1. In “Adaptive Cooperative Systems” Summarized by Ho-Sik Seok
3.3 The Ising Model 3.4 The Ising Model in One Dimension 3.5 The Ising Model in Two Dimensions
In “Adaptive Cooperative Systems”
Summarized by Ho-Sik Seok

2. © 2009 SNU CSE Biointelligence Lab
3.3.1 Ferromagnetism
Ferromagnetism
An alignment of the moments can be sustained in the absence of an external field.
The ferromagnetism behavior posses a thermal dependence.
Spontaneous magnetization: when the external field strength is zero
Hysteresis: in a ferromagnetism material, the magnetization is not a unique function of the external, applied field, but rather depends on its previous history.
The response is delayed, but once started, the change is rapid until the saturation plateau is reached.
The response curve for a single ferromagnetic domain: the changes in magnetization are reversable, and there is no hysteresis.
Pauli exchange coupling
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3. 3.3.2 The Classical Limit and XY-Model
The total energy E: adding together the contributions from all pairwise interactions over the lattice.
Heisenberg XY model
Coupling interaction i: the angle between the direction of the ith spin vector in the lattice and the x-axis.
Projection of the spin along an z-axis
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4. 3.3.4 The Ising Hamiltonian and Partition Function
Up and down orientations of the spin elements corresponds to spin values of +1 and -1.
The interaction energy between atoms located at the jth and kth lattice sites.
J: a measure of the strength of the exchange interaction.
For a ferromagnetic interaction, J is positive.
For J positive, the energy is lowered wherever neighboring atoms have the same spin and raised whenever their spins are in opposition.
s: an electronic spin coordinate , (j,k): j and k are nearest-neighbor lattice.
Eext: an energy term representing the interaction of the external magnetic field with the lattice atoms.
The total energy E(H)
Partition function
: Langrange multiplier ( ), H: the external magnetic field
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5. 3.3.5 Thermodynamic Parameters
The internal energy <E>
The magnetization <M>
The heat capacity can be determined from the internal energy.
The spontaneous magnetization is the value in the limit of zero external magnetic field of the magnetization M(, 0).
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6. 3.4 The Ising Model in One Dimension (1/4)
The interaction energy for a one-dimensional lattice.
Transfer matrix method
The partition function is related to the largest characteristic value (eigenvalue) of a matrix whose elements represent single nearest-neighbor interactions.
The partition function
For a pair of nearest neighbors s and s’, there are four possible values of the exponent
The periodic boundary condition: SN+1 = S1
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7. 3.4 The Ising Model in One Dimension (2/4)
The partition function can be rewritten in terms of transfer matrix P.
The eigenvalues of (nn) matrix A are the values  that are solutions of the equation
This set of equations will poses a nontrivial solution provided that
The n roots n of this equation are the eigenvalues.
These eigenvalues are independent of the choice of basis.
Tr: the trace (sum of the diagonal elements) of a matrix
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8. 3.4 The Ising Model in One Dimension (3/4)
It is possible to select a basis in which the transfer matrix P is diagonal with matrix elements equal to the eigenvales
The eignevalues + and - given as solutions to the secular equation
With eigenvalues obtained by solving the secular equation
For + > -
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9. 3.4 The Ising Model in One Dimension (4/4)
Magnetization per spin M(, H)
The internal energy per spin in the one-dimensional Ising model in the absence of an external magnetic field.
The heat capacity: by differentiating with respect to the temperature
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10. 3.5 The Ising Model in Two Dimensions(1/2)
Constructing a 2D lattice from a series of m 1D Ising chains.
The total interaction energy
The first term: the interactions between the layers of the lattice.
The second term: the internal energy within the jth layer.
The partition function
Solution summarization
The solution for the internal energy.
Recasting into the alternative form.
The first term is identical to that for the one-dimensional situation.
The second term takes into account the cooperativity arising in the two-dimensional system.
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11. 3.5 The Ising Model in Two Dimensions(2/2)
In the vicinity of the critical point
The critical temperature corresponds to which yields the critical value
The spontaneous magnetization for a two-dimensional Ising lattice
The nonzero for this quantity, the long-range order parameter for a ferromagnet, is evidence that a continuous phase transition occurs in this system
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