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The @mpoham macro

When dealing with (quasi-) one-dimensional systems that are defined by a sum of local operators, a convenient representation exists in terms of a sparse matrix product operator with an upper diagonal structure (MPOHamiltonian). The generation of such an object starting from a sum of local operators is facilitated by the macro @mpoham, which provides several syntactic sugar features.

MPSKitModels.@mpoham — Macro
@mpoham(block)

Specify a Matrix Product Operator that is represented by a sum of local operators.

This macro converts expressions of the form O{i...} to an operator acting on sites i... where O is an operator, and i can be an integer or a lattice point. The macro recognizes expressions of the following forms:

  • O{i...} to indicate local operators O acting on sites i...
  • -Inf:Inf, -∞:∞, -Inf:step:Inf, -∞:step:∞ to indicate infinite chains.
  • 1:L to indicate finite chains.
  • ∑ to represent sums.

Examples

H_ising = @mpoham sum(σᶻᶻ{i, i+1} + h * σˣ{i} for i in -Inf:Inf)
H_heisenberg = @mpoham ∑(sigma_exchange(){i,j} for (i,j) in nearest_neighbours(-∞:∞))
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Internally, the macro generates operators that have some knowledge of the lattice structure, through the following structures:

MPSKitModels.LocalOperator — Type
struct LocalOperator{T,G}

N-body operator acting on N sites, indexed through lattice points of type G. The operator is represented as a vector of MPOTens, instantiate_operatorors, each of which acts on a single site.

Fields

  • opp::Vector{T}: N-body operator represented by an MPO.
  • inds::Vector{G}: N site indices.
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