Hamiltonians

In addition to states, we provide some convenience classes to represent quantum Hamiltonians acting on composite quantum systems. These Hamiltonians can be constant, time-dependent or translationally invariant. They can be converted to matrices, tensors or matrix-product operators.

The basic class is the NNHamiltonian, an abstract object representing a sum of nearest-neighbor operators \(H = \sum_{i=0}^{N-2} h_{i,i+1}\) where each \(h_{i,i+1}\) acts on a different, consecutive pair of quantum objects. This class is extended by different convenience classes that simplify the construction of such models, or provide specific, well-known ones:

NNHamiltonian(size)	Abstract class representing a Hamiltonian for a 1D system with nearest-neighbor interactions.
ConstantNNHamiltonian(size, dimension)	Nearest-neighbor 1D Hamiltonian with constant terms.
ConstantTIHamiltonian(size[, interaction, ...])	Translationally invariant Hamiltonian with constant nearest-neighbor interactions.
HeisenbergHamiltonian(size[, field])	Nearest-neighbor Hamiltonian with constant Heisenberg interactions over 'size' S=1/2 spins.

As example of use, we can inspect the HeisenbergHamiltonian class, which creates the model \(\sum_i \vec{S}_i\cdot\vec{S}_{i+1}\) more or less like this:

>>> SdotS = 0.25 * (sp.kron(σx, σx) + sp.kron(σy, σy) + sp.kron(σz, σz))
>>> ConstantTIHamiltonian(size, SdotS)