Public interfaces

runMC(param::Parameter)
runMC(params::AbstractArray{Parameter}
      ;
      parallel::Bool=false,
      autoID::Bool=true)

Runs Monte Carlo simulation(s) and returns calculated observables.

Restart

If a checkpoint file named "$(param["Checkpoint Filename Prefix"])_$(param["ID"]).dat" exists and param["Checkpoint Interval"] > 0.0, runMC loads this file and restarts the pending simulation. NOTE: Restart will fail if the version or the system image of julia change (see the doc of Serialization.serialize ).

Keyward arguments

• autoID: If true, "ID"s will be set (overwritten) as params[i]["ID"] = i.
• parallel: If true, runs simulations in parallel (uses pmap instead of map).

Required keys in param

• "Model"
• "Lattice"
• "Update Method"

Optional keys in param

• "MCS": The number of Monte Carlo steps after thermalization
  • Default: 8192
• "Thermalization": The number of Monte Carlo steps for thermalization
  • Default: MCS>>3
• "Seed": The initial seed of the random number generator, MersenneTwister
  • Default: determined randomly (see the doc of Random.seed!)
• "Checkpoint Filename Prefix": See the "Restart" section.
  • Default: "cp"
• "ID": See the "Restart" section.
  • Default: 0
• "Checkpoint Interval": Time interval between writing checkpoint file in seconds.
  • Default: 0.0, this means that NO checkpoint file will be loaded and saved.

dim(lat::Lattice)
dim(model::Model)

Returns the dimension of lattice.

size(lat::Lattice, [dim::Integer])
size(model::Model, [dim::Integer])

Returns the size of lattice.

sites(lat::Lattice, [sitetype::Integer])
sites(model::Model, [sitetype::Integer])

Returns an iterator over sites with sitetype (if omitted, over all sites)

bonds(lat::Lattice, [bondtype::Integer])
bonds(model::Model, [bondtype::Integer])

Returns an iterator over bonds with bondtype (if omitted, over all bonds)

numsites(lat::Lattice)
numsites(model::Model)

Returns the number of all sites.

numsites(lat::Lattice, sitetype::Integer)
numsites(model::Model, sitetype::Integer)

Returns the number of sitetype sites.

numbonds(lat::Lattice)
numbonds(model::Model)

Returns the number of all bonds.

numbonds(lat::Lattice, bondtype::Integer)
numbonds(model::Model, bondtype::Integer)

Returns the number of bondtype bonds.

neighbors(site::Site)
neighbors(lat::Lattice, site::Integer)
neighbors(model::Model, site::Integer)

Returns the index pairs of neighbor sites and bonds of site.

neighborsites(site::Site)
neighborsites(lat::Lattice, site::Integer)
neighborsites(model::Model, site::Integer)

Returns the indecies of neighbor sites of site.

neighborbonds(site::Site)
neighborbonds(lat::Lattice, site::Integer)
neighborbonds(model::Model, site::Integer)

Returns the indecies of neighbor bonds of site.

source(bond::Bond)
source(lat::Lattice, bond::Integer)
source(model::Model, bond::Integer)

Returns the source site index of bond.

target(bond::Bond)
target(lat::Lattice, bond::Integer)
target(model::Model, bond::Integer)

Returns the target site of bond.

sitetype(site::Site)
sitetype(lat::Lattice, site::Integer)
sitetype(model::Model, site::Integer)

Returns the type of site

bondtype(bond::Bond)
bondtype(lat::Lattice, bond::Integer)
bondtype(model::Model, bond::Integer)

Returns the type of bond.

sitecoordinate(site::Site)
sitecoordinate(lat::Lattice, site::Integer)
sitecoordinate(model::Model, site::Integer)

Returns the coordinate of the site in the Cartesian system

bonddirection(bond::Bond)
bonddirection(lat::Lattice, bond::Integer)
bonddirection(model::Model, bond::Integer)

Returns the unnormalized direction of the bond as vector in the Cartesian system

cellcoordinate(site::Site)
cellcoordinate(lat::Lattice, site::Integer)
cellcoordinate(model::Model, site::Integer)

Returns the coordinate of the cell including site in the Lattice system

Ising model with energy $E = -\sum_{ij} J_{ij} \sigma_i \sigma_j$, where $\sigma_i$ takes value of 1 (up spin) or -1 (down spin).

