Monte carlo simulation of 3D Ising model

1. 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 ΣSi
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 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 shows
divergence at critical temperature (Tc). 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
<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 the
critical 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 the
phase transition occurs in thermodynamic limit (L3
) and in this limit 2nd
order quantity diverges at Tc
.
U4
= 1 -

2. To work in large size system is computationally too hard and time consuming so to mimic the
behaviour of phase transition in finite size system we use some sort of scaling analysis. We
simulated the system for 15,000 Monte Carlo steps(MCS); out of which first 10,000 MCS
considered for thermalization and rest of which collected for measurement of desired quantities.

3. In the next two figures I have shown the zoomed image of the intersecting curves region.

4. T=0
T = 3.5 J/Kb
T = 4.5 J/Kb
I.e;
near Tc
T = 5.5 J/Kb
T = 4.5 J/Kb
i.e;
near Tc

5. up-spin
down-spin
These snapshots are taken at 15,000th
MCS for 103
size system only. We have summarized the
number 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 and
proceeds to paramagnetic phase. For T<Tc , up-spins(or down-spins) are in majority and show
ferromagnetic behaviour and after Tc up and down-spins are approximately equal in number . As a
result of this show paramagnetic phase behaviour (up and down spin nullify each other and so
M≈0).
Conclusion:
We simulate the Ising model in 3D with Monte Carlo and we use the Metropolis algorithm to update
the distribution of spins. The behaviour of magnetization, specific heat, susceptibility, and Binder
ratio (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-size
systems.
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