VSOP19, Quy Nhon 3-18/08/2013. Ngo Van Thanh, Institute of Physics, Hanoi, Vietnam.

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1 VSOP19, Quy Nhon 3-18/08/2013 Ngo Van Thanh, Institute of Physics, Hanoi, Vietnam.

2 Part II. Monte Carlo Simulation Methods II.1. The spin models II.2. Boundary conditions II.3. Simple sampling Monte Carlo methods II.4. Importance sampling Monte Carlo methods

A Guide to Monte Carlo Simulations in Statistical Physics D. Landau and K. Binder, (Cambridge University Press, 2009). Monte Carlo Simulation in Statistical Physics: An Introduction K. Binder and D.

3 A Guide to Monte Carlo Simulations in Statistical Physics D. Landau and K. Binder, (Cambridge University Press, 2009). Monte Carlo Simulation in Statistical Physics: An Introduction K. Binder and D. W. Heermann (Springer-Verlag Berlin Heidelberg, 2010). Understanding Molecular Simulation : From Algorithms to Applications D. Frenkel, (Academic Press, 2002). Frustrated Spin Systems H. T. Diep, 2 nd Ed. (World Scientific, 2013). Lecture notes PDF files : Example code :

4 II.1. The spin models Introduction Spin model = spin kind + lattice structure (with including the dimension) Spin kinds : Ising XY Heisenberg Lattice structure : Simple cubic (SC) Body center cubic (BCC) Face center cubic (FCC) Stacked triangular (hexagonal) Dimensions : 2D, 3D or films

5 n-vector model The hamiltonian (2.1) n : number of conponent n = 0 : The Self-Avoiding Walks n = 1 : Ising model n = 2 : XY model n = 3 : Heisenberg model exchange interaction J The sum is taken over nearest neighbours spin pair : Ferromagnetic interaction : Antiferromagnetic interaction : Spin glass next nearest neighbours spin pair

6 Ising spin model The hamiltonian (2.2) the probability that the system is in state s (2.3) For the ferromagnetic Ising model on 2D simple cubic lattice The energy (2.4)

7 XY spin model This is the special case of the n-vector model, for n = 2 The hamiltonian (2.5) Using polar coordinate, with and (2.6) the system in a external magnetic field (2.7)

8 Heisenberg spin model This is the special case of the n-vector model, for n = 3 The hamiltonian with (2.8) The equation of motion in the continuum limit The Potts model The standard Potts model, spin (2.9) (2.10) : is the Kronecker delta (2.11)

9 II.2. Boundary conditions Periodic boundary condition (PBC)

10 film thickness Free boundary condition Using for a thin-film geometry Applying the free boundary condition on z-direction only Use PBC on (x, y) plane

11 II.3. Simple sampling Monte Carlo methods Comparisons of methods for numerical integration of given functions: Simple methods Consider a definite integral (2.12) draw a box extending from a to b and from 0 to y 0 Using random numbers drawn from a uniform distribution drop N points randomly into the box count the number N 0 which fall below f(x) An estimate for the integral y y 0 f(x) (2.13) 0 a b x

12 Crude method choose N values of x randomly, then calculate f(x) (2.14) Intelligent methods : control variate method (2.15) where selects a integrable function

13 Simulation of radioactive decay : Consider sample of N nuclei which decay at rate the rate of decay (2.16) the nuclei is chosen randomly The number of undecayed nuclei (2.17) N 0 : the initial number of nuclei, related to the half-life of the system Simulation Dividing the time into discrete intervals during the time interval, test for decay of each undecayed nucleus Determine the number of undecayed nuclei Increasing the time step, then repeat the process

14 Program structure N = N 0 ; NP = N 0 LOOP from (t = dt) to T max, step dt LOOP from 1 to NP! Loop over each remaining parent nucleus R = uniform_randomnumber()! 0 R 1 IF ( R < λ dt ) THEN N = N-1 END IF END LOOP over nuclei NP = N Record t, N END LOOP over time

