LECTURE 1 WORM ALGORITHM FOR CLASSICAL STATISTICAL MODELS I
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1 LECTURE 1 WORM ALGORITHM FOR CLASSICAL STATISTICAL MODELS I
2 LECTURE 1 WORM ALGORITHM FOR CLASSICAL STATISTICAL MODELS I General idea of extended configuration space; Illustration for closed loops.
3 LECTURE 1 WORM ALGORITHM FOR CLASSICAL STATISTICAL MODELS I General idea of extended configuration space; Illustration for closed loops. Ising model: mapping to closed loop representation + disconnected loop spin-spin correlation function
4 LECTURE 1 WORM ALGORITHM FOR CLASSICAL STATISTICAL MODELS I General idea of extended configuration space; Illustration for closed loops. Ising model: mapping to closed loop representation + disconnected loop spin-spin correlation function Worm Algorithm configuration space. Complete algorithm and code for Ising model.
5 LECTURE 1 WORM ALGORITHM FOR CLASSICAL STATISTICAL MODELS I General idea of extended configuration space; Illustration for closed loops. Ising model: mapping to closed loop representation + disconnected loop spin-spin correlation function Worm Algorithm configuration space. Complete algorithm and code for Ising model. Finite-size scaling
6 Why bother with mappings and algorithms?
7 Why bother with mappings and algorithms?
8 Why bother with mappings and algorithms? Efficiency
9 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams
10 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams PhD while still young
11 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams New quantities, more theoretical tools to address physics PhD while still young
12 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams New quantities, more theoretical tools to address physics Grand canonical ensemble Off-diagonal correlations Single-particle and/or condensate wave functions Winding numbers and PhD while still young
13 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams New quantities, more theoretical tools to address physics Grand canonical ensemble Off-diagonal correlations Single-particle and/or condensate wave functions Winding numbers and PhD while still young New physics
14 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams New quantities, more theoretical tools to address physics Grand canonical ensemble Off-diagonal correlations Single-particle and/or condensate wave functions Winding numbers and PhD while still young New physics
15 Why bother with mappings and algorithms? Efficiency Better accuracy Large system size More complex systems Finite-size scaling Critical phenomena Phase diagrams New quantities, more theoretical tools to address physics Grand canonical ensemble Off-diagonal correlations Single-particle and/or condensate wave functions Winding numbers and PhD while still young New physics
16 Overall Monte carlo scheme - configuration space (essentially anything you can explain in a lifetime) - each cnf. has a weight factor associated with it - quantity of interest defined on this cnf. space
17 Overall Monte carlo scheme - configuration space (essentially anything you can explain in a lifetime) - each cnf. has a weight factor associated with it - quantity of interest defined on this cnf. space
18 Overall Monte carlo scheme - configuration space (essentially anything you can explain in a lifetime) - each cnf. has a weight factor associated with it - quantity of interest defined on this cnf. space MC
19 Overall Monte carlo scheme - configuration space (essentially anything you can explain in a lifetime) cnf = state of Ising spins - each cnf. has a weight factor associated with it - quantity of interest defined on this cnf. space MC
20 Overall Monte carlo scheme - configuration space (essentially anything you can explain in a lifetime) cnf = state of Ising spins - each cnf. has a weight factor associated with it - quantity of interest defined on this cnf. space MC
21 Overall Monte carlo scheme - configuration space (essentially anything you can explain in a lifetime) - each cnf. has a weight factor associated with it - quantity of interest defined on this cnf. space MC
22 Overall Monte carlo scheme - configuration space (essentially anything you can explain in a lifetime) anything you can draw without loose ends - each cnf. has a weight factor associated with it - quantity of interest defined on this cnf. space MC
23 Overall Monte carlo scheme - configuration space (essentially anything you can explain in a lifetime) anything you can draw without loose ends - each cnf. has a weight factor associated with it you tell me - quantity of interest defined on this cnf. space MC
24 conventional sampling scheme: local shape change Add/delete small loops No sampling of topological classes (non-ergodic) can not evolve to Critical slowing down (large loops are related to critical modes) dynamical critical exponent in many cases
25 conventional sampling scheme: local shape change Add/delete small loops No sampling of topological classes (non-ergodic) can not evolve to Critical slowing down (large loops are related to critical modes) dynamical critical exponent in many cases
26 conventional sampling scheme: local shape change Add/delete small loops No sampling of topological classes (non-ergodic) can not evolve to Critical slowing down (large loops are related to critical modes) dynamical critical exponent in many cases
