX • Making of Graphical Plots • Some experience with C++ (or similar) • Statistical Analysis (Averaging,
Distributions) • Basic Statistical Physics
16
Literature • H.Gould and J. Tobochnik: „Introduction to Computer
Simulation Methods“ (Wesley, 1996) • D. Landau and K. Binder: „A Guide to Monte Carlo
Simulations in Statistical Physics“ (Cambridge, 2000) • D. Stauffer, F.W. Hehl, V. Winkelmann and J.G.
Zabolitzky: „Computer Simulation and Computer Algebra“ (Springer, 1988)
• K. Binder and D.W. Heermann: „Monte Carlo Simulation in Statistical Physics“ (Springer; 1997)
• N.J. Giordano: „Computational Physics“ (Pearson, 1996) • J.M. Thijssen: „Computational Physics“ (Cambridge,
1999) • M. P. Allen and D.J. Tildesley: Computer Simulation of
Liquids (Oxford 1987)
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Classical Statistical Mechanics
We consider a many body system of N classical particles i each having n degrees of freedom pi
(j) (discrete or continuous). One configuration X is given by X = pi
(j), i = 1,…N, j = 1,…n. The set of all possible configurations is called the „phase space“.
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Hamiltonian
The time evolution of the system should be described by a Hamiltonian H (that should not explicitely depend on time) through the Liouville equation:
ρρ ,),( HtXt
−=∂∂
where ρ is the distribution of configurations.
21
Thermal equilibrium
0=∂∂
tρThe steady state of this equation:
defines the „thermal equilibrium“.
)()(1 XXQQX
ρ∑Ω=
The thermal average over a quantity Q is
where Ω is the volume of the phase space.
22
Ensembles
• Microcanonical ensemble: fix E, V, N • Canonical ensemble: fix T, V, N • Grandcanonical ensemble: fix T, V, μ • Canonical pressure ensemble: fix T, p, N
We can fix either volume V or pressure p, either energy E or temperature T, either particle number N or chemical potential μ, either magnetization M or magnetic field H.
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E(X) = energy of configuration X is fixed and probability for system to be in X is equal for all E:
Microcanonical Ensemble
))((1)( EXHZ
Xpmk
eq −= δ
[ ]∑ −=−=X
mk EXHTrEXHZ ))(())(( δδ
Zmk is the partition function:
24
Temperature T is fixed and the probability for system to be in X given by Boltzmann factor: E(X) = energy of configuration X
Canonical Ensemble
kTXE
Teq e
ZXp
)(1)(−
=
∑−
=X
kTXE
T eZ)(
∑ =X
eq Xp 1)(ZT is the partition function:
∑−
=X
kTXE
T
eXQZ
TQ)(
)(1)(Thermal average of quantity Q is:
25
The Ising Model
• Magnetic Systems • Opinion models • Binary mixtures
Ernst Ising (1900-1998)
Spins on a lattice
26
The Ising Model
Nii ,....1 ,1 =±=σ
∑∑=
−−==N
ii
N
nnjiji HJE
1:,σσσ H
Binary variables:
on a graph of N sites interacting via the Hamiltonian:
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Order parameter
0 1
1( ) limN
s iH iM T
Nσ
→=
= ∑spontaneous magnetization:
ordered phase
disordered phase critical temperature
( )s cM T T β∝ −
β = 1/8 (2d) β ≈ 0.326 (3d)
s
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Response functions
γχ −
=
−∝∂∂
= cHT
TTHMT
0,
)(
,
( )v cV H
EC T T TT
α−∂= ∝ −∂
susceptibility:
specific heat:
both diverge at Tc
29
Response as fluctuation Derive fluctuation-dissipation theorem for the susceptibility:
01
01
1
0
( ) 0
0, :
( , )( )
1with ,
N
ii
N
ii
T
N H
iX i
HH
X
Z H HN
i ji j nn
eM T HT
H He
JkT
β σ
β σ
σχ
β σ σ β
=
=
+
=
+=
= =
∑∂ ∂
= =∂ ∂ ∑
= =
∑∑
∑
∑
H
H
H
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Fluctuation-dissipation theorem
( )
00 1
1
2
2
11
2
00
22
( )( ) ( )
( ) ( ) ( )
NN
ii i
i
N HN H
ii X i
X i
T T
HH
eeT
Z H Z H
T M T M T
β σβ σ β σβ σ
χ
χ β
==
++
==
==
∑∑
= −
⇒ = −
∑∑∑ ∑H
H
0)( ≥⇒ Tχ
22 2V BC k E Eβ = −
Analogously one can show for the specific heat:
These formulas are used in MC simulations.
31
Specific heat
α−−∝ cv TTTC )(
α = 0 (2d) α ≈ 0.11 (3d)
comparing with experimental data for binary mixture
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Susceptibility
γχ −−∝ cTTT )(
γ = 7/4 (2d) γ ≈ 1.24 (3d)
numerical data from a finite system
33
Correlation length )()0()( RRC σσ=correlation function:
2−
∝ +( )R
C R M ae ξ
For T ≠ Tc and for large R:
where ξ is the correlation length.
