3D ISING MODEL
6
Narendra Kumar Course: Computational Condensed Matter Three-dimensional Ising Model project :1 credit Introduction: I have studied 3D Ising model using Metropolis algorithm . The Ising model has been interesting (due to its simplicity) since its formulation by Ernest Ising. In this model spin variables take only two values, +1 and -1. Further, each of these variables interact only with the adjacent former. This model became the basis for phase transition and critical phenomena. We can model several systems into Ising system (e.g; lattice gas model) and then can be solved easily. Ising model in 3D: In 3D , the Hamiltonian of the model can be written as N N H = -J Σ S(i,j,k)[S(i-1,j,k) + S(i+1,j,k) + S(i,j-1,k) + S(i,j+1,k) + S(i,j,k-1) + S(i,j,k+1)] - h ΣS i i,j,k=1 i Here I have kept spins on the corners of a simple cubic lattice. I have considered three systems of sizes 36 3 , 40 3 and 44 3 . I have not considered any external field, so h=0. The magnetization(M), susceptibility(χ) and specific heat (C v ) are obtained by using the following relations: N M = (1/N) Σ S i , i χ = (J/K b T)(<M 2 > - <M> 2 ) , C v = (J/K b T 2 )(<E 2 > - <E> 2 ) Susceptibility and specific heat provides the information about phase-transition and shows divergence at critical temperature (T c ). Kurt Binder suggested an another parameter to estimate critical point. This is called Binder ratio. This is a standard observational tool for estimating critical point and defined as <M 4 > 3 <M 2 > 2 where <M 4 > is the 4 th cumulant of magnetisation. For different system sizes , U 4 curve intersects each other at a fixed point which coincides with the critical point. Numerical Results: Magnetisation (M) is the order-parameter in ferromagnetic system. Before critical temperature(T c ) system is ferromagnetic and after T c systems becomes paramagnetic (M = 0). We know that the phase transition occurs in thermodynamic limit (L 3 ) and in this limit 2 nd order quantity diverges at T c . U 4 = 1 -
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Transcript of 3D ISING MODEL
Narendra Kumar Course: Computational Condensed Matter
Three-dimensional Ising Model project :1 credit
Introduction:I have studied 3D Ising model using Metropolis algorithm . The Ising model has been interesting(due to its simplicity) since its formulation by Ernest Ising. In this model spin variables take onlytwo values, +1 and -1. Further, each of these variables interact only with the adjacent former. Thismodel became the basis for phase transition and critical phenomena. We can model several systemsinto Ising system (e.g; lattice gas model) and then can be solved easily.
Ising model in 3D: In 3D , the Hamiltonian of the model can be written as N N
H = -J Σ S(i,j,k)[S(i-1,j,k) + S(i+1,j,k) + S(i,j-1,k) + S(i,j+1,k) + S(i,j,k-1) + S(i,j,k+1)] - h ΣSi
i,j,k=1 i
Here I have kept spins on the corners of a simple cubic lattice. I have considered three systems ofsizes 363, 403 and 443 . I have not considered any external field, so h=0. The magnetization(M),susceptibility(χ) and specific heat (Cv) are obtained by using the following relations:
N
M = (1/N) Σ Si , i χ = (J/KbT)(<M2> - <M>2) , Cv = (J/KbT2)(<E2> - <E>2)
Susceptibility and specific heat provides the information about phase-transition and showsdivergence at critical temperature (Tc). Kurt Binder suggested an another parameter to estimatecritical point. This is called Binder ratio. This is a standard observational tool for estimating criticalpoint and defined as <M4> 3 <M2>2
where <M4> is the 4th cumulant of magnetisation.
For different system sizes , U4 curve intersects each other at a fixed point which coincides with thecritical point.
Numerical Results:Magnetisation (M) is the order-parameter in ferromagnetic system. Before critical temperature(Tc)system is ferromagnetic and after Tc systems becomes paramagnetic (M = 0). We know that thephase transition occurs in thermodynamic limit (L3) and in this limit 2nd order quantity diverges at Tc
.
U4 = 1 -
To work in large size system is computationally too hard and time consuming so to mimic thebehaviour of phase transition in finite size system we use some sort of scaling analysis. Wesimulated the system for 15,000 Monte Carlo steps(MCS); out of which first 10,000 MCSconsidered for thermalization and rest of which collected for measurement of desired quantities.
In the next two figures I have shown the zoomed image of the intersecting curves region.
T=0T = 3.5 J/K
b
T = 4.5 J/Kb I.e;
near Tc
T = 5.5 J/Kb
T = 4.5 J/Kb
i.e;
near Tc
up-spin
down-spin
These snapshots are taken at 15,000th MCS for 103 size system only. We have summarized thenumber of up-spin and down-spin atoms in the following table.
Orientation of T = 0 T = 3.5 J/Kb T = 4.5 J/Kb T = 5.5J/Kb
spin
No. of up 1000 948 384 454 spin
No. of down 0 52 616 546 spin
We see that as we increase the temperature of the system some atomic spins get flipped andproceeds to paramagnetic phase. For T<Tc , up-spins(or down-spins) are in majority and showferromagnetic behaviour and after Tc up and down-spins are approximately equal in number . As aresult of this show paramagnetic phase behaviour (up and down spin nullify each other and soM≈0).
Conclusion:We simulate the Ising model in 3D with Monte Carlo and we use the Metropolis algorithm to updatethe distribution of spins. The behaviour of magnetization, specific heat, susceptibility, and Binderratio (for different lattice sizes) versus temperature suggest a phase transition around T = 4.5 J/Kb
(literature value of critical temperature). We could get better plots if we worked in large-sizesystems.
References:(1) Critical Behavior of a Cubic-Lattice 3D Ising Model for Systems with Quenched Disorder by A.K. Murtazaev et. al (2) Computational Analysis of 3D Ising Model Using Metropolis Algorithms by A. F. Sonsin et. al(3) [BOOK] Understanding molecular simulation by Frenkel & Smith(4) Solving the 3D Ising Model with the Conformal Bootstrap by Sheer El-Showk et. al