15 Source : radioactive_decay.f90 PROGRAM RADIOACTIVE_DECAY IMPLICIT NONE REAL :: t,tmax,dt,r,lambda INTEGER :: n,n0,np,nt,i! CALL RANDOM_SEED() n0 = 1000; tmax = 50.0 dt = 0.1; lambda = 0.1 n = n0; t = 0.0 DO WHILE (t <= tmax.and. n > 0) t = t + dt np = n DO i = 1,np CALL RANDOM_NUMBER(r) IF (r < lambda*dt) n = n - 1 END DO nt = NINT(n0*exp(-t*lambda)) print*,t,n,nt END DO END PROGRAM RADIOACTIVE_DECAY

17 Finding the groundstate of a hamiltonian Consider a system of Ising spins Algorithm Randomly chosen an initial state of the system Choose a random site on the lattice Try to overturn the spin, determining the change in energy if the energy is lowered, then accept the overturn of the spin otherwise it is left unchanged, go to the next site. the system is then either in the groundstate or in some metastable state. We can use different initial configurations, one tests to see if the same state is reached as before if a lower energy state is found.

18 Source : groundstate.f90 PROGRAM GROUNDSTATE IMPLICIT NONE INTEGER, PARAMETER :: N = 10 REAL,DIMENSION(n,n) :: S INTEGER :: i,j,k,l,ip,im,jp,jm REAL :: r,e1,e2,temp! initial configuration CALL RANDOM_SEED() CALL RANDOM_NUMBER(S) DO i = 1,n DO j = 1,n IF (S(i,j) > 0.5) THEN S(i,j) = 1.0 ELSE S(i,j) = -1.0 ENDIF print*,0,etot()

19 DO l = 1,60! Simulation DO k = 1,40 CALL RANDOM_NUMBER(r) i = INT(r*float(n))+1 CALL RANDOM_NUMBER(r) j = INT(r*float(n))+1! periodic boundary condition ip = i + 1; im = i - 1 jp = j + 1; jm = j - 1 IF (ip > n) ip = 1; IF (im < 1) im = n IF (jp > n) jp = 1; IF (jm < 1) jm = n! calculate the energy e1 = -S(i,j)*(S(ip,j)+S(im,j)+S(i,jp)+S(i,jm)) e2 = -e1 IF (e2 <= e1) S(i,j) = -S(i,j) print*,l,etot()

20 CONTAINS! list of subroutine/function REAL FUNCTION etot temp = 0.0 DO i = 1,n ip = i + 1; im = i - 1 IF (ip > n) ip = 1 IF (im < 1) im = n DO j = 1,n jp = j + 1; jm = j - 1 IF (jp > n) jp = 1 IF (jm < 1) jm = n temp = temp & - S(i,j)*(S(ip,j)+S(im,j)+S(i,jp)+S(i,jm)) etot = 0.5*temp/float(n*n) END FUNCTION etot! END PROGRAM GROUNDSTATE

22 Generation of random walks Introduction A random walk consists of a connected path formed by randomly adding new bonds to the end of the existing walk The mean-square end-to-end distance (2.18) : critical exponent : non-universal constants : correction to scaling exponent The partition function (2.19) : an effective coordination number

23 Random walks (RW) used for the description of diffusion phenomena the end-to-end distance (2.20) Simulation of the simple random walk Picking a starting point Generating a random number to determine the direction of each subsequent Calculating the end-to-end distance Performing a statistical analysis of the resultant distribution non-reversal random walk (RNNW) The choice of the (n + 1) step from the n th step of a return to the point reached at the (n - 1) step is forbidden

24 Algorithm of random walks DO i = 1 to ns x = 0; y = 0 DO j = 1 to N id = INT(r*4.0) SELECT CASE(id) CASE (0) x = x + 1 CASE (1) x = x - 1 CASE (2) y = y + 1 CASE (3) y = y - 1 END SELECT accumulate-results the end-to-end distance! number-of-samples! initial position! number of step

