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Generator of operators

In previous pages we have introduced the essential elements in detail of a quantum lattice system in the unitcell description framework. In this page, we will discuss how such elements can be incorporated to get the operator representation of the lattice Hamiltonian.

Operator-formed lattice Hamiltonians

The essential elements to obtain an operator-formed lattice Hamiltonian are 1) terms, 2) bonds and 3) Hilbert space.

Accordingly, OperatorGenerator is the type to incorporate all these elements:

OperatorGenerator(terms::Tuple{Vararg{Term}}, bonds::Vector{<:Bond}, hilbert::Hilbert)

Then, the operator representation of the lattice Hamiltonian can be obtained by the expand function:

expand(gen::OperatorGenerator)

which is based on the expansion of terms introduced in the last section of the previous page Couplings among different degrees of freedom.

Now, let's go back to the example proposed in the page of Introduction:

julia> lattice = Lattice([zero(Sym)], [one(Sym)]);
julia> hilbert = Hilbert(site=>Fock{:f}(1, 2) for site=1:length(lattice));
julia> t = Hopping(:t, symbols("t", real=true), 1);
julia> U = Hubbard(:U, symbols("U", real=true));
julia> operators = expand(OperatorGenerator((t, U), bonds(lattice, 1), hilbert))Operators with 6 Operator
  Operator(t, CompositeIndex(Index(2, FID{:f}(1, 1//2, 2)), SymPyCore.Sym{PyCall.PyObject}[1], SymPyCore.Sym{PyCall.PyObject}[0]), CompositeIndex(Index(1, FID{:f}(1, 1//2, 1)), SymPyCore.Sym{PyCall.PyObject}[0], SymPyCore.Sym{PyCall.PyObject}[0]))
  Operator(U, CompositeIndex(Index(2, FID{:f}(1, 1//2, 2)), SymPyCore.Sym{PyCall.PyObject}[1], SymPyCore.Sym{PyCall.PyObject}[0]), CompositeIndex(Index(2, FID{:f}(1, 1//2, 1)), SymPyCore.Sym{PyCall.PyObject}[1], SymPyCore.Sym{PyCall.PyObject}[0]), CompositeIndex(Index(2, FID{:f}(1, -1//2, 2)), SymPyCore.Sym{PyCall.PyObject}[1], SymPyCore.Sym{PyCall.PyObject}[0]), CompositeIndex(Index(2, FID{:f}(1, -1//2, 1)), SymPyCore.Sym{PyCall.PyObject}[1], SymPyCore.Sym{PyCall.PyObject}[0]))
  Operator(U, CompositeIndex(Index(1, FID{:f}(1, 1//2, 2)), SymPyCore.Sym{PyCall.PyObject}[0], SymPyCore.Sym{PyCall.PyObject}[0]), CompositeIndex(Index(1, FID{:f}(1, 1//2, 1)), SymPyCore.Sym{PyCall.PyObject}[0], SymPyCore.Sym{PyCall.PyObject}[0]), CompositeIndex(Index(1, FID{:f}(1, -1//2, 2)), SymPyCore.Sym{PyCall.PyObject}[0], SymPyCore.Sym{PyCall.PyObject}[0]), CompositeIndex(Index(1, FID{:f}(1, -1//2, 1)), SymPyCore.Sym{PyCall.PyObject}[0], SymPyCore.Sym{PyCall.PyObject}[0]))
  Operator(t, CompositeIndex(Index(2, FID{:f}(1, -1//2, 2)), SymPyCore.Sym{PyCall.PyObject}[1], SymPyCore.Sym{PyCall.PyObject}[0]), CompositeIndex(Index(1, FID{:f}(1, -1//2, 1)), SymPyCore.Sym{PyCall.PyObject}[0], SymPyCore.Sym{PyCall.PyObject}[0]))
  Operator(t, CompositeIndex(Index(1, FID{:f}(1, -1//2, 2)), SymPyCore.Sym{PyCall.PyObject}[0], SymPyCore.Sym{PyCall.PyObject}[0]), CompositeIndex(Index(2, FID{:f}(1, -1//2, 1)), SymPyCore.Sym{PyCall.PyObject}[1], SymPyCore.Sym{PyCall.PyObject}[0]))
  Operator(t, CompositeIndex(Index(1, FID{:f}(1, 1//2, 2)), SymPyCore.Sym{PyCall.PyObject}[0], SymPyCore.Sym{PyCall.PyObject}[0]), CompositeIndex(Index(2, FID{:f}(1, 1//2, 1)), SymPyCore.Sym{PyCall.PyObject}[1], SymPyCore.Sym{PyCall.PyObject}[0]))

Note to run the above codes, SymPy.Sym and SymPy.symbols should be imported first.

