Ising Model of Ferromagnetism

Tyler Brau and Xiaoqi Yu

Ferromagnetism

➔Why certain materials form permanent

magnets?

➔At what temperature do they cease to exhibit

spontaneous magnetization?

➔What are their behavior under an external

magnetic field?

The Ising Model

Energy Difference between

a lattice spin and its nearest

neighbors

Project Goals

➔Extend 2D Ising model to 3D

➔Utilize Block Spin Transform (BST) to find the

critical temperature for a 3D lattice

➔Examine the behavior of the lattice about the

critical temperature to determine if phase

transition is first or second order

3D Ising Model

➔Initialize the Lattice to all +1

➔Add front and back lattice points as two

additional nearest neighbor:

➔For each lattice point, If , flip the lattice to

opposite spin

➔Otherwise, flip it with probability of

Lattice Size= 81*81*81

Calculated Magnetization for all

temperatures from 0 to 8, with

step size of 0.1

Run Time: 7.79*10^3 seconds, or

2.2 hrs

Critical temperature was found at

4.5, at which the magnetization

drops the fastest

3D magnetization vs. Temperature

Block Spin Transformation

1 BST

Max at 2.27

Difference in Magnetization after

BST

2 BST

3 BST

Algorithm to find Critical

Temperature Using BST

Algorithm was a stochastic optimization

algorithm which found the maximum of the

peak

The initial guess at the critical temperature

was that for a 2D lattice, or 2.27

Specific implementation used was the Kiefer-

Wolfowitz algorithm

Algorithm Results

With a0 = 1 and c0 = ¼, algorithm converged to a

temperature of 4.3 as the critical temperature

Expansion around this point gave the critical

temperature to be 4.50 ±0.05

Total run time was 2.01*10^3 s, or 33 mins

BST approach was 3.9 times faster

The Behavior of Ferromagnet under an external Field

Take-Aways

➔At critical temperature, ferromagnet experiences a

phase transition and loses its spontaneous

magnetization

➔Block Spin Transformation greatly reduced the

computational time. Only one transform is necessary

to find the critical temperature

➔Phase transition is second order, not first order