25 Source : randomwalk.f90 INTEGER, PARAMETER :: ns = 1000,n = 100 INTEGER :: i,j,x,y,id REAL :: r,d,d2,temp CALL RANDOM_SEED() d = 0.0; d2 = 0.0 DO i=1,ns x = 0; y = 0 DO j = 1,n CALL RANDOM_NUMBER(r) id = INT(r*4.0) SELECT CASE (id) CASE (0) x = x + 1 CASE (1) x = x - 1 CASE (2) y = y + 1 CASE (3) y = y - 1 END SELECT temp = float(x**2+y**2) d = d + sqrt(temp); d2 = d2 + temp d = d/float(ns); d2 = d2/float(ns) print*,ns,n,d,d2

26 Algorithm of non-reversal random walks DO i = 1 to ns! number-of-samples x = 0; y = 0! initial position xp = 0; yp = 0! previous position DO j = 1 to N! number of step repeat xt = x; yt = y! template position generate-one-step (x; y) until ((x xp) or (y yp)) xp = xt yp = yt accumulate-results

27 Source : nr_randomwalk.f90 PROGRAM NR_RANDOM_WALKS IMPLICIT NONE INTEGER, PARAMETER :: ns = 1000,n = 100 INTEGER :: i,j,x,y,id,xp,yp,xt,yt REAL :: r,d,d2,temp LOGICAL :: test! CALL RANDOM_SEED() d = 0.0; d2 = 0.0 CALL NRRW()! non-reversal random walks d = d/float(ns); d2 = d2/float(ns) print*,ns,n,d,d2 CONTAINS SUBROUTINE NRRW END SUBROUTINE NRRW END PROGRAM NR_RANDOM_WALKS

28 SUBROUTINE NRRW DO i=1,ns x = 0; y = 0; xp = x; yp = y DO j = 1,N xt = x; yt = y; test =.FALSE. DO WHILE(.NOT.test) CALL RANDOM_NUMBER(r) id = INT(r*4.0) IF (id == 0) THEN x = xt + 1; y = yt ELSEIF (id == 1) THEN x = xt - 1; y = yt ELSEIF (id == 2) THEN y = yt + 1; x = xt ELSEIF (id == 3) THEN y = yt - 1; x = xt ENDIF IF ((x.ne. xp).or.(y.ne.yp)) test =.TRUE. xp = xt; yp = yt temp = float(x**2+y**2) d = d + sqrt(temp); d2 = d2 + temp END SUBROUTINE NRRW

29 Self-avoiding walks (SAW) Used to probe the configurations of flexible macromolecules in the solvents The walker dies when attempting to intersect a portion of the already completed walk Simulations Picking a starting point Generating a random number to the different possible choices for the next step If the new site is one which already contains a portion of the walk, then the process is terminated at the N th step. effective exponent (2.21)

30 For, the effective exponent is then related to the true value (2.22) the current estimates for (2.23) K. Kremer and K. Binder, Comput. Phys. Rep. 7, 261 (1988) The exponent Using the equation (2.19), we have (2.24) using symmetric values in step number (2.25)

31 Algorithm of Self-avoiding walks integer lattice(-n:n;-n:n); do sample = 1 to n-of-samples step = 0; x = 0; y = 0; xc = 0; yc = 0; repeat repeat generate-one-step(xnew, ynew); until lattice(xnew, ynew) sample + 1; if lattice(xnew, ynew) = sample then terminate = true else lattice(x, y) = sample; x = xc; y = yc; lattice(x, y) = sample + 1; xc = xnew; yc = ynew; step = step + 1; endif until (terminate or step = N); accumulate-results enddo

32 Source: sa_walk.f90 PROGRAM SELF_AVOIDING_WALKS IMPLICIT NONE INTEGER, PARAMETER :: ns = 100,n = 100 INTEGER, DIMENSION(-n:n,-n:n) :: lattice INTEGER :: i,x,y,xt,yt,id,step,nn REAL :: r,d,d2,temp LOGICAL :: terminate,newsite! CALL RANDOM_SEED() d = 0.0; d2 = 0.0 CALL SAW()! self avoiding walks d = d/float(ns); d2 = d2/float(ns) print*,ns,n,d,d2! CONTAINS SUBROUTINE SAW END SUBROUTINE SAW! END PROGRAM SELF_AVOIDING_WALKS

