Transcript Monte Carlo Simulation of Ising Model and Phase Transition

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
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Describe the Ising model for magnetism;
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
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Magnetic susceptibility χ :
Types of magnetic materials:
1. Diamagnetic: χ<0 and constant (Helium);
2. Paramagnetic: magnetic susceptibility
χ>0 and χ∝1/T (Rare earth);
3. Ferromagnetic: Iron. Below a critical
temperature (Curie temperature), χ
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, χ ∝T; above this temperature,
the material becomes paramagnetic. (MnO)
Hysteresis loop
Ising Model(2D)
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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 σ(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. σ=+1 or -1.
Ising Model(2D)
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Thermal properties are defined, and computed, by
the partition function, which is the normalization
factor of the probability of a thermodynamic state:
Z (T )   exp[ E / k BT ]

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p
1
exp[ E / k BT ]
Z (T )
Using Z(T), we can calculate the specific heat C ,
and magnetic susceptibility χ
Phase transitions
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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
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C and χ 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 χ , because they diverge
at some critical value of temperature (Tc≈2.269 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
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Monte Carlo: A method using pseudorandom number to simulate the
random thermal fluctuation from state to state of a system;
The probability of a particular state αfollows Boltzmann distribution:
1
p
exp[ E / k BT ]
Z (T )
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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
δE=E(new)-E(old);
3. If δE>0, the probability to accept the new state P(old->new) = exp[δE/kT], otherwise P(old->new) = 1.
Simulation settings
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Set the spin-spin interaction energy J=1,
Boltzmann constant k=1, Bohr magneton
B 
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5.7884
8.617
0.67
The unit of Energy is J; the unit of
temperature T is 1/ kB
Simulation interface
Results: Energy per spin versus Temperature
(Zero external field). The derivative C=dE/dT
diverges at around Tc≈2.269.
Results: C versus T. Specific heat divergence is shown more
clearly at Tc≈2.269 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
1
0.8
0.6
Magnetization per Site
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
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
Tc≈2.269. For T far above Tc, M oscillates around 0.
2D Ising Model: T=2.25, L=20 square lattice
1
0.8
0.6
Magnetization per Site
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
0
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
0.9
0.8
|Magnetization per Site|
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
-0.1
1
1.5
2
2.5
reduced temperature
3
3.5
4
Results: Magnetic susceptibility χ versus T. χ diverges at
around Tc≈2.269. 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
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Demonstrate that second order phase transition of
specific heat C and magnetic susceptibility χ occur at
Tc≈2.269, as predicted by Onsager’s exact solution.
Demonstrate the existence of spontaneous
magnetization and hysteresis loop below Tc≈2.269
(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.