T > Tc
T < Tc
34
Correlation length
νξ −−∝ cTT
η−−∝ dRRC 2)(
The correlation length diverges at Tc as:
with a critical exponent ν. ν = 1 (2d) ν ≈ 0.63 (3d)
At Tc we have for large R:
with η = 1/4 (2d) η ≈ 0.05 (3d)
35
Exponent relations
γνηνα
γβα
=−=−
=++
)2( 2
22d
Exponents are related through:
scaling
hyperscaling
so that only two exponents are independent. H.E.Stanley, „Introduction to Phase Transitions and Critical Phenomena“ (Clarendon, Oxford, 1971)
36
Monte Carlo Method
Nicholas Constantine Metropolis
Stanislaw Ulam
37
Simulates an experimental measuring process with sampling and averaging. Big advantages: Systematic improvement by increasing the number of samples M. Error goes like:
Monte Carlo Method (MC)
M1
∝∆
38
MC strategy
• Choose randomly a new configuration.
• If the „equilibrium condition“ is not fulfilled then reject, otherwise accept.
• Calculate physical properties and add to the averaging loop.
39
Problem of sampling
∑>=<X
eq XpXQTQ )()()(
The distribution of average energy <E> gets sharper with increasing size.
Choosing configurations equally distributed over energy would be very ineffective.
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M(RT)2 algorithm
N.C. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller and E. Teller (1953)
importance sampling through a Markov chain: X1 → X2 → ….. which goes into a steady state in which the probability for a configuration is pst(X).
Markov chain: Xi only depends on Xi-1.
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1. Ergodicity: One must be able to reach any configuration Y after a finite number of steps.
2. Normalization:
3. Reversibility:
Properties of Markov chain
1( )Y
T X Y→ =∑
Start in configuration X and propose a new configuration Y with probability T(X→Y).
( ) ( )T X Y T Y X→ = →
42
Transition probability The proposed configuration Y will be accepted with probability A(X→Y). Transition probability of the Markov chain is:
W(X→Y) = T(X→Y) ⋅ A(X→Y)
Master equation:
( , ) ( ) ( ) ( ) ( )Y Y
dp X t p Y W Y X p X W X Ydt
= → − →∑ ∑
43
Properties of W(X→Y)
• Ergodicity:
• Normality:
• Homogeneity:
0, : ( )X Y W X Y∀ → >
1( )Y
W X Y→ =∑
→ =∑ ( ) ( ) ( )st stY
p Y W Y X p X
44
In stationary state one should have equilibrium distribution (Boltzmann).
Detailed balance ( , ) ( ) ( ) ( ) ( )
Y Y
dp X t p Y W Y X p X W X Ydt
= → − →∑ ∑
0= ⇔ =( , ) ( ) ( )st eq
dp X t p X p Xdt
( ) ( ) ( ) ( )eq eqY Y
p Y W Y X p X W X Y⇒ → = →∑ ∑sufficient condition is detailed balance:
( ) ( ) ( ) ( )eq eqp Y W Y X p X W X Y→ = →
46
Metropolis (M(RT)2)
1( )
( ) min ,( )
eq
eq
p YA X Y
p X
→ =
1 ( )
( )E X
kTeq
T
p X eZ
−=
1 1− ∆
− − → = =
( ) ( )
( ) min , min ,E Y E X E
kT kTA X Y e e
Boltzmann:
if energy decreases always accept increases accept with probability
47
MC of the Ising Model
• Choose one site i (having spin σi). • Calculate ΔE = E(Y) - E(X) = 2Jσihi. • If ΔE ≤ 0 then flip spin: σi → -σi. • If ΔE > 0 flip with probability exp(-βΔE)
Single flip Metropolis:
σ= ∑
i jnn of i
hwhere hi is the local field at site i
, :
N
i ji j nn
E J σ σ= − ∑
48
Implementation on computer
4
10 2 4
2 0 4 8
( ) , ,
, ,
i jj nn of i
i i
h
E hJ
σ
σ
=
= = ± ±
∆⇒ = = ± ±
∑
Look-up tables: Consider the Ising model on a square lattice
Since for ΔE ≤ 0 we accept move with probability 1 we only need to store two values: 4( ) JkP k e β−= with k = ½σihi
(i.e. k = 1,2)
49
Multi-spin coding Technique to increase speed and reduce memory space for Boolean variables.
Consider Ising model on simple cubic lattice. There we have 6 nearest neighbors, i.e. the energies can have 7 different values (0,…,6). Therefore we need 3 bits per site.
One computer word has 64 bits.
50
Multi-spin coding
Use the bitwise logical functions, example: (0,1,0,0,1) XOR (1,1,0,1,1) = (1,0,0,1,0)
Store neighboring sites in different words Nj . Calculate energy of 21 sites simultaneously:
E = N XOR N1 + … + N XOR N6
Define i th site in a word (i = 1,…21): Ni = (0,…,0,1,0,…,0,0) ↑ ↑ ↑ ↑
64 3i -2 2 1 position
51
Multi-spin coding
Result: One updates 21 sites simultaneously and reduces memory requirement by a factor 21.
The „changer word“ cw is "1" if the spin is flipped and is "0" if the spin is not flipped.
7 = (0,…,0,1,1,1) is a mask to extract the last 3 bits of E through E & 7.