33 SUBROUTINE SAW DO i = 1,ns lattice = 0 x = 0; y = 0 step = 0; terminate =.FALSE. DO WHILE ((.NOT. terminate).and. (step <= n)) xt = x; yt = y nn = lattice(x+1,y) + lattice(x-1,y) & + lattice(x,y+1) + lattice(x,y-1) IF (nn == 4) THEN terminate =.TRUE. ELSE newsite =.FALSE. DO WHILE (.NOT. newsite) CALL RANDOM_NUMBER(r) id = INT(r*4.0)

34 IF (id == 0) THEN x = xt + 1; y = yt ELSEIF (id == 1) THEN x = xt - 1; y = yt ELSEIF (id == 2) THEN y = yt + 1; x = xt ELSEIF (id == 3) THEN y = yt - 1; x = xt ENDIF IF (lattice(x,y) == 0) newsite =.TRUE. step = step + 1 lattice(x,y) = 1 ENDIF temp = float(x**2+y**2) d = d + sqrt(temp) d2 = d2 + temp END SUBROUTINE SAW

35 Thermal Averages by the Simple Sampling Method The model Consider a polymer chain, : attractive energy between two nearest neighbor monomers Defination The distribution (2.26) number of SAW configurations of N steps with n nearest-neighbor contacts (2.27) number of SAW configurations of N steps with n nearest-neighbor contacts and an end-to-end vector.

36 The averages of interest (2.28) (2.29) The specific heat C per bond of the chain (2.30) Using Eq. (2.28), we have (2.31)

37 II.4. Importance sampling Monte Carlo methods Introduction used for the study of phase transitions at finite temperature Consider a spin model, the Hamiltonian is given by (2.32) the averages (2.33) the thermal average of any observable (2.34)

38 probability density (2.35) integrates Eq. (2.34) over all states with their proper weights (2.36) We consider a process where the phase space points are selected according to some probability (2.37) A natural choice for simple average (2.38)

39 Algorithm (Metropolis method) The configurations are generated from a previous state using a transition probability which depends on the energy difference between the initial and final states The master equation of probability of the system that is in the state n at time t (2.39) : is the transition rate for In the equilibrium process, the condition of detailed balance: We have (2.40)

40 The probability of the n th state occurring in a classical system (2.41) If we produce the n th state from the m th state From Eqs. (2.40) and (2.41), we have (2.42) The choice of rate (Metropolis form) (2.43) : is the time required to attempt a spin-flip

41 Metropolis importance sampling Monte Carlo scheme Ising model (1) Choose an initial state (2) Choose a site i (3) Calculate the energy change E which results if the spin at site i is overturned (4) Generate a random number r such that 0 < r < 1 (5) If exp(- E/k B T) > r, flip the spin (6) Go to the next site and go to (3)

42 Program structure initialize the lattice SUBROUTINE equilibrating! Equilibrating process DO ieq = 1 to N_eq CALL monte_carlo_step() END SUBROUTINE SUBROUTINE averaging! Averaging process DO iav = 1 to N_av CALL monte_carlo_step() do-analysis END SUBROUTINE SUBROUTINE monte_carlo_step generate-one-sweep END SUBROUTINE

43 Source : ising_2dsl.f90 Fortran code (ferromagnetic Ising spin model on 2D square lattice) PROGRAM ISING_2D_SL IMPLICIT NONE INTEGER, PARAMETER :: n = 10,nms = n*n,nt = 20 REAL,DIMENSION(n,n) :: S INTEGER :: i,j,ip,im,jp,jm,ie,ia,ms,it,neq,nav REAL :: r,tmin,tmax,t,dt,e1,e2,temp REAL :: ener,magn! CALL RANDOM_SEED() neq = 2000 nav = 4000 tmin = 0.5; tmax = 4.0

44 dt = (tmax-tmin)/float(nt-1) t = tmin! DO it = 1,nt CALL spin_conf() CALL equilibrating() ener = 0.0 magn = 0.0 CALL averaging() ener = ener/float(nav) magn = magn/float(nav) print*, t,ener,magn t = t + dt CONTAINS END PROGRAM ISING_2D_SL! initial configuration! equilibrating process! averaging process! list of subroutines/functions

