Two-Spin Ising Correlations

The Ising model describes interacting binary spins. In a simple two-spin system, the full partition function and resulting thermodynamic observables can be calculated exactly without approximations.

States and Energies

For two spins s_1, s_2 ∈ {-1, 1}, there are four possible microstates: (+,+), (+,-), (-,+), and (-,-). The energy of each state is given by the Hamiltonian:

E = -J * s₁ * s₂ - h * (s₁ + s₂)

where J is the coupling constant (J > 0 favors alignment, ferromagnetic) and h is the external magnetic field. To compute ensemble averages at temperature T (with k_B = 1), we use the Gibbs weights w = e^(-(E - E_(min))/T). Shifting by E_min prevents underflow and simplifies the T=0 limit.

Calculating observables at J=1, h=1, T=1

Let's calculate the properties for the ferromagnetic case with a positive external field:

State (+,+): E = -1*(1) - 1*(2) = -3
State (+,-): E = -1*(-1) - 1*(0) = 1
State (-,+): E = -1*(-1) - 1*(0) = 1
State (-,-): E = -1*(1) - 1*(-2) = 1

The minimum energy is E_min = -3. The shifted energies are 0, 4, 4, 4.

Z_shifted = e⁰ + e^-4 + e^-4 + e^-4 = 1 + 3e^-4 ≈ 1.0549469

Weights: w(+,+) = 1 / 1.0549 ≈ 0.9479; w(others) = e^-4 / 1.0549 ≈ 0.0173

The mean energy is E = -3*(0.9479) + 1*(3*0.0173) = -2.7916599. The magnetization per spin is the mean of (s₁+s₂)/2, giving 0.930553. The ground-state degeneracy is 1 (only the (+,+) state).

Exploring phase space and zero-temperature limit

To see how interactions affect the system, open the Two-Spin Ising Correlations module and try the following:

Set Coupling J to -1 (antiferromagnetic) and Field h to 0. Notice the ground-state degeneracy is 2, and magnetization per spin is 0.
Change Temperature to 0. The Heat capacity drops to 0, and Total susceptibility becomes exactly 0 (divergent if fluctuations remain, but here var(M)=0). Entropy reflects the log of the degeneracy.
Set Field h to 1 and observe the shift in probabilities towards the state aligned with the field.

Model Limitations

This is a finite two-spin system. It does not exhibit a true thermodynamic phase transition at finite temperature, which requires the thermodynamic limit (N → ∞). The variables presented (susceptibility, heat capacity) are calculated for the total system of two spins or per spin as explicitly defined, rather than an extensive macroscopic quantity.

Primary References

For an introduction to exactly solvable small systems in statistical mechanics, see Statistical Mechanics (Wolschin) .

Two-Spin Ising Correlations