27 Global updates:
28 Global updates: Why not?
29 Global updates: Why not?
30 Global updates: Why not?
31 Global updates: Why not? easy to match, if simple
32 This is why Typical, incomprehensible
33 This is why Typical, incomprehensible
34 This is why Typical, incomprehensible
35 This is why Typical, incomprehensible simple (human generated)
36 This is why Typical, incomprehensible Complicated, impossible to plan for ahead of time simple (human generated)
37 This is why Typical, incomprehensible Complicated, impossible to plan for ahead of time simple (human generated) Not optimized => exponentially small!
38 This is why Typical, incomprehensible Complicated, impossible to plan for ahead of time simple (human generated) Not optimized => exponentially small!
39 This is why Typical, incomprehensible Complicated, impossible to plan for ahead of time simple (human generated) Not optimized => exponentially small!
40 This is why Typical, incomprehensible not nice and round Complicated, impossible to plan for ahead of time simple (human generated) Not optimized => exponentially small!
41 Worm algorithm idea draw and erase: + or keep drawing
42 Worm algorithm idea draw and erase: Masha Ira or + keep drawing
43 Worm algorithm idea draw and erase: Masha Ira or Ira Masha + keep drawing
44 Worm algorithm idea draw and erase: Masha Ira or Ira Masha + keep drawing Ira
45 Worm algorithm idea draw and erase: Ira Masha Ira or Ira Masha + keep drawing
46 Worm algorithm idea draw and erase: Ira Masha Ira or Ira Masha + keep drawing Topological classes are sampled efficiently (whatever you can draw!)
47 Worm algorithm idea draw and erase: Ira Masha Ira or Ira Masha + keep drawing Topological classes are sampled efficiently (whatever you can draw!) No critical slowing down in most cases Disconnected loops relate to important physics (correlation functions) and are not merely an algorithm trick!
48 Worm algorithm idea Extended configuration space = closed loop + open ( incomplete ) loop A Physical cnf. space Extended cnf. space
49 Worm algorithm idea Extended configuration space = closed loop + open ( incomplete ) loop A Physical cnf. space Extended cnf. space
50 Worm algorithm idea Extended configuration space = closed loop + open ( incomplete ) loop A Physical cnf. space Extended cnf. space
51 Worm algorithm idea Extended configuration space = closed loop + open ( incomplete ) loop A Physical cnf. space Extended cnf. space
52 Worm algorithm idea Extended configuration space = closed loop + open ( incomplete ) loop A B Physical cnf. space Extended cnf. space
53 Worm algorithm idea Extended configuration space = closed loop + open ( incomplete ) loop Conventional schemes: Cycle (cnf. update measure) A B Physical cnf. space Extended cnf. space
54 Worm algorithm idea Extended configuration space = closed loop + open ( incomplete ) loop A B Physical cnf. space Extended cnf. space
55 Worm algorithm idea Extended configuration space = closed loop + open ( incomplete ) loop Extended schemes: Cycle (cnf. update if (physical) then measure) A B Physical cnf. space Extended cnf. space
56 High-T expansion for the Ising model
57 High-T expansion for the Ising model where
58 High-T expansion for the Ising model where
59 High-T expansion for the Ising model where number of lines (thickness); enter/exit rule
60 Spin-spin correlation function: same as before I 1 4 M 3 4 2
61 Spin-spin correlation function: same as before I 1 4 M 3 4 2
62 Spin-spin correlation function: same as before I 1 4 M 3 4 2
63 Spin-spin correlation function: same as before I 1 Worm algorithm cnf. space = Same as for generalized partition 4 M 3 4 2
64 Spin-spin correlation function: same as before I 1 Worm algorithm cnf. space = Same as for generalized partition 4 M Can be used to control relative time spent in Z vs. G
65 Spin-spin correlation function: same as before I 1 Worm algorithm cnf. space = Same as for generalized partition 4 M 3 4 2
66 Getting more practical: since
67 Getting more practical: since I no-overlaps M
68 Getting more practical: since
69 Getting more practical: since I Complete algorithm : M - If, select a new site for them at random - select direction to move, let it be bond - If accept with prob.