45 CONTAINS! list of subroutines/functions! SUBROUTINE spin_conf CALL RANDOM_NUMBER(S) DO i = 1,n DO j = 1,n IF (S(i,j) > 0.5) THEN S(i,j) = 1.0 ELSE S(i,j) = -1.0 ENDIF END SUBROUTINE spin_conf! SUBROUTINE equilibrating DO ie = 1, neq CALL monte_carlo_step() END SUBROUTINE equilibrating

46 SUBROUTINE averaging DO ia = 1, nav CALL monte_carlo_step()! Calculate the average energy per spin temp = 0.0 DO i = 1,n ip = i + 1; im = i - 1 IF (ip > n) ip = 1 IF (im < 1) im = n DO j = 1,n jp = j + 1; jm = j - 1 IF (jp > n) jp = 1 IF (jm < 1) jm = n temp = temp -S(i,j)*& (S(ip,j)+S(im,j)+S(i,jp)+S(i,jm)) ener = ener + 0.5*temp/float(n*n)

47 ! Calculate the average magnetization per spin temp = 0.0 DO i = 1,n DO j = 1,n temp = temp + S(i,j) magn = magn + abs(temp)/float(n*n) END SUBROUTINE averaging

48 SUBROUTINE monte_carlo_step DO ms = 1,nms! number of Monte Carlo step CALL RANDOM_NUMBER(r) i = INT(r*float(n))+1 CALL RANDOM_NUMBER(r) j = INT(r*float(n))+1! periodic boundary condition ip = i + 1; im = i - 1 jp = j + 1; jm = j - 1 IF (ip > n) ip = 1 IF (im < 1) im = n IF (jp > n) jp = 1 IF (jm < 1) jm = n! calculate the different energy e1 = -S(i,j)*(S(ip,j)+S(im,j)+S(i,jp)+S(i,jm)) e2 = -e1 CALL RANDOM_NUMBER(r) IF (r < exp(-(e2-e1)/t)) S(i,j) = -S(i,j) END SUBROUTINE monte_carlo_step

49 Ferromagnetic Ising spin model on square lattice Energy vs temperature for N = 10

50 Ferromagnetic Ising spin model on square lattice Magnetization vs temperature for N = 10

51 Exercises modify the program «ising_2dsl.f90» : Adding the calculations for specific heat and susceptibility ener2 = ener2 + (0.5*temp/float(n*n))**2 magn2 = magn2 + (temp/float(n*n))**2 cv = float(n*n)*(ener2 ener*ener)/t/t chi = float(n*n)*(magn2 magn*magn)/t For the case of three dimensions simple cubic lattice Local energy : e1 = -S(i,j,k)*(S(ip,j,k)+S(im,j,k)+S(i,jp,k)+S(i,jm,k) +S(i,j,kp)+S(i,j,km))

52 Ferromagnetic Ising spin model on square lattice Specific heat vs temperature for N = 10

53 Ferromagnetic Ising spin model on square lattice Magnetic susceptibility vs temperature for N = 10

54 Ferromagnetic Ising spin model on simple cubic lattice Energy, magnetization, specific heat and susceptibility vs temperature for N = 12

55 Write a new program for ferromagnetic XY spin model on square lattice Including the calculations for specific heat and susceptibility The local energy e1 = -SX(i,j)*(SX(ip,j)+SX(im,j)+SX(i,jp)+SX(i,jm)) -SY(i,j)*(SY(ip,j)+SY(im,j)+SY(i,jp)+SY(i,jm))

56 Generate a new XY spin with random orientation CALL RANDOM_NUMBER(r) PHI = 2.0* *r SXN = COS(PHI) SYN = SIN(PHI)! orientation

57 Source : xy_2dsl.f90 Fortran code (ferromagnetic XY spin model on square lattice) PROGRAM XY_2D_SL IMPLICIT NONE REAL,PARAMETER:: pi2 = 2.0* INTEGER, PARAMETER :: n = 10,nms = n*n,nt = 20 REAL,DIMENSION(n,n) :: sx,sy INTEGER :: i,j,ip,im,jp,jm,ie,ia,ms,it,neq,nav REAL :: r,tmin,tmax,t,dt,e1,e2,si,phi,sxn,syn,temp REAL :: sxt,syt,ener,magn! CALL RANDOM_SEED() neq = nav = tmin = 0.1; tmax = 2.0

