HeisenbergModel class

class HeisenbergModel(N, J)

A class that simulates the 1D Heisenberg model, a fundamental model in quantum mechanics for understanding magnetic properties in materials.

Parameters:

• N (int) – The number of spins in the 1D Heisenberg chain.

• J (float) – The exchange interaction strength between neighboring spins.

N

The number of spins in the Heisenberg chain.

J

The coupling constant representing the strength of the exchange interaction.

Sx

The Pauli X matrix used to represent spin interactions along the x-axis.

Sy

The Pauli Y matrix used to represent spin interactions along the y-axis.

Sz

The Pauli Z matrix used to represent spin interactions along the z-axis.

I

The identity matrix representing no spin interaction.

build_hamiltonian()

Constructs the Hamiltonian matrix for the Heisenberg model using the tensor product of Pauli matrices to represent spin interactions.

Returns:

The Hamiltonian matrix of the system.

Return type:

torch.Tensor

solve_eigenvalues()

Diagonalizes the Hamiltonian matrix to find the eigenvalues and eigenvectors, which represent the energy levels and the corresponding quantum states of the system.

Returns:

A tuple containing an array of eigenvalues and a matrix of eigenvectors.

Return type:

(torch.Tensor, torch.Tensor)

plot_energy_levels(eigenvalues)

Generates a plot of the energy levels of the Heisenberg model.

Parameters:

eigenvalues (torch.Tensor) – The array of energy levels obtained from diagonalizing the Hamiltonian.

Example Usage

To instantiate the HeisenbergModel, calculate the energy levels, and plot them, you can use the following code:

model = HeisenbergModel(N=4, J=1)
eigenvalues, eigenvectors = model.solve_eigenvalues()
model.plot_energy_levels(eigenvalues)

The output will be a matplotlib plot displaying the energy levels of a 1D Heisenberg chain with 4 spins and an exchange interaction strength of 1.