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# Monte Carlo Simulation of

## Ising Model and Phase

Transition Studies

## Yu Sun*, Yilin Wu**

*Department of Electric Engineering, University of Notre
Dame
**Department of Physics, University of Notre Dame
Instructor: Prof. Mark Alber, Department of Mathematics,
University of Notre Dame

Outline

## Introduce the Monte Carlo simulation method

as well as the Metropolis algorithm;

## Present our Monte Carlo simulation results for

Ising model and discuss its properties,
especially the phase transition behavior.

Introduction to Magnetism

Magnetic susceptibility :

## Types of magnetic materials:

1. Diamagnetic: <0 and constant
(Helium);
2. Paramagnetic: magnetic susceptibility
>0 and 1/T (Rare earth);
3. Ferromagnetic: Iron. Below a critical
temperature (Curie temperature),
depends on magnetic field, and the M-H
diagram shows a hysteresis loop; above
this temperature, the material becomes
paramagnetic;
4. Anti-Ferromagnetic: Below a critical
temperature, T; above this
temperature, the material becomes
paramagnetic. (MnO)

Hysteresis
loop

Ising Model(2D)

## A lattice model proposed to interpret

ferromagnetism in materials(1925).

## Basic idea: Elementary particles have

an intrinsic property called spin.
Spins carry magnetic moments. The
magnetism of a bulk material is made
up of the magnetic dipole moments of
the atomic spins inside the material.

## Ising model postulates a lattice with a

spin (or magnetic dipole moment) on
each site, defining the following
Hamiltonian:

## E is total energy of the system, J is the

nearest spin-spin interaction energy, H
is external magnetic field. =+1 or -1.

Ising Model(2D)

## Thermal properties are defined, and

computed, by the partition function, which is
the normalization factor of the probability of a
1
thermodynamic
state:
p

exp[ E / k T ]
Z (T ) exp[ E / k T ]

Z (T )

## Using Z(T), we can calculate the specific heat

C , and magnetic susceptibility

Phase transitions

## The abrupt sudden change in physical properties of the

thermodynamic system around some critical value of
thermodynamic variables (such as temperature). A particular
quantity is the specific heat.

## Ehrenfest classification of Phase Transition:

First-order phase transitions exhibit a discontinuity in the
first derivative of the chemical potential with a thermodynamic
variable. Such as solid/liquid/gas transitions.

## Second-order phase transitions (also called continuous

phase transition) have a discontinuity or divergence in a second
derivative of the chemical potential with thermodynamic
variables.

Phase transitions

## C and are second derivative of chemical potential with

T and H separately.

## Onsager (1944) obtained the exact solution for 2D Ising

model without external field. The solution shows that
there exists second order phase transition in C and ,
because they diverge at some critical value of
temperature (Tc2.269 in unit of (1/Boltzmann
constant)). The studies can explain the ferromagnetic to
paramagnetic transition of materials.

## Monte Carlo simulations also reveal the phase transition

properties of Ising model.

## Monte Carlo method and

Metropolis Algorithm

## simulate the random thermal fluctuation from state to state of a

system;
The probability of a particular state follows Boltzmann
1
distribution:
p
exp[ E / k T ]

Z (T )

## In theory, sum over all possible states to calculate the statistical

mean values of a physical quantity, weighing each state based
on its Boltzmann factor;

## Metropolis algorithm (importance sampling technique):

1.Flip one randomly picked spin;
2.Calculate the total energy difference between new and old spin
state E=E(new)-E(old);
3. If E>0, the probability to accept the new state P(old->new) =
exp[-E/kT], otherwise P(old->new) = 1.

Simulation settings

## Set the spin-spin interaction energy

J=1, Boltzmann constant k=1, Bohr
magneton
5.7884

; 0.67
B

8.617

## The unit of Energy

J; the unit of
1/ kis
B
temperature T is

Simulation interface

## Results: Energy per spin versus

Temperature (Zero external field). The
derivative C=dE/dT diverges at around
Tc2.269.

## Results: C versus T. Specific heat divergence is shown

more clearly at Tc2.269 in this figure. Second
order phase transition occurs.

## Results: Magnetization per spin (Zero external field),

T=1.5, 2.0. The figures show spontaneous
magnetization (most of the spins align in the same
direction).
2D Ising Model: T=1.5, L=20 square lattice

e
t
i
S
r
e
p

0.8
0.6
0.4

n
o
i 0.2
t
a
z
0
i
t
e
n
g -0.2
a
M
-0.4
-0.6

-0.8
-1

500

1000

1500

2000

2500
MC step

3000

3500

4000

4500

5000

## Results: Magnetization per spin (Zero external

field), T=2.25, 4.0. Fluctuations become more
significant near Tc2.269. For T far above Tc, M
oscillates around 0.
2D Ising Model: T=2.25, L=20 square lattice

e
t
i
S
r
e
p
n
o
i
t
a
z
i
t
e
n
g
a
M

0.8
0.6
0.4

0.2

0
-0.2

-0.4
-0.6

-0.8
-1

500

1000

1500
MC step

2000

2500

3000

## Results: Magnetization per spin versus

Temperature (Zero external field).
2D Ising Model: L= 20 square lattice, 1000 MC cycles

|
e
t
i
S
r
e
p
n
o
i
t
a
z
i
t
e
n
g
a
M
|

0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1

1.5

2.5
reduced temperature

3.5

## Results: Magnetic susceptibility versus T.

diverges at around Tc2.269. It is second order
phase transition. Above Tc, it is paramagnetic.

## Results: Magnetization per spin versus

External field H at T= 0.2. It shows a
hysteresis loop, characteristic of
ferromagnetic materials.

Summary of Results

## Demonstrate that second order phase transition of

specific heat C and magnetic susceptibility occur
at Tc2.269, as predicted by Onsagers exact
solution.

## Demonstrate the existence of spontaneous

magnetization and hysteresis loop below Tc2.269
(J>0). These show that the system is
ferromagnetic below Tc.

## Combing these results, the ferromagnetic to

paramagnetic phase transition of 2D Ising model
is demonstrated.
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