Hamiltonian

Hamiltonian of many-body system usually has the form of so called $k$-local Hamiltonian, which means each local hamiltonian only involves k bodies (bits).

for example, the Hamiltonian of Traverse Field Ising Model

\[H = \sum_i \sigma_x^i + \sum_{<i,j>}\sigma_z^i\sigma_z^j\]

More generally, a local Hamiltonian has the form

\[H = \sum_{l_1,l_2,\dots,l_{m_1} \in region(1)} H_{l_1, l_2, \dots, l_{m_1}} + \sum_{l_1, l_2, \dots, l_{m_2} \in region(2)} H_{l_1, l_2, \dots, l_{m_2}} + \dots + \sum_{l_1, l_2, \dots, l_{m_k} \in region(k)} H_{l_1, l_2, \dots, l_{m_k}}\]

where $l_i$ denotes the $i$th body/site/bit/... of the system, $region(k)$ denotes the $k$th region of the system, $m_k$ denotes the number of body/site/bit/... involves in this region (e.g for nearest neighbor m_k is 2).

Thus the form of hamiltonian can be define in a program without knowing the exact form of lattice/geometry.

we offer a macro @ham to accomplish this, e.g

We can define a J1J2 hamiltonian on arbitrary lattice by

julia> h = @ham sum(:nearest, S * S) + sum(:nextnearest, S * S)
ERROR: MethodError: no method matching QMTK.LocalHamiltonian(::Symbol, ::SparseMatrixCSC{Complex{Int64},Int64})
Closest candidates are:
  QMTK.LocalHamiltonian(!Matched::Type{R<:QMTK.AbstractRegion}, ::AbstractSparseArray{Tv,Ti,2} where Ti where Tv) where R<:QMTK.AbstractRegion at /home/travis/.julia/v0.6/QMTK/src/Hamiltonian/LocalHamiltonian.jl:15
  QMTK.LocalHamiltonian(!Matched::Type{T<:QMTK.SiteLabel}, ::AbstractSparseArray{Tv,Ti,2} where Ti where Tv) where T<:QMTK.SiteLabel at /home/travis/.julia/v0.6/QMTK/src/Hamiltonian/LocalHamiltonian.jl:17

The region is described by the type Region

Region
Nearest
NextNearest

we provide two shorthands for regions in @ham macro, one is :nearest denotes Region{1} and :nextnearest denotes Region{2}.

TODO:

allow use Region{N} in @ham
allow to parse Region{0} to traverse sites

Access the Hamiltonian

Only when the lattice is defined, the matrix form of a certain Hamiltonian is determined. Then we could access it.

We offer two ways to access the matrix form of a Hamiltonian.

The recommended way is to use iterators, all the instance of hamiltonian is callable with input of lattice and left-hands basis (site/bit/...).

julia> lattice = Chain(Fixed, 4)
QMTK.Chain{QMTK.Fixed}(4)

julia> lhs = Sites(Bit, 4)
4-element QMTK.Sites{QMTK.Bit,Float32,1}:
 0.0
 0.0
 0.0
 0.0

julia> for (val, rhs) in h(lattice, lhs)
           println(val, ", ",rhs)
       end
ERROR: UndefVarError: h not defined

It will return an iterator that traverse all non-zeros values of left-hands basis.

If you just want to use its matrix form rather than access part of its value, you can directly use mat

julia> mat(h, lattice)
ERROR: UndefVarError: mat not defined