52
Multi-spin coding cw=0; for(i=1;i<=21;i++) z=ranf(); if(z<P(E&7))cw=(cw|1); cw=ror(cw,3); E=ror(E,3); cw=ror(cw,1); N=(N^cw);
& = AND | = OR ^ = XOR
ror = circular right shift
53
Sampling
Each time we accept a spin-flip we generate a new configuration which is however very similar to the previous one. So the samples in our Markov chain are very correlated. But we need statistically uncorrelated configurations to make averages ! We also need to get decorrelated of the initial configuration of the Markov chain.
54
Dynamic interpretation of MC
( , ) ( ) ( ) ( ) ( )Y Y
dp X t p Y W Y X p X W X Ydt
= → − →∑ ∑
0( ) ( , ) ( ) ( , ) ( ( ))X X
A t p X t A X p X t A X t= =∑ ∑The time evolution of a quantity A is
with
0
( ) ( )( )
( ) ( )nlA
A t At
A t A− ∞
Φ =− ∞
Suppose that the configuration at t0 is not at equilibrium, then define „non-linear correlation function“:
55
Non-linear correlation time ln ΦA(t)
-0.4
-0.3
-0.2
-0.1 slope = -1/τnlA
T ≠ Tc
nl
56
Non-linear correlation time
0
τ∞
≡ Φ∫ ( )nl nlA A t dt
( )nlA
tnlA t e τ
−
Φ =
Define the non-linear correlation time τnl
A as: example:
nlAnl z
A cT Tτ −∝ −Critical slowing down:
znlA is the non-linear dynamical critical exponent.
Describes relaxation towards equilibrium.
57
Linear correlation function
0( ) ( )( )AB
A t B t A Bt
AB A B−
Φ =−
With two quantities A and B in equilibrium define the „linear time correlation function“:
with
0 0 0( ) ( ) ( , ) ( ( )) ( ( ))X
A t B t p X t A X t B X t=∑
58
Auto correlation function
20 0
220 0
( ) ( ) ( )( )
( ) ( )
t t tt
t tσ
σ σ σ
σ σ
−Φ =
−
for example the spin-spin correlation:
If A = B we have an „auto correlation“
59
Time autocorrelation function
Φσ(t)
60
Linear correlation time
0
τ∞
≡ Φ∫ ( )AB AB t dt
( ) AB
t
AB t e τ−
Φ =
Define the linear correlation time τAB as:
example:
AB
AB czT Tτ −
∝ −Critical slowing down:
zAB is the dynamical critical exponent.
Describes relaxation in equilibrium.
61
Dynamical critical exponents
zσ = 2.16 (2d) zσ = 2.09 (3d)
α
βσσ
−=−
=−
1
nlEE
nl
zzzz
kinetic Ising model
conjectured relations:
62
Finite size effects
problem when: system size L < correlation length ξ i.e. close to the critical point:
L critical region
round-off
Tc T
63
Critical dynamics in finite sizes
( ) cL T T T νξ −= ∝ −
ABAB
zz
AB cT T L ντ −∝ − ∝
⇒
Number of discarded samples grows like power law of the system size.
at Tc
ντ =ABz
AB AL
or
64
Decorrelated configurations To calculate averages one wants to have configurations that are statistically not correlated.
• First to reach equilibrium, throw away n0 = c τnl(T) many configurations. • Then only take every neth configuration, with ne = c τ(T). • At Tc use:
0ν ν= =and nlz z
en cAL n cAL
c ≈ 3 is a safety factor.
66
Glauber dynamics
1( )
EkT
EkT
eA X Ye
∆−
∆−
→ =+
Roy C. Glauber (1963) (Nobel prize 2005)
fulfills detailed balance
67
Glauber dynamics 2
2 , 1
i i
i i
J h
i i jJ hj nn
eA he
βσ
βσ σ−
−=
≡ =+ ∑
( )( 1) ( ) zσ σ+ = − ⋅ −i i it t sign A
2
21
β
β=+
i
i
J h
i J hep
e1
1 1for
for
( )
i i
flip i ii i
pp A
pσ
σσ
= −= = − = +
1 11
1
for
for
( )
i i
no flip i ii i
pp A
pσ
σσ
− = −= − = = +
Spin-flip probability for Ising model:
Implementation using a random number z:
with
11 1
with
with
i i
i i
pp
σσ
= +
= − −
⇒
68
Heat bath method
2
2 1
i
i
J h
i J hep
e
β
β=+
11 1
with probability
with probability
i i
i i
pp
σσ
= +
= − −
with
Choose site i and set:
Is equivalent to Glauber dynamics.
69
Binary mixtures (lattice gas)
Consider two species A and B distributed with given concentrations on the sites of a lattice. EAA is energy of A-A bond EBB is energy of B-B bond EAB is energy of A-B bond
Set EAA = EBB = 0 and EAB = 1.
⇒ Ising model with J = 1 and constant M.
70
Kawasaki dynamics
Kyozi Kawasaki
• Choose any A-B bond. • Calculate ΔE for A-B → B-A. • Metropolis: If ΔE ≤ 0 flip, else
flip with p = exp(-βΔE). • Glauber: Flip with probability
p = exp(- βΔE)/(1+ exp(- βΔE)).
zσ = 2.32 (2d) for Ising model
71
Kawasaki dynamics
Link zum Applet
72
Microcanonical Monte Carlo
Keep energy constant.
• Creutz algorithm • Q2R • Kadanoff – Swift
73
Creutz algorithm
Michael Creutz (1983)
Introduce a small energy reservoir Ed called „demon“ which can store a maximum energy Emax .