70
71 I=M
72 I M
73 I M
74 I M
75 I
76 I
77 I
78 I M
79 I M
80 I M Correlation function: Magnetization fluctuations: Energy: either or
81 I M
82 Demo:
83 Demo: Link the lattice with periodic boundary conditions site2 dir=3 dir=2 site dir=1 site1 dir=1 dir=4
84 Demo: Link the lattice with periodic boundary conditions site2 dir=3 dir=2 site dir=1 site1 dir=1 dir=4 create an array nn(site,dir) which returns the nearest neighbor of site = 1, 2, N in the direction dir = 1, 2,, d, d+1,, d+d for example: site1= nn(site,1) and site= nn(site1,1+dim) Periodic boundary conditions mean that: site2= nn(site1,1) and site1= nn(site2,1+dim)
85 integer, parameter :: L=10, dim=2, N=L**dim, dd=2*dim double precision, parameter :: JT=0.3d0 double precision :: ratio integer :: nn(n,dd), back(dd) integer :: site,site1 integer :: ivec(dim), i ratio=tanh(jt) DO site=1,n site1=site-1 DO i=dim,1,-1 ivec(i)=site1 / L**(i-1) site1=site1-ivec(i)*l**(i-1) ENDDO ivec(2) 1 site ivec(1) ENDDO DO i=1,dim site1=site+l**(i-1) IF(ivec(i)==L-1) site1=site1-l**i nn(site,i)=site1 nn(site1,i+dim)=site ENDDO DO k=1, dd IF(k>dim) back(k)=k-dim IF(k<dim+1) back(k)=k+dim ENDDO
86 integer, parameter :: L=10, dim=2, N=L**dim, dd=2*dim double precision, parameter :: JT=0.3d0 double precision :: ratio integer :: nn(n,dd), back(dd) integer :: site,site1 integer :: ivec(dim), i ratio=tanh(jt) DO site=1,n site1=site-1 DO i=dim,1,-1 ivec(i)=site1 / L**(i-1) site1=site1-ivec(i)*l**(i-1) ENDDO ivec(2) 1 site ivec(1) ENDDO DO i=1,dim site1=site+l**(i-1) IF(ivec(i)==L-1) site1=site1-l**i nn(site,i)=site1 nn(site1,i+dim)=site ENDDO A B=nn(A,k) DO k=1, dd IF(k>dim) back(k)=k-dim IF(k<dim+1) back(k)=k+dim ENDDO
87 integer, parameter :: L=10, dim=2, N=L**dim, dd=2*dim double precision, parameter :: JT=0.3d0 double precision :: ratio integer :: nn(n,dd), back(dd) integer :: site,site1 integer :: ivec(dim), i ratio=tanh(jt) DO site=1,n site1=site-1 DO i=dim,1,-1 ivec(i)=site1 / L**(i-1) site1=site1-ivec(i)*l**(i-1) ENDDO ivec(2) 1 site ivec(1) ENDDO DO i=1,dim site1=site+l**(i-1) IF(ivec(i)==L-1) site1=site1-l**i nn(site,i)=site1 nn(site1,i+dim)=site ENDDO A B=nn(A,k) DO k=1, dd IF(k>dim) back(k)=k-dim IF(k<dim+1) back(k)=k+dim ENDDO A=nn(B,back(k))
88 Now, we need to initialize the configuration and counters for Z,G,M2,E
89 Now, we need to initialize the configuration and counters for Z,G,M2,E Configuration is kept in the array bond(site,dd), the sum of all bond numbers is NB, do not forget Ira and Masha and the distance between them IM(dim) And, of course, we need a random number generator (I will not mention it below).
