Simple code to generate ground state wavefunctions of the 2D Ising model using (Lancoz) exact diagonalization. The nearest-neighbour Hamiltonian is applied sequentially to each coupling in the lattice. This is done by noting that the full Hamiltonian is a sum of operator strings such as H = 1 * 1 * 1 * h * 1... 1 * 1 * 1 * 1 * h... We only actually store h1 1h, and perform the operation Hv by applying hv, which implements one of the operator strings, then cyclically permuting v so that h*v now implements the next one. This is done until all the cyclic permutations are exhausted. Periodic boundary conditions and translation invariance are both assumed.
For a 1D lattice, i.e.
_ _ _ _
1 2 3 4
we need to apply the permutation (1 2 3 4) for a full cycle: i.e. the action of the local hamiltonian on each of [1 2 3 4], [4 1 2 3], [3 4 1 2], and [2 3 4 1] needs to be summed. For a 2D lattice, say
_ _ _
1 4 7
_ _ _
2 5 8
_ _ _
3 6 9
(note Julia arrays are column-major), we need to do this for each row and once for each column of the lattice: i.e. (123)(4)(5)(6)(7)(8)(9) for a full cycle, then (456)..., then (789), then (147), (456), and (789). The present code handles this by first generating all of the necessary permutations in the arrays permuteidxs and permutevals, then applying a full cycle of each one.
The wavefunction is then saved to disk as an array with shape (2, 2, 2, ... N). Each index runs over the two spin states of a particular site.
As a quick test, with hfield=0 the energy density should be -J*dim, where dim is the dimensionality of the lattice.
Unlike Morningstar's code, no effort is made to find a compressed basis using symmetry structures (e.g. the wavefunction has 2^N elements). This makes the code much easier to write, but considerably less efficient; about 16 lattice sites is probably the most it can handle.
This code is a slight modification of Markus Hauru's implementation, which covers the 1D case. That code can be found at https://github.com/mhauru/MPS-in-Julia-minicourse/blob/master/meeting_01.ipynb.
To install dependencies, run dependencies.jl:
julia dependencies.jl
If errors occur, try updating Julia to v0.6.3 (the version in the Ubuntu repos did not work)