• Choose randomly a site. • Calculate ΔE for spin flip • Accept flip if:
0max dE E E−∆≥ ≥
74
Creutz algorithm
Algorithm is determinsitic, i.e. no random numbers. Algorithm is reversible, i.e. there exist no transients. Perfect for multi-spin coding and for parallelisation, in that case use one demon per processor (e.g. 64).
Obtain temperature T through the histogram P(Ed) of the energies Ed of the demons since it should follow a Boltzmann distribution:
( )dE
kTdP E e
−∝
75
Q2R
Case Emax = 0 of Creutz algorithm on square lattice G. Vichniac (1984)
( ) ( )1 1 1 1
1
1 20 2
ifif
( )
( )
ij ij ij
ij i j i j ij ij
t f x t
x
xf x
x
σ σ
σ σ σ σ− + − +
+ = ⊕
= + + +
== ≠
totalistic cellular automaton
σij = 1,0
Applet
76
Q2R
( ) ( ) ( ) ( )( ) ( ) ( )( )( )1 2 3 4 1 3 2 41t tσ σ σ σ σ σ σ σ σ σ+ = ⊕ ⊕ ∧ ⊕ ∨ ⊕ ∧ ⊕Can also be expressed as logical function:
σ σ3
σ4
σ2 σ1
i jnn
E σ σ= ⊕∑
deterministic and reversible
Energy
64 updates in about 12 cycles !!
is a conserved quantity.
77
Implementation of Q2R
σσ ˆ and Divide lattice in two sub-lattices:
( ) ( )( ) ( )
( ) ( )( ) ( )( ) ( )
1
1
( )
( ) ( )
ˆ
ˆ ˆ ˆ
i j ij nn i
i k j ik nn j j nn i
t f t tR
t f f t t t
σ σ σ
σ σ σ σ
=
= =
+ = ⊕=
+ = ⊕ ⊕
Use multi-spin coding to implement reversible bitwise logical automaton:
but not ergodic 20.27N cycles of size 20.73N
for lattice of N sites.
78
Boundary conditions
• Open boundaries: no neighbor • Fixed boundaries: neighbor with fixed spin • Periodic boundaries: define index vectors
11
1 11
if ( )
if
if ( )
if
i i LIP i
i L
i iIM i
L i
+ <= =
− >= =
finite system L × L
79
Helical boundary conditions
( )1= + −k i j L
index system as a one dimensional string
1± ±and k k L
neighbors are:
80
Finite size scaling
V.Privman (ed.) «Finite-Size Scaling and Numerical Simulations of Statistical Systems» (World Scientific, Singapore, 1990) K.Binder in Festkörperprobleme (Advances in Solid State Physics), Vol. 26, ed. P. Grosse, (Vieweg, Braunschweig, 1986), p. 133
81
Tc for finite sizes νξ −−∝= )()( 11 cTTTL
)(2 121 cTTTT −≈−
ν1
21
−∝− LTTL
T1 T2
)1()(1ν−
−= aLTLT ceff
size of critical region:
T
83
χ for finite sizes
84
Finite size scaling for χ
Maximum of χ at Tc: νγ
χ LL ∝)(max
slope = γ / ν
85
L
χ L
Finite size scaling for χ
])[(),(1ν
χνγ
χ LTTLLT c−= F
data collapse
A.E.Ferdinand and M.E Fisher (1967)
|T-Tc|
slope = - γ
Fχ is a universal scaling function.
86
Binder cumulant
Kurt Binder (1981)
4
221
3L
L
L
MU
M≡ −
CCP 2006, Corea
87
Binder cumulants
4 1
4 4 1
2 22 2 1
2
( )( )
( )
M cL
C c
LM c
L T T LMT T L
ML T T L
βν ν
ν
βν ν
ℑ −
= = ℑ −
ℑ −
At Tc this quantity is independent of L.
88
Binder cumulants
2
( ) d
L
d
L
M L
LLP M e σ
πσ
−
= 2L B Lk Tσ χ=
24 23L L
M M=⇒
T > Tc Magnetization follows a Gaussian distribution:
⇒ UL = 0
89
Binder cumulants
2 2( ) ( )12
( )d d
s sd
L L
L
M M L M M L
LLP M e eσ σ
πσ
− +− −
= +
24 2=L L
M M
T < Tc
⇒
The distribution of the magnetization is the superposition of two Gaussians
M. Rovere et al, J.Phys.2, p. 7009 (1990)
⇒ UL = 2/3
90
Binder cumulants 4
22
0 for 1 2 for 3 3
cL
L Lc
L
T TMU
T TM→∞
>= − →
<
91
Binder cumulants
92
Corrections to scaling 1
1( ) ( ) ( ) ....c cM T A T T A T T ββ= − + − +
1 1 , β β ν ν> <1 1
1( , ) ( ) ( ) ....M c M cM T L L T T L L T T Lβν ν ν
− = ℑ − + ℑ − +
x-
1
1
max , , 1β β βν ν ν
= −
x
( ) ( ) 1
1( ) ...c cT C T T C T Tν νξ − −= − + − +
universal correction to scaling exponents
with
93
For T < Tc the Ising model has at H = 0 a phase transition in H of first order. This means that one has a jump in magnetization ΔM and entropy ΔS and a latent heat ΔE. Consequently susceptibility and specific heat exhibit delta functions at the transition.