The LaTeX formatted outputs will be discussed in the page of LaTeX formatted outputs.

Here, we have two comments:

  • In the construction of OperatorGenerator, a vector of Bonds other than a Lattice is required. Indeed, it is more flexible to control the generated operators by passing a vector of Bonds. By restricting the range of the input bonds, a certain subset of the lattice Hamiltonian can be easily obtained, which is sometimes useful in some quantum many-body algorithms.
  • The Hilbert space is essential because in general not all information of the local generators of a concrete quantum lattice system is contained in the terms. Remind: 1) the statistics of particles or the total spin of local spins is usually omitted in the coupling pattern of a term, and 2) the total number of local orbital/spin degrees of freedom, or the dimension of lattice vibrations, cannot be determined solely by the information provided by the coupling pattern of a term. Therefore, in order to obtain the lattice Hamiltonian, such incomplete information must be supplemented by the Hilbert space.

Parameters tuning of lattice Hamiltonians

It is customary to tune the parameters of a lattice Hamiltonian. This can be achieved by the update! function exported by this package:

update!(gen::OperatorGenerator; termid₁=termvalue₁, termid₁=termvalue₂, ...)

Here, termidᵢ (i = 1, 2, ...) is the id of a term in the lattice Hamiltonian, and termvalueᵢ is the new overall coefficient of this term.

The parameters of the terms in an OperatorGenerator can be requested by the type Parameters (a type alias of NamedTuple):

Parameters(gen::OperatorGenerator) -> Parameters

Let's see an example.

julia> lattice = Lattice([0.0], [1.0]);

julia> hilbert = Hilbert(site=>Fock{:f}(1, 2) for site=1:length(lattice));

julia> t = Hopping(:t, 1.0, 1);

julia> gen = OperatorGenerator((t,), bonds(lattice, 1), hilbert);

julia> Parameters(gen)
(t = 1.0,)

julia> expand(gen)
Operators with 4 Operator
  Operator(1.0, CompositeIndex(Index(2, FID{:f}(1, -1//2, 2)), [1.0], [0.0]), CompositeIndex(Index(1, FID{:f}(1, -1//2, 1)), [0.0], [0.0]))
  Operator(1.0, CompositeIndex(Index(1, FID{:f}(1, -1//2, 2)), [0.0], [0.0]), CompositeIndex(Index(2, FID{:f}(1, -1//2, 1)), [1.0], [0.0]))
  Operator(1.0, CompositeIndex(Index(2, FID{:f}(1, 1//2, 2)), [1.0], [0.0]), CompositeIndex(Index(1, FID{:f}(1, 1//2, 1)), [0.0], [0.0]))
  Operator(1.0, CompositeIndex(Index(1, FID{:f}(1, 1//2, 2)), [0.0], [0.0]), CompositeIndex(Index(2, FID{:f}(1, 1//2, 1)), [1.0], [0.0]))

julia> update!(gen; t=2.0);

julia> Parameters(gen)
(t = 2.0,)

julia> expand(gen)
Operators with 4 Operator
  Operator(2.0, CompositeIndex(Index(2, FID{:f}(1, -1//2, 2)), [1.0], [0.0]), CompositeIndex(Index(1, FID{:f}(1, -1//2, 1)), [0.0], [0.0]))
  Operator(2.0, CompositeIndex(Index(1, FID{:f}(1, -1//2, 2)), [0.0], [0.0]), CompositeIndex(Index(2, FID{:f}(1, -1//2, 1)), [1.0], [0.0]))
  Operator(2.0, CompositeIndex(Index(2, FID{:f}(1, 1//2, 2)), [1.0], [0.0]), CompositeIndex(Index(1, FID{:f}(1, 1//2, 1)), [0.0], [0.0]))
  Operator(2.0, CompositeIndex(Index(1, FID{:f}(1, 1//2, 2)), [0.0], [0.0]), CompositeIndex(Index(2, FID{:f}(1, 1//2, 1)), [1.0], [0.0]))