90 Now, we need to initialize the configuration and counters for Z,G,M2,E Configuration is kept in the array bond(site,dd), the sum of all bond numbers is NB, do not forget Ira and Masha and the distance between them IM(dim) And, of course, we need a random number generator (I will not mention it below). integer :: bond(n,dd), NB, Ira, Masha, IM(dim) double precision :: Z, G(0:L-1,0:L-1), M2, E Collect all definitions in one place Z=1.d-15 M2=0.d0 E=0.d0 G=0.d0 NB=0 bond=0 Ira=1 Masha=1 IM=0
91 Now, we need to initialize the configuration and counters for Z,G,M2,E Configuration is kept in the array bond(site,dd), the sum of all bond numbers is NB, do not forget Ira and Masha and the distance between them IM(dim) And, of course, we need a random number generator (I will not mention it below). integer :: bond(n,dd), NB, Ira, Masha, IM(dim) double precision :: Z, G(0:L-1,0:L-1), M2, E Z=1.d-15 M2=0.d0 E=0.d0 G=0.d0 NB=0 Collect all definitions in one place partition function bond=0 Ira=1 Masha=1 IM=0
92 Now, we need to initialize the configuration and counters for Z,G,M2,E Configuration is kept in the array bond(site,dd), the sum of all bond numbers is NB, do not forget Ira and Masha and the distance between them IM(dim) And, of course, we need a random number generator (I will not mention it below). integer :: bond(n,dd), NB, Ira, Masha, IM(dim) double precision :: Z, G(0:L-1,0:L-1), M2, E Z=1.d-15 M2=0.d0 E=0.d0 G=0.d0 NB=0 Collect all definitions in one place partition function magnetization squared bond=0 Ira=1 Masha=1 IM=0
93 Now, we need to initialize the configuration and counters for Z,G,M2,E Configuration is kept in the array bond(site,dd), the sum of all bond numbers is NB, do not forget Ira and Masha and the distance between them IM(dim) And, of course, we need a random number generator (I will not mention it below). integer :: bond(n,dd), NB, Ira, Masha, IM(dim) double precision :: Z, G(0:L-1,0:L-1), M2, E Z=1.d-15 M2=0.d0 E=0.d0 G=0.d0 NB=0 Collect all definitions in one place partition function magnetization squared energy bond=0 Ira=1 Masha=1 IM=0
94 Now, we need to initialize the configuration and counters for Z,G,M2,E Configuration is kept in the array bond(site,dd), the sum of all bond numbers is NB, do not forget Ira and Masha and the distance between them IM(dim) And, of course, we need a random number generator (I will not mention it below). integer :: bond(n,dd), NB, Ira, Masha, IM(dim) double precision :: Z, G(0:L-1,0:L-1), M2, E Z=1.d-15 M2=0.d0 E=0.d0 G=0.d0 NB=0 Collect all definitions in one place partition function magnetization squared energy correlation function bond=0 Ira=1 Masha=1 IM=0
95 Now, we need to initialize the configuration and counters for Z,G,M2,E Configuration is kept in the array bond(site,dd), the sum of all bond numbers is NB, do not forget Ira and Masha and the distance between them IM(dim) And, of course, we need a random number generator (I will not mention it below). integer :: bond(n,dd), NB, Ira, Masha, IM(dim) double precision :: Z, G(0:L-1,0:L-1), M2, E Z=1.d-15 M2=0.d0 E=0.d0 G=0.d0 NB=0 Collect all definitions in one place partition function magnetization squared energy correlation function number of bond lines bond=0 Ira=1 Masha=1 IM=0
96 Now, we need to initialize the configuration and counters for Z,G,M2,E Configuration is kept in the array bond(site,dd), the sum of all bond numbers is NB, do not forget Ira and Masha and the distance between them IM(dim) And, of course, we need a random number generator (I will not mention it below). integer :: bond(n,dd), NB, Ira, Masha, IM(dim) double precision :: Z, G(0:L-1,0:L-1), M2, E Z=1.d-15 M2=0.d0 E=0.d0 G=0.d0 NB=0 Collect all definitions in one place partition function magnetization squared energy correlation function number of bond lines bond=0 Ira=1 Masha=1 IM=0 This is it for preparations! We are all set to do the simulation. Just one update!