First order transition
94
First order transition We have hysteresis and for small systems the magnetization jumps after the „ergodic time“ Te.
M
95
2 2( ) ( )12
( )d d
s sd
L L
L
M M L M M L
LLP M e eσ σ
πσ
− +− −
= +
The distribution of the magnetization is the superposition of two Gaussians
FSS of 1st order transition
Consider times much larger than Te.
96
FSS of 1st order transition
2
2
( ) tanh( )
( )cosh ( )
χ β
βχ χβ
= +
∂= = +∂
D dL L L
dD L
L L dL
M H H M HM LM LMH
H HM L
From the distribution of the magnetization of two Gaussians one can then derive:
( 0)
χ
χ −
=
∆
dL
dL
H LL
⇒ maximum of susceptibility: and width of the peak:
(K.Binder)
97
FSS of 1st order transition
Ising model on square lattice at kT/J = 2.1
98
The Potts model
, 1 1,
states 1,...,
σ σ σ
σ
δ δ=
=
= = − −∑ ∑i j i
i
i j nn i
q qE J HH
q = 2 corresponds to the Ising model. For q → 1 one obtains bond percolation due to the theorem of Kasteleyn and Fortuin (1969).
Potts (1952)
99
The Potts model
(a) columnar grain growth during zone annealing (b) coarsening of an eutectic microstructure (c) grain growth in a polycrystal (d) porosity in a sintered ceramics
Applications in surface science, biology, sociology, material science, QCD, etc
Material science:
100
Potts model The Potts model has a first order transition in T in 2d for q > 4 and in d > 2 for q > 2.
specific heat for q = 10 (2d)
101
Percolation
bond percolation on square lattice p is the probability to occupy a site. Neighboring occupied sites are „connected“ and belong to the same cluster.
103
Kasteleyn and Fortuin Consider Potts model on an arbitrary graph with bonds ν: Define on bond ν0 operators of Contraction C and Deletion D:
0
1 v v
vE J ε ε
= =
∑
if endpoints are in same statewith
if endpoints are in different states
104
Kasteleyn and Fortuin
( )v
v vJ
JE X
X X X v
Z e e eβ ε
β εβ−
−−∑
= = =∑ ∑ ∑∏
( ) ( )
1
1 1 1 : :
1 (1 )
v v v v
i j i j
J J J JJ
X X Xv v v v v v
J J JC D C C D C D
Z e e e e e
Z e Z Z e Z e pZ Z Zp
β ε β ε β ε β εβ
σ σ σ σ
β β β
− − − −−
≠ ≠ ≠= ≠
− − −
= = +
= + − = − + = + −
∑ ∑ ∑∏ ∏ ∏Consider bond ν1 with endpoints i and j .
ZC and ZD are the partition functions of the graphs contracted and deleted at ν1 .
1 β−≡ − Jp e
The partition function can be transformed:
105
Kasteleyn and Fortuin
1 1(1 )
v vC DZ pZ p Z= + −
1 21 2 1 2 1 2
2 2(1 ) (1 ) (1 )v vv v v v v v D DC C C D D CZ p Z p p Z p pZ p Z= + − + − + −
For bond ν1 we found:
Now do the same also with bond ν2 . ⇒
Now do it for all edges. Then the graph is reduced to a set of separated points corresponding to connected contracted (occupied) bonds (clusters). Each can be in q different states.
# c d(1 )Z q p p= −∑ of clusters
configurationsbond percolation
of⇒ c and d are the
#s of contracted and deleted bonds.
1 β−≡ − Jp e
106
The Potts model
, 1 1,
states 1,...,
σ σ σ
σ
δ δ=
=
= = − −∑ ∑i j i
i
i j nn i
q qE J HH
q = 2 corresponds to the Ising model. For q → 1 one obtains bond percolation due to the theorem of Kasteleyn and Fortuin (1969).
Potts (1952)
107
Kasteleyn and Fortuin
... ... (1 )b
p p= −∑ # of occupied bonds # of empty bonds
bond percolation configurations
# of clustersq b
Z q=
The partition function of the q-state Potts model is :
with
This is a fundamental relation between magnetic phase transitions and geometry (percolation).
1 β−≡ − Jp e
109
Coniglio-Klein Clusters Consider a unit of all connected sites that are in the same state and remove the bonds between them with probability:
The resulting cluster of bonds is called a Coniglio-Klein cluster.
1 β−≡ − Jp e
Bill Klein Antonio Coniglio
110
Cluster algorithms
Single flip is slow for T < Tc. Probability to flip a group of s sites simultaneously is:
( )21 0
sJse β− →
i.e. it is even much smaller.
111
Cluster algorithms
#0
( , ) (1 ) (1 )c d c dp C p p q p pσ = − −∑
bond percolationon graph without
C C
C
1 β−≡ − Jp e
( ) ( )1 1 2 2 2 1( , ) ( , ) ( , ) ( , ) ( , ) ( , )σ σ σ σ σ σ→ = →p C W C C p C W C C
Probability that cluster C is in state σ0:
is independent on σ0 . Detailed balance for a change σ1 → σ2 of cluster C:
1 2( , ) ( , )σ σ=p C p Cis easy to fulfill, because
112
Cluster algorithms
( ) 21 2
1 2
( , ) 1( , ) ( , )( , ) ( , ) 2
σσ σσ σ
→ = =+
p CW C Cp C p C
Glauber:
Metropolis:
( ) 21 2
1
( , )( , ) ( , ) min ,1 1( , )
σσ σσ
→ = =
p CW C Cp C
i.e. choose new state always with probability ½.
i.e. always choose new state.
