HierarchicalLattices.jl

This library provides several functions to generate hierarchical lattices (Griffiths, Kaufman). The primary idea to do this is to define a transformation on a part of a graph and then iterate said transformation across all the subgraphs. This library utilizes Graphs.jl to construct graphs and MetaGraphs.jl to store spin metadata on the vertices and bond strength on the edges. At the moment, this library also includes implementations of the Metropolis and Wolff algorithms for simulating spin-1/2 Ising systems on these lattices, however this will be moved to a separate library in the future. Implementations and interfaces are in a nascent stage, thus expect breaking changes in both. At the moment, only the diamond hierarchical lattice has been implemented.

Code structure

For every lattice type, it's implementation will be stored in src/NameLattice. Any MCMC algorithm can also be stored in this directory. At the moment the DiamondLattice implements lattice generation methods in Lattice.jl

Usage

Warning
This documentaiton may be out of date if breaking commits are made after 29 May, 2023. Raise an issue in such a case.

Warning
This library currently has a plotting recipe for a MetaGraph, which means it can clash with another graph plotting library implementing the same. Use with caution.

There are two main data structures exported for the diamond lattice, the DiamondLattice and StackedDiamondLattice types. Refer to the doc strings for how to use each function. A typical use case of simulating a lattice can be as follows.

julia> using HierarchicalLattices

julia> order = 5; b = 2; initstate = :zero
:zero

julia> lattice = DiamondLattice(diamond_ising_lattice(order, b, initstate), order)
DiamondLattice(5, {684, 1024} undirected Int64 metagraph with Float64 weights defined by :weight (default weight 1.0), {684, 1024} undirected Int64 metagraph with Float64 weights defined by :weight (default weight 1.0), 1.0)

julia> ID = IsingData(lattice, Float64[], Float64[], Int64[])
IsingData(DiamondLattice(5, {684, 1024} undirected Int64 metagraph with Float64 weights defined by :weight (default weight 1.0), {684, 1024} undirected Int64 metagraph with Float64 weights defined by :weight (default weight 1.0), 1.0), Float64[], Float64[], Int64[])

julia> metropolis!(ID, 1000, 1.64)

julia> ID.lattice.initial_state.vprops
Dict{Int64, Dict{Symbol, Any}} with 684 entries:
  319 => Dict(:val=>1)
  185 => Dict(:val=>1)
  420 => Dict(:val=>1)
  525 => Dict(:val=>1)
  365 => Dict(:val=>1)
  638 => Dict(:val=>1)
  263 => Dict(:val=>1)
  422 => Dict(:val=>1)
  242 => Dict(:val=>1)

julia> ID.lattice.final_state.vprops
Dict{Int64, Dict{Symbol, Any}} with 684 entries:
  319 => Dict(:val=>1)
  185 => Dict(:val=>1)
  420 => Dict(:val=>-1)
  525 => Dict(:val=>1)
  365 => Dict(:val=>1)
  638 => Dict(:val=>1)
  263 => Dict(:val=>1)
  422 => Dict(:val=>1)
  242 => Dict(:val=>-1)

Similar usage for a StackedDiamondLattice can be seen (this time with the Wolff algorithm).

julia> using HierarchicalLattices

julia> order = 5; depth = 15; stackingweight = 1.5; initstate = :zero
:zero

julia> stackedlattice = StackedDiamondLattice(order, depth, stackingweight, initstate)
StackedDiamondLattice(5, 1.5, {10260, 24936} undirected Int64 metagraph with Float64 weights defined by :weight (default weight 1.0), {10260, 24936} undirected Int64 metagraph with Float64 weights defined by :weight (default weight 1.0), Dict(4986 => Dict(1.5 => [4302, 5670], 1.0 => [4789, 4845]), 7329 => Dict(1.5 => [6645, 8013], 1.0 => [6860, 6975]), 4700 => Dict(1.5 => [4016, 5384], 1.0 => [4137, 4250]), 4576 => Dict(1.5 => [3892, 5260], 1.0 => [4122, 4160]), 7144 => Dict(1.5 => [6460, 7828], 1.0 => [6845, 6950]), 6073 => Dict(1.5 => [5389, 6757], 1.0 => [5506, 5619]), 2288 => Dict(1.5 => [1604, 2972], 1.0 => [2054, 2128]), 1703 => Dict(1.5 => [1019, 2387], 1.0 => [1375, 1494]), 1956 => Dict(1.5 => [1272, 2640], 1.0 => [1400, 1492]), 8437 => Dict(1.5 => [7753, 9121], 1.0 => [8210, 8281])…))

julia> WD = WolffData(lattice = stackedlattice, magnetization_history=Float64[], internalenergy_history=Float64[], thermalization_steps=1000, saveinterval=20)
WolffData{StackedDiamondLattice}(StackedDiamondLattice(5, 1.5, {10260, 24936} undirected Int64 metagraph with Float64 weights defined by :weight (default weight 1.0), {10260, 24936} undirected Int64 metagraph with Float64 weights defined by :weight (default weight 1.0), Dict(4986 => Dict(1.5 => [4302, 5670], 1.0 => [4789, 4845]), 7329 => Dict(1.5 => [6645, 8013], 1.0 => [6860, 6975]), 4700 => Dict(1.5 => [4016, 5384], 1.0 => [4137, 4250]), 4576 => Dict(1.5 => [3892, 5260], 1.0 => [4122, 4160]), 7144 => Dict(1.5 => [6460, 7828], 1.0 => [6845, 6950]), 6073 => Dict(1.5 => [5389, 6757], 1.0 => [5506, 5619]), 2288 => Dict(1.5 => [1604, 2972], 1.0 => [2054, 2128]), 1703 => Dict(1.5 => [1019, 2387], 1.0 => [1375, 1494]), 1956 => Dict(1.5 => [1272, 2640], 1.0 => [1400, 1492]), 8437 => Dict(1.5 => [7753, 9121], 1.0 => [8210, 8281])…)), Float64[], Float64[], 20, 1000)

julia> wolff!(WD, 100, 1.64; showprogress = true)
Progress: 100%|████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| Time: 0:02:31 ( 0.15  s/it)
Progress: 100%|████████████████████████████████████████████████████████████████████████████████████████████████████████████████████████| Time: 0:00:14 ( 0.15  s/it)
([-29469.0, -29387.0, -29486.0, -29497.0, -29386.0], [10194.0, 10176.0, 10204.0, 10202.0, 10174.0])

The StackedDiamondLattice constructor can construct a MetaGraph that is stacking diamond lattices and provides you with a single resulting MetaGraph. This has the advantage that, for the most part, the algorithms do not change, and one can just implement the algorithms for a general MetaGraph with spin stored in the :val property and interaction strength stored in the edge weight (if nonexistent, then this is the default weight of the graph edges).