97 SUBROUTINE UPDATE integer :: k, site1, N_old IF(Ira==Masha) THEN Ira=rndm()*N+1 ; Masha=Ira ; ENDIF k=rndm()*dd+1 ; site1=nn(ira,k) N_old=bond(Ira,k) IF(N_old==0.AND. rndm()>ratio) RETURN bond(ira,k)=mod( N_old+1, 2) bond(site1,back(k))=bond(ira,k) The update itself If in Z configuration move Ira and Masha elsewhere at random select direction at random accept/reject configuration change NB=NB-N_old+bond(Ira,k) IF(k>dim) THEN ; IM(back(k))=MOD(IM(back(k))-1,L) ELSE ; IM(k)=MOD(IM(k)+1,L) ; ENDIF Ira=site1 END SUBROUTINE UPDATE
98 SUBROUTINE UPDATE integer :: k, site1, N_old IF(Ira==Masha) THEN Ira=rndm()*N+1 ; Masha=Ira ; ENDIF k=rndm()*dd+1 ; site1=nn(ira,k) N_old=bond(Ira,k) IF(N_old==0.AND. rndm()>ratio) RETURN bond(ira,k)=mod( N_old+1, 2) bond(site1,back(k))=bond(ira,k) The update itself If in Z configuration move Ira and Masha elsewhere at random select direction at random accept/reject configuration change NB=NB-N_old+bond(Ira,k) IF(k>dim) THEN ; IM(back(k))=MOD(IM(back(k))-1,L) ELSE ; IM(k)=MOD(IM(k)+1,L) ; ENDIF keep useful counters available Ira=site1 END SUBROUTINE UPDATE
99 SUBROUTINE UPDATE integer :: k, site1, N_old IF(Ira==Masha) THEN Ira=rndm()*N+1 ; Masha=Ira ; ENDIF k=rndm()*dd+1 ; site1=nn(ira,k) N_old=bond(Ira,k) IF(N_old==0.AND. rndm()>ratio) RETURN bond(ira,k)=mod( N_old+1, 2) bond(site1,back(k))=bond(ira,k) NB=NB-N_old+bond(Ira,k) IF(k>dim) THEN ; IM(back(k))=MOD(IM(back(k))-1,L) ELSE ; IM(k)=MOD(IM(k)+1,L) ; ENDIF The update itself If in Z configuration move Ira and Masha elsewhere at random select direction at random accept/reject configuration change total bond number keep useful counters available Ira=site1 END SUBROUTINE UPDATE
100 SUBROUTINE UPDATE integer :: k, site1, N_old IF(Ira==Masha) THEN Ira=rndm()*N+1 ; Masha=Ira ; ENDIF k=rndm()*dd+1 ; site1=nn(ira,k) N_old=bond(Ira,k) IF(N_old==0.AND. rndm()>ratio) RETURN bond(ira,k)=mod( N_old+1, 2) bond(site1,back(k))=bond(ira,k) NB=NB-N_old+bond(Ira,k) IF(k>dim) THEN ; IM(back(k))=MOD(IM(back(k))-1,L) ELSE ; IM(k)=MOD(IM(k)+1,L) ; ENDIF The update itself If in Z configuration move Ira and Masha elsewhere at random select direction at random accept/reject configuration change total bond number keep useful counters available Distance between Ira and Masha Ira=site1 END SUBROUTINE UPDATE
101 SUBROUTINE UPDATE integer :: k, site1, N_old IF(Ira==Masha) THEN Ira=rndm()*N+1 ; Masha=Ira ; ENDIF k=rndm()*dd+1 ; site1=nn(ira,k) N_old=bond(Ira,k) IF(N_old==0.AND. rndm()>ratio) RETURN bond(ira,k)=mod( N_old+1, 2) bond(site1,back(k))=bond(ira,k) NB=NB-N_old+bond(Ira,k) IF(k>dim) THEN ; IM(back(k))=MOD(IM(back(k))-1,L) ELSE ; IM(k)=MOD(IM(k)+1,L) ; ENDIF The update itself If in Z configuration move Ira and Masha elsewhere at random select direction at random accept/reject configuration change total bond number keep useful counters available Distance between Ira and Masha Ira=site1 END SUBROUTINE UPDATE Simpler/shorter then single spin flip, but more efficient then cluster algorithms!