113
Swendsen-Wang
• Occupy bond with probability p if states are equal, otherwise do leave empty.
• Identify the clusters with Hoshen-Kopelman algorithm.
• Flip each cluster with probability ½ for Ising or choose always a new state for q > 2.
R.H. Swendsen and J.-S. Wang (1987)
( 1 )Jp e β−≡ −
114
Swendsen- Wang Critical slowing down is substantially reduced.
z ≈ 0.3 in 2d z ≈ 0.55 in 3d
Link zum Applet
115
Wolff algorithm
• Choose a site randomly. • If neighboring site is in same state add it to
the cluster with probability p. • Repeat this until every site on the boundary
of the cluster has been checked exactly once. • Choose any new state for the cluster (with
probability one).
Ulli Wolff (1989)
( 1 )Jp e β−≡ −
116
General formalism
( ) ( )( , ) ( , ) ( , ) ( , ) ( , ) ( , )p X G W X G X G p X G W X G X G′ ′ ′→ = →
( )e.g.( , ) ( ) , ( ) E X
X G XZ p X G p X p X e β−= = =∑∑ ∑
( ) ( , )( , ) ( , )( , ) ( , )
′′→ =
′+p X GW X G X G
p X G p X G
D. Kandel, E. Domany and A. Brandt (1989)
( ) ( , )( , ) ( , ) min ,1( , )
p X GW X G X Gp X G
′ ′→ =
1( , ) ( , ) ( ) with ( , )
0p X G X G V G X G
= ∆ ∆ =
detailed balance
Glauber
Metropolis
algorithm simplifies when:
117
Improved estimators
( ) 0C i ii C
M σ σ∈
= − =∑
1 0 i jσ σ
=
if in the same clusterotherwise
i, j
( )22M Mχ β= − 2 22 2
,
1 1σ σ σ= =∑ ∑i j ii j cluster
MN N
From one configuration one can already get an average over many states because one can flip any subset of clusters. For example one gets for the magnetization of one cluster:
the correlation function:
and the susceptibility:
118
Vectorization
A vectorized loop is an assembly line.
I = 1,10000 A(I) = B(I) * (C(I) + D(I))
ideal case:
= multiple instruction – single data (MISD)
119
Vectorization
Problematic: • Conditional branchings like if-statements • Indirect addressing • Short loops Examples for big vector machines: SX-9 from NEC, VP2200 from Fujitsu, S-810 from Hitachi, Y-MP from Cray
120
Vectorization of MC
• Make update in inner loop, i.e. no loops inside this loop.
• Replace if (P(I) > z) s = -s by s = s * sign ( z – P (I)).
• Use a vectorized random number generator. • If a loop does not vectorize split it up in
several loops. • Make one dimensional indexing and use
helical boundary conditions.
121
Parallelization
SIMD ↔ MIMD
shared memory ↔ distributed memory
coarse grained ↔ fine grained
Exist many different architectures:
122
Parallelization of MC
Simplest parallelization is „farming“ where each processor executes the same program with different data (SIMD). Here each processor must get a different seed for the random number generator.
123
Domain decomposition
124
Domain decomposition
Dynamic load sharing
125
126
Parallelization of MC with domain decomposition
• MC on a regular lattice is well suited for parallelization because it is local.
• Put neighbors in different sublattices. • Use standard domain decomposition and
distribute using in MPI (block,block). • Use logical mask to extract one sublattice. • Use periodic shift (CSHIFT) to get neighbors
for periodic boundary conditions.
127
Parallelization of MC
Metropolis Monte Carlo for the Ising model on the square lattice using „CMFortran“
128
MPI for MC
MPI = message passing instructions shift automatically does message passing if value is on a different processor. size automatically refers to the size of the subsystem which is on one processor.
129
MPI for MC DO n=1,iterations
DO i=1,size DO j=1,size
old_spin = spin(i,j) new_spin = -old_spin
CC -------- Get neighboring spins.C shift is a function defined to handle theC periodic boundary conditions and passing ofC data between processors.C spin1 = shift(i-1,j) spin2 = shift(i+1,j) spin3 = shift(i,j-1) spin4 = shift(i,j+1)CC -------- Sum neighboring spins to get energy.C
spin_sum = spin1 + spin2 + spin3 + spin4 old_energy = old_spin * spin_sum new_energy = - old_energy energy_diff = new_energy - old_energy
CC -------- Metropolis accept/reject step.C
IF ( ( energy_diff.LE.0 ) .OR. & ( EXP(-beta*energy_diff).GT.random() ) ) THEN
spin(i,j) = new_spin ENDIF
ENDDO ENDDO
ENDDO
130
Efficiency of Parallelization Bottleneck is communication between processors.
