The Ising Model

1
The Ising Model

• Mathematical Biology Lecture 5
• James A. Glazier
• (Partially Based on Koonin and Meredith,
  Computational Physics, Chapter 8)

2
Ising Model Basics

• A Simple, Classical Model of a Magnetic Material.
• A Lattice (Usually Regular) with a Magnet or
  Classical Spin at Each Site, Aligned Either Up
  or Down (in Quantum Mechanics
  Would be ).
• The Spins Interact with Each Other Via a
  Coupling of Strength J and to an External Applied
  Magnetic Field B.
• The Two Spin Interactions are

1 1
-1 1
1 -1
-1 -1
J
-J
-J
J
3
Ising Model BasicsContinued

• The Total Energy of the Spins is the
  Hamiltonian
• If Jgt0 have a Ferromagnet. Energy is Lowest if
  all s are the Same. Favored.
• If Jlt0 have an Antiferromagnet. Energy is Lowest
  if neighboring s are Opposite.
  Favored.

N
S
N
S
N
S
S
N
4
Ising Model BasicsConclusion

• The One-Dimensional Ising Model is Exactly
  Soluble and Is a Homework Problem in
  Graduate-Level Statistical Mechanics.
• The Two-Dimensional Ising Model is Also Exactly
  Soluble (Onsager) but is Impressively Messy.
• The Three-Dimensional Ising Model is Unsolved.
• Can Have Longer-Range Interactions, Which can
  have Different J for Different Ranges. Can Result
  in Complex Behaviors, E.g. Neural Networks.
• Similarly, Triangular Lattices and Jlt0 can
  Produce Complex Behaviors, E.g. Frustration and
  Spin-Glasses. Cant
  Satisfy All Bonds.

1
-1
1
-1
1
5
Examples
-1 -1 -1
-1 1 -1
-1 -1 -1
1 1 1
1 1 1
1 1 1
1 -1 1
-1 1 -1
1 -1 1
Note Maximum Energy is 24J and Minimum -24J
for 3 x 3 Lattice (Absorbing Boundaries).
6
Thermodynamics

• What are The Statistical Properties of the
  Lattice at a Given Temperature T?
• Define the State of the Lattice as a Vector

Example
(i,j) (1,j) (2,j) (3,j)
(i,1) -1 -1 -1
(i,2) -1 1 -1
(i,3) -1 -1 -1
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Statistical Mechanics

• In Thermodynamics All Statistical Properties Are
  Determined by the Partition Function Z

• The Probability of a Particular Configuration
  is

• For bltbcritical (I.e. TgtTcritical) Spins are
  Essentially Random. I.e. the Probability of
• All Configurations is Essentially Equal.
• For bgtbcritical (I.e. TltTcritical), then for Jgt0,
  Configurations with Almost All Spins Aligned are
  Much More Probable.
• Tcritical is the Neél or Curie Temperature. For
  J1, bcritical0.44 or Tcritical1.6
• As Magnets are Heated, their Magnetization
  Disappears.

8
Degeneracy

• In the Low Temperature Limit, Can have Multiple
  Equivalent of Degenerate States with the Lowest
  Energy. These will be Equally Probable.
• The Change from a Large Number of Equiprobable
  Random States to Picking (Randomly) One of
  Several Degenerate States is a Spontaneous
  Symmetry Breaking.
• Example
• For Very Low Temperatures, the Probability of
  Flipping Between
• the Two States is Near 0. For Higher Temperatures
  , Flipping Occurs (Causes Problems for Small
  Magnets, E.g. in Disk Drives)

1 1 1
1 1 1
1 1 1
-1 -1 -1
-1 -1 -1
-1 -1 -1
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Ising Metropolis-Boltzmann Dynamics

• Pick a Lattice Site at Random and Try to Swap the
  Spin Between 1 ? -1.
• Example ?
• If HtltH0 then Accept the Swap.
• If HtgtH0 then Accept the Swap with Probability
• 
  a Boltzmann Factor.
• Making as Many Spin-Flip Attempts as Lattice
  Sites Defines One Monte Carlo Step (MCS).

1 1 1
1 1 1
1 1 1
1 1 1
1 -1 1
1 1 1
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Alternative Dynamics

• Generating the Trial States Optimally is Complex.
• Both Deterministic and Random Algorithms.
• Alternative Dynamics Include Kawasaki (Pick Two
  Sites at Random and Swap Their Spins).
  Fundamentally Different From Metropolis Since
  Total Number of 1s and -1s is Conserved. Thus
  Samples a Different Configuration Space From
  Single-Spin Dynamics.