58 dt = (tmax-tmin)/float(nt-1) t = tmin! DO it = 1,nt CALL spin_conf() CALL equilibrating() ener = 0.0; magn = 0.0 CALL averaging() ener = ener/float(nav) magn = magn/float(nav) print*, t,ener,magn t = t + dt! initial configuration CONTAINS! list of subroutines/functions

59 SUBROUTINE spin_conf! DO i = 1,n DO j = 1,n CALL RANDOM_NUMBER(r) phi = pi2*r sx(i,j)=cos(phi) sy(i,j)=sin(phi) END SUBROUTINE spin_conf! SUBROUTINE equilibrating DO ie = 1, neq CALL monte_carlo_step() END SUBROUTINE equilibrating

60 SUBROUTINE averaging DO ia = 1, nav CALL monte_carlo_step() temp = 0.0 DO i = 1,n ip = i + 1; im = i - 1 IF (ip > n) ip = 1 IF (im < 1) im = n DO j = 1,n jp = j + 1; jm = j - 1 IF (jp > n) jp = 1 IF (jm < 1) jm = n temp = temp - sx(i,j)*(sx(ip,j) + sx(im,j) & + sx(i,jp) + sx(i,jm)) & - sy(i,j)*(sy(ip,j) + sy(im,j) & + sy(i,jp) + sy(i,jm))

61 ener = ener + 0.5*temp/float(n*n) sxt = 0.0; syt = 0.0 DO i = 1,n DO j = 1,n sxt = sxt + sx(i,j) syt = syt + sy(i,j) magn = magn + sqrt(sxt**2 + syt**2)/float(n*n) END SUBROUTINE averaging

62 SUBROUTINE monte_carlo_step DO ms = 1,NMS CALL RANDOM_NUMBER(r) i = INT(r*float(n))+1 CALL RANDOM_NUMBER(r) j = INT(r*float(n))+1 ip = i + 1; im = i - 1 jp = j + 1; jm = j - 1 IF (ip > n) ip = 1 IF (im < 1) im = n IF (jp > n) jp = 1 IF (jm < 1) jm = n! periodic boundary condition

63 ! calculate the local energy sxt = sx(ip,j)+sx(im,j)+sx(i,jp)+sx(i,jm) syt = sy(ip,j)+sy(im,j)+sy(i,jp)+sy(i,jm) e1 = -sx(i,j)*sxt - sy(i,j)*syt CALL RANDOM_NUMBER(r) phi = pi2*r sxn=cos(phi) syn=sin(phi) e2 = -sxn*sxt -syn*syt CALL RANDOM_NUMBER(r) IF (r < exp(-(e2-e1)/t)) THEN sx(i,j) = sxn sy(i,j) = syn ENDIF END SUBROUTINE monte_carlo_step! END PROGRAM XY_2D_SL

64 Ferromagnetic XY spin model on square lattice Energy and magnetization vs temperature for N = 10 There exist two phases A low-temperature-phase with a quasi-long range order, where most spins are aligned and the correlation-function decays with a power law An unordered high-temperature-phase where the correlation-function decays exponentially The phase transition is called «Kosterlitz-Thouless transition»

65 Counterclockwise vortices

66 Write a new program for ferromagnetic Heisenberg spin model on two dimensions square lattice The local energy e1 = -SX(i,j)*(SX(ip,j)+SX(im,j)+SX(i,jp)+SX(i,jm)) -SY(i,j)*(SY(ip,j)+SY(im,j)+SY(i,jp)+SY(i,jm)) -SZ(i,j)*(SZ(ip,j)+SZ(im,j)+SZ(i,jp)+SZ(i,jm))

67 Generate a new Heisenberg spin with random module and orientation CALL RANDOM_NUMBER(r) PHI = 2.0* *r CALL RANDOM_NUMBER(r) SZN = 2.0*r-1.0 SI = SQRT(1-SZN*SZN) SXN = SI*COS(PHI) SYN = SI*SIN(PHI)! orientation! projection of SZ