Gene Amdahl
fraction of parallelized time to total time
131
Histogram methods Aim is to obtain functions at one temperature from a simulation at another temperature. Z.W. Salsburg, J.D. Jacobson, W. Fickett and W.W. Wood J. Chem Phys. 30, 65 (1959) (also Ferrenberg and Swendsen, 1989)
1( ) ( ) ( ) , ( )
( ) ( ) , ( )
T T TE ET
EkT
T
Q T Q E p E Z p EZ
p E g E e g E−
= =
= =
∑ ∑
density of states
132
Histogram methods
*
*
* 1( ) ( ) ( )T
ET
Q T Q E p EZ
= ∑
**
* ( ) ( ) ( )E EE
kTkTkTTT
p E g E e p E e − +− = =
*
*,( )
− + ≡
E EkTkT
T Tf E e
*
*
,*
,
( ) ( ) ( )( )
( ) ( )=∑∑
T T TE
T T TE
Q E p E f EQ T
p E f E
We want to calculate:
Defining:
we obtain:
133
Problem of sampling
< >=∑( ) ( ) ( )TE
Q T Q E p E
The distribution of energy E around the average < E >T gets sharper with increasing size.
134
Broad histogram method The problem of the method before is that the values of Q(E) were sampled close to the maximum of pT(E) which for large systems is very peaked. If T and T * are not too close the overlap between the distributions is very small so that very few configurations are sampled around the maximum of T *. Consequently one has very bad statistics. Solution → Broad histogram method
(de Oliveira, Penna, HJH, 1996)
135
Broad histogram method
Make Markov process in energy space. Be Nup the number of all processes that increase the energy: E → E + ΔE and Ndown the number of processes that decrease the energy: E → E - ΔE. Then the equivalent condition to detailed balance to reach a homogeneous steady state would be:
( ) ( ) ( ) ( )down upg E E N E E g E N E+ ∆ + ∆ =
136
Broad histogram method Metropolis:
Choose a new configuration for instance by flipping randomly a spin. If E → E – ΔE then accept , if E → E + ΔE then accept with probability: ( )
( )down
up
N E EN E
+ ∆
Check for each site of a configuration if a change of state would increase or decrease the energy ⇒ Nup and Ndown.
137
Broad histogram method
( ) ( ) ( ) ( )down upg E E N E E g E N E+ ∆ + ∆ =
log ( ) log ( ) log ( ) log ( )
( )log ( ) 1 log( )
up down
up
down
g E E g E N E N E EN Eg E
E E N E E
+ ∆ − = − + ∆
∂⇒ =
∂ ∆ + ∆
Take logarithm, divide by ΔE and consider small ΔE:
138
Broad histogram method
Ising model on square lattice
( )log ( ) 1 log( )
∂=
∂ ∆ + ∆up
down
N Eg EE E N E Euse:
⇒ g(E )
139
Broad histogram method
Ising model on square lattice
140
Broad histogram method Choose a site randomly and change state if energy is decreased and if energy would be increased change with probability Ndown/Nup. At each step one accumulates the values for Nup(E ), Ndown(E ) and Q(E ). Finally calculate:
( ) ( )( )
( )
EkT
EEkT
E
Q E g E eQ T
g E e
−
−= ∑
∑
141
Broad histogram method Ising model on square lattice
32 × 32 lattice crosses = BHMC circles = usual histogram continuous line = exact
142
Broad histogram method
( ) ( )
0.701
c
c
EkT
T
c
p E g E e
T
−
=
≈
BHM has been particularly useful for first order transitions.
(F.Wang and Landau, 2001)
q = 10 Potts model on a square lattice
( )EcTp
143
Histogram methods
• Multiple histogram method • Multcanonical MC • Flat histograms • Umbrella sampling • ....
Other variants:
144
Flat Histogram
• Start with g(E) = 1 and set f ≡ e. • Make MC update with p(E) = 1/g(E). • If attempt succesful at E: g(E) = f · g(E). • Obtain a histogram of the energies H(E). • If H(E) flat enough, then . • Stop when .
„Flatness“ can be measured as the ratio of the minimum to maximum value, „enough“ could be multiple of f .
f = f81 10−≈ +f
Jian-Sheng Wang (1999)
145
Umbrella sampling Torrie and Valleau (1977)
In order to overcome energy barriers Multiply transition probability with function which is large at the barrier and later remove this function again at the averaging.
( ) ( )( )
( )( )
/ ,
1/
E CkT
wE C
wkT
C
A ww C ep C A
ww C e
−
−= =
∑
Glenn Torrie John Valleau
146
Other Ising-like models • Antiferromagnetic models:
• Ising spin glass:
• ANNNI model:
• Metamagnets:
, :ij i j
i j nnJ σ σ= ∑H
1 2 2, : :
i j i ii j nn i nnn
J Jσ σ σ σ += − +∑ ∑H
, :( 1)i
i j ii j nn i
J Hσ σ σ= + −∑ ∑H
1 2, : , :
i j i j ii j nn i j nnn i
J J Hσ σ σ σ σ= − −∑ ∑ ∑H
staggered field
random interaction
frustration in x-direction
incomensurate phases with „Lifshitz point“
tricritical point
147
The ANNNI model
148
The O(n) model
( ),
with =1, ,x x yXY i j x i i i i i
i j nn iJ S S H S S S S S
=
= − − =∑ ∑
H
( )1 11
,with 1, ,..., n
n vector i j i i i i ii j nn i
J S S H S S S S S−=
= − − = =∑ ∑
H
( ),
with =1, , ,x x y zHeisenberg i j x i i i i i i
i j nn iJ S S H S S S S S S
=
= − − =∑ ∑
H
O(n) model:
n = 1 is the Ising model, n = 2 is the XY-model:
n = 3 is the Heisenberg model:
n = ∞ is the „spherical model“.
