Question

The energy of the Ising model. Derive an expression for the energy of the Ising model from the partition function.

Short answer

\( \langle E \rangle = - \frac{\partial \ln Z}{\partial \beta} \)

Step-by-step solution

- Understand the Hamiltonian

The Ising model Hamiltonian is given by \( H = -J \sum_{\langle i, j \rangle} s_i s_j \), where \(J\) is the interaction energy between neighboring spins and \(s_i\) represents the spin at site \(i\).

- Write the Partition Function

The partition function is defined as \( Z = \sum_{\{s_i\}} \exp(-\beta H) \), where \( \{s_i\} \) denotes the sum over all spin configurations and \( \beta = \frac{1}{k_B T} \) (with \( k_B \) being Boltzmann's constant and \( T \) the temperature).

- Define the Expectation Value of Energy

The expectation value of the energy is given by \( \langle E \rangle = \frac{1}{Z} \sum_{\{s_i\}} H \exp(-\beta H) \).

- Express the Sum in Terms of Partition Function

Rewrite the expression for the expectation value of energy using the partition function: \( \langle E \rangle = \frac{-1}{Z} \frac{\partial Z}{\partial \beta} \).

- Differentiate the Partition Function

Differentiate \( Z \) with respect to \( \beta \): \( \frac{\partial Z}{\partial \beta} = \sum_{\{s_i\}} H \exp(-\beta H) \), using the chain rule.

- Substitute Back

Substitute the differentiated partition function back into the expression for \( \langle E \rangle \): \( \langle E \rangle = - \frac{1}{Z} \frac{\partial Z}{\partial \beta} \). Simplifying this, we get \( \langle E \rangle = - \frac{\partial \ln Z}{\partial \beta} \).

Key concepts

Hamiltonian

The Hamiltonian in the Ising model is an essential concept that defines the energy dynamics of a system of spins. It is expressed as \( H = -J \sum_{\langle i, j \rangle} s_i s_j \). Let's break this down:

• Interaction energy \(J\): This is the energy associated with the interaction between neighboring spins. It determines whether the spins prefer to align (ferromagnetic interaction) or anti-align (antiferromagnetic interaction).
• Spins \(s_i\) and \(s_j\): These represent the spin states at sites \(i\) and \(j\). Each spin can be either +1 or -1, corresponding to up or down spin states.
• Sum over neighbors: The summation \( \sum_{\langle i, j \rangle} \) indicates that we sum over all pairs of neighboring spins.

In simple terms, the Hamiltonian captures how spins interact with each other in the Ising model. The negative sign ensures that spin alignment (same direction) lowers the energy when \(J\) is positive.

Partition Function

The partition function \(Z\) is a fundamental quantity in statistical mechanics that encapsulates all the possible states of a system. For the Ising model, it is written as \( Z = \sum_{\{s_i\}} \exp(-\beta H) \), where:

• Sum over all configurations: \(\{s_i\}\) denotes all possible configurations of the spins. Since each spin can be either +1 or -1, there are \( 2^N \) possible configurations for \(N\) spins.
• Boltzmann factor: \(\exp(-\beta H)\) weighs the contribution of each configuration to the partition function based on its energy \(H\). Higher energy states contribute less due to the negative exponent.
• Temperature dependence: \(\beta = \frac{1}{k_B T}\) introduces the temperature dependence, with \( k_B \) being Boltzmann's constant and \(T\) the temperature. This relationship shows how thermal fluctuations affect the system.

The partition function is crucial because it allows us to calculate various thermodynamic quantities and the probabilities of different states. Essentially, it serves as a gateway to understanding the statistical properties of the Ising model.

Expectation Value

The expectation value of the energy \( \langle E \rangle \) in the Ising model provides the average energy of the system at equilibrium. It is given by \( \langle E \rangle = \frac{1}{Z} \sum_{\{s_i\}} H \exp(-\beta H) \). Here's a closer look:

• Average over configurations: The sum \(\sum_{\{s_i\}}\) denotes averaging over all possible spin configurations.
• Weighted by probability: Each configuration is weighted by \( H \exp(-\beta H) \), reflecting its likelihood based on its energy and the temperature.
• Normalization by \( Z \): Dividing by the partition function \(Z\) ensures that the probabilities sum to 1, providing a proper average.

Moreover, using the relationship with the partition function, we can express this expectation value as \( \langle E \rangle = - \frac{\partial \ln Z}{\partial \beta} \). This formulation is particularly useful because it links thermodynamic quantities directly to the partition function, making calculations more straightforward.