102 Main Monte Carlo cycle integer :: i Z DO i=1,10000 call UPDATE ENDDO DO i=1, call UPDATE IF(Ira==Masha) THEN Z=Z+1.d0 E=E+NB ENDIF G( IM(1), IM(2) )=G( IM(1), IM(2) )+1.d0 M2=M2+1.d0 IF(MOD(i, )==0) print*, E/Z, M2/ ENDDO thermolize! collect statistics Z configuration G configuration; G(0)=Z print something from time to time
103 Solving the critical slowing down problem: Question: What are the signatures of the phase transition (critical modes)? spin representation large domains of similarly oriented spins of linear size ~ L single-spin flips are not efficient in updating them! loop representation I M large loops of linear size ~ L (long-range correlations between spins = large distance between I and M) draw large loops!
104 Finite size scaling
105 Finite size scaling FACT: simulations of systems of linear size L=100 pin-point critical points with accuracy of SEVEN decimal places! but
106 Finite size scaling FACT: simulations of systems of linear size L=100 pin-point critical points with accuracy of SEVEN decimal places! but Consider some scale invariant property, e.g.
107 Finite size scaling FACT: simulations of systems of linear size L=100 pin-point critical points with accuracy of SEVEN decimal places! but Consider some scale invariant property, e.g.
108 Finite size scaling FACT: simulations of systems of linear size L=100 pin-point critical points with accuracy of SEVEN decimal places! but Consider some scale invariant property, e.g.
109 Finite size scaling FACT: simulations of systems of linear size L=100 pin-point critical points with accuracy of SEVEN decimal places! but Consider some scale invariant property, e.g. Plot curves
110 Finite size scaling FACT: simulations of systems of linear size L=100 pin-point critical points with accuracy of SEVEN decimal places! but Consider some scale invariant property, e.g. Plot curves Read the critical point from the intersection
111
112
113
114 If it was just then intersection at would be perfect.
115 If it was just then intersection at would be perfect. Corrections to scaling with reasonably large exponent
116 If it was just then intersection at would be perfect. Corrections to scaling with reasonably large exponent Analyticity at the origin:
117 If it was just then intersection at would be perfect. Corrections to scaling with reasonably large exponent Analyticity at the origin: Intersection of two curves defines :
118 If it was just then intersection at would be perfect. Corrections to scaling with reasonably large exponent Analyticity at the origin: Intersection of two curves defines :
119 If it was just then intersection at would be perfect. Corrections to scaling with reasonably large exponent Analyticity at the origin: Intersection of two curves defines :
120 If it was just then intersection at would be perfect. Corrections to scaling with reasonably large exponent Analyticity at the origin: Intersection of two curves defines :
121 If it was just then intersection at would be perfect. Corrections to scaling with reasonably large exponent Analyticity at the origin: Intersection of two curves defines : relatively large power
122
123 Data collapse
124 Data collapse
125 Data collapse attempt joint fitting of all curves with unknown
126 Data collapse attempt joint fitting of all curves with unknown
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