149
Phase transitions Mermin-Wagner theorem (1966): In two dimensions a system with continuous degrees of freedom and short range interactions has no phase transition which involves long range order.
Heisenberg model in three dimensions:
150
Continuous degrees of freedom
Phase space is not discrete anymore. Monte Carlo move: Choose for site i a new spin:
with small random , i i iS S S S S S′ = + ∆ ∆ ∆ ⊥
Use cluster methods by making a projection on a plane. Then flipping means a reflection with respect to this plane.
151
Continuous degrees of freedom
Broad histogram method for the XY model in 3d.
152
Real space Renormalization
At critical point we have scale invariance. Renormalize system by changing scale by l.
LL l=
Niemeijer and van Leeuwen (1976)
System size changes by l variables (spins) must be redefined, new effective interactions ⇒ new Hamitonian but free energy density stays constant .
153
Real space Renormalization
dcF H l F H T Tε ε ε− =with
( , )= ( , ) -
,
,
,
T H
T H
T H
y yd
y y
y y
F H l F l l H
F H F l l H
l H l H
ε ε
ε ε
ε ε
⇒
⇒
( , )= ( )
( , )= ( )
= =
homogeneous scaling law close to critical point:
renormalized The free energy is extensive and therefore to keep its density constant it scales as:
154
Real space Renormalization νξ ε −
ν ξε ξ− =
l
1
1Tyl lν
νε ε εε= = =
-l
1ν
=Ty
correlation length:
rescaling with l:
⇒ ⇒
155
Real space Renormalization
6 2LL L l= ⇒ = =
l = 3
Example: majority rule
1
sign
i
iii cell
σ
σ σ∈
= ±
=
∑
156
Real space Renormalization
Example: renormalization by decimation
2=l
The renormalized Hamiltonian also has next-nearest neighbor interactions.
157
Decimation of 1D Ising Model
)
158
Decimation of 1D Ising Model
159
Decimation of 1D Ising Model
160
2=l
Next-nearest neighbor interaction induced by the decimation of a site.
Proliferation of interactions
In general the renormalized Hamiltonian has longer range interactions than the original one. A simple example is decimation on the square lattice:
161
Real space Renormalization
1( )
M
i ki k c
K O Oα
α α αα
σ σ += ∈
= =∑ ∑∏withH, :
H σ σ= ∑ i ji j nn
K
( ( )) ( )
( , ) ( , ) 1Ge P e Pσ σ
σ σ
σ σ σ σ+ = =∑ ∑with
H H
( ( )) ( )
GZ c Z e eσ σ
σ σ
+= ⇔ =∑ ∑
H H
general Hamiltonian ( K = - J / k T ) example:
definition of renormalized Hamiltonian:
in order to fulfill the conservation of free energy density:
162
Real space Renormalization
1( )
M
i ki k c
K O Oα
α α αα
σ σ += ∈
= =∑ ∑∏with
H
* * *1( ,..., )MK K K Kα α=
ααβ
β
∂=∂
KTK
* *( )K K T K Kα α αβ β ββ
− = −∑
*K
1( ,..., ) , 1,...,MK K K Mα α =
renormalized Hamiltonian
renormalization:
critical point is fixed point:
Jacobi matrix linearization of transformation at K*:
163
Real space Renormalization Flow diagram of renormalization
fixed point (K1*, K2
*) K1
K2
164
Real space Renormalization 1 1 M,..., ,..., Mλ λ φ φ and
1αλ >
* αβ KT
α α αφ λ φ=
1 ln ln
ε ε λ νλ
= ⇒ = ⇒ = =
T Ty yT
T T
ll ly
eigenvalues and eigenvectors of
relevant eigenvalue ⇒ fixed point unstable
calculate critical exponents through:
scaling field
165
MCRG
β ββ
β ββ
ασα
ασ
∑∂
= =∑ ∂∑∑
K O
K O
O e FOKe
= Monte Carlo Renormalization Group Ma (1977), Swendsen (1979)
ln=F Z free energy
measure:
166
MCRG
ααβ α β α β
β
ααβ α β α β
β
χ
χ
∂≡ = −
∂
∂≡ = −
∂
OO O O O
K
OO O O O
K
and measure the response functions:
167
MCRG
, αααβ αβ
β β
χ χ∂∂
≡ ≡∂ ∂
OOK K
( ) ( )( ) ( 1)
n nn n
O OKT
K K Kα αγ
αβ γβ αγγ γβ β γ
χ χ +∂ ∂∂
= = =∂ ∂ ∂∑ ∑
using the chain rule we obtain for the n-th iteration:
⇒ we obtain Tγβ from the correlation functions by solving a set of M coupled linear equations.
168
Strategy in MCRG
renormalization renormalization
169
Example for MCRG
ν of 4-state Potts model in two dimensions
170
Errors in MCRG
• Statistical • Truncation of Hamiltonian • Finite number of iterations • Finite size • Imprecision in K*