68 For two dimensions triangular lattice The local energy e1 = -S(i,j)*(S(ip,j)+S(im,j)+S(i,jp)+S(i,jm) + S(ip,jm)+S(im,jp)) For 3D stracked triangular lattice e1 = -S(i,j,k)*(S(ip,j,k)+S(im,j,k)+S(i,jp,k)+S(i,jm,k) +S(ip,jm,k)+S(im,jp,k) +S(i,j,kp) +S(i,j,km))

69 Source : ising_2dtl.f90 Fortran code (ferromagnetic Ising spin model on triangular lattice) PROGRAM ISING_2D_TL IMPLICIT NONE INTEGER, PARAMETER :: n = 10,nms = n*n,nt = 40 REAL,DIMENSION(n,n) :: S INTEGER :: i,j,ip,im,jp,jm,ie,ia,ms,it,neq,nav REAL :: r,tmin,tmax,t,dt,e1,e2,temp REAL :: ener,magn! CALL RANDOM_SEED() neq = nav = tmin = 1.0; tmax = 6.0

70 dt = (tmax-tmin)/float(nt-1) t = tmin! DO it = 1,nt CALL spin_conf()! initial configuration CALL equilibrating() ener = 0.0; magn = 0.0 CALL averaging() ener = ener/float(nav) magn = magn/float(nav) print*, t,ener,magn t = t + dt CONTAINS! list of subroutine/function

71 SUBROUTINE spin_conf! CALL RANDOM_NUMBER(S) DO i = 1,n DO j = 1,n IF (S(i,j) > 0.5) THEN S(i,j) = 1.0 ELSE S(i,j) = -1.0 ENDIF END SUBROUTINE spin_conf! SUBROUTINE equilibrating DO ie = 1, neq CALL monte_carlo_step() END SUBROUTINE equilibrating

72 SUBROUTINE averaging DO ia = 1, nav CALL monte_carlo_step() temp = 0.0 DO i = 1,n ip = i + 1; im = i - 1 IF (ip > n) ip = 1 IF (im < 1) im = n DO j = 1,n jp = j + 1; jm = j - 1 IF (jp > n) jp = 1 IF (jm < 1) jm = n temp = temp - S(i,j)*(S(ip,j)+S(im,j) & +S(i,jp)+S(i,jm) + S(ip,jm)+S(im,jp) ) ener = ener + 0.5*temp/float(n*n)

73 temp = 0.0 DO i = 1,n DO j = 1,n temp = temp + S(i,j) magn = magn + abs(temp)/float(n*n) END SUBROUTINE averaging

74 SUBROUTINE monte_carlo_step DO ms = 1,NMS CALL RANDOM_NUMBER(r); i = INT(r*float(n))+1 CALL RANDOM_NUMBER(r); j = INT(r*float(n))+1! periodic boundary condition ip = i + 1; im = i - 1 jp = j + 1; jm = j - 1 IF (ip > n) ip = 1 IF (im < 1) im = n IF (jp > n) jp = 1 IF (jm < 1) jm = n! calculate the energy e1 = -S(i,j)*(S(ip,j)+S(im,j)+S(i,jp)+S(i,jm) & + S(ip,jm)+S(im,jp) ) e2 = -e1 CALL RANDOM_NUMBER(r) IF (r < exp(-(e2-e1)/t)) S(i,j) = -S(i,j) END SUBROUTINE monte_carlo_step END PROGRAM ISING_2D_TL

75 Ferromagnetic Ising spin model on 2D triangular lattice Energy vs temperature for N = 10

76 Ferromagnetic Ising spin model on 2D triangular lattice Magnetization vs temperature for N = 10

Hanoi 7/11/2018. Ngo Van Thanh, Institute of Physics, Hanoi, Vietnam.

Hanoi 7/11/2018. Ngo Van Thanh, Institute of Physics, Hanoi, Vietnam. Hanoi 7/11/2018 Ngo Van Thanh, Institute of Physics, Hanoi, Vietnam. Finite size effects and Reweighting methods 1. Finite size effects 2. Single histogram method 3. Multiple histogram method 4. Wang-Landau