Q state Potts model with energy $E = -\sum_{i,j} \delta_{\sigma_i, \sigma_j}$, where $\sigma_i$ takes an integer value from $1$ to $Q$ and $\delta$ is a Kronecker's delta. Order parameter (total magnetization) is defined as \begin{equation} M = \frac{Q-1}{Q}N1 - \frac{1}{Q}(N-N1), \end{equation} where $N$ is the number of sites and $N_1$ is the number of $\sigma=1$ spins.

Q state clock model with energy $E = -\sum_{ij} J_{ij} \cos(\theta_i - \theta_j)$, where $\theta_i = 2\pi \sigma_i/Q$ and $\sigma_i$ takes an integer value from $1$ to $Q$.

XY model with energy $E = -\sum_{ij} J_{ij} \cos(\theta_i - \theta_j)$, where $\theta_i = 2\pi \sigma_i$ and $\sigma_i \in [0, 1)$.

AshkinTeller model with energy $E = -\sum_{ij} J^\sigma_{ij} \sigma_i \sigma_j + J^\tau_{ij} \tau_i \tau_j + K_{ij} \sigma_i \sigma_j \tau_i \tau_j$, where $\sigma_i$ and $\tau_i$ takes value of 1 (up spin) or -1 (down spin).

Spin-$S$ XXZ model represented as the following Hamiltonian,

$\mathcal{H} = \sum_{i,j} \left[ J_{ij}^z S_i^z S_j^z + \frac{J_{ij}^{xy}}{2} (S_i^+ S_j^- + S_i^-S_j^+) \right] - \sum_i \Gamma_i S_i^x,$

where $S^x, S^y, S^z$ are $x, y$ and $z$ component of spin operator with length $S$, and $S^\pm \equiv S^x \pm iS^y$ are ladder operator. A state is represented by a product state (spins at $\tau=0$) of local $S^z$ diagonal basis and an operator string (perturbations). A local spin with length $S$ is represented by a symmetrical summation of $2S$ sub spins with length $1/2$.

local_update!(model, param)

Updates spin configuration by local spin flip and Metropolice algorithm

SW_update!(model, param::Parameter)

Updates spin configuration by Swendsen-Wang algorithm

Wolff_update!(model, param::Parameter)

Updates spin configuration by Wolff algorithm

loop_update!(model, param::Parameter)

Updates spin configuration by loop algorithm

loop_update!(model, param::Parameter)
loop_update!(model, T::Real,
             Jz::AbstractArray,
             Jxy::AbstractArray,
             Gamma:AbstractArray)

Updates spin configuration by loop algorithm under the temperature T = param["T"] and coupling constants Jz, Jxy and transverse field Gamma

simple_estimator(model::Ising, T::Real, Js::AbstractArray)

Returns the following observables as Dict{String, Any}

Observables

• "Energy"
  • energy density
• "Energy^2"
  • square of energy density
• "Magnetization"
  • magnetization density
• "|Magnetization|"
  • absolute value of magnetization density
• "Magnetization^2"
  • square of magnetization density
• "Magnetization^4"
  • quadruple of magnetization density

simple_estimator(model::Clock, T::Real, Js::AbstractArray)
simple_estimator(model::XY, T::Real, Js::AbstractArray)

Returns the following observables as Dict{String, Any}

Observables

• "Energy"
  • Energy per spin (site)
• "Energy^2"
• "|Magnetization|"
  • Absolute value of total magnetization per spin (order paremeter)
• "|Magnetization|^2"
• "|Magnetization|^4"
• "Magnetization x"
  • x component of total magnetization per spin (order paremeter)
• "|Magnetization x|"
• "Magnetization x^2"
• "Magnetization x^4"
• "Magnetization y"
  • y component of total magnetization per spin (order paremeter)
• "|Magnetization y|"
• "Magnetization y^2"
• "Magnetization y^4"
• "Helicity Modulus x"
• "Helicity Modulus y"

simple_estimator(model::AshkinTeller, T::Real, Jsigma, Jtau, K)

Returns the following observables as Dict{String, Any}

Observables

• "Energy"
  • energy density
• "Energy^2"
  • square of energy density
• "|Magnetization|"
  • absolute value of magnetization density, $|m| = \sqrt{ (\sum_i \sigma_i )^2 + (\sum_i \tau_i)^2 } / N$
• "|Magnetization|^2"
  • square of magnetization density
• "|Magnetization|^4"
  • quadruple of magnetization density
• "Magnetization sigma"
  • magnetization density (sigma spin)
• "|Magnetization sigma|"
• "Magnetization sigma^2"
• "Magnetization sigma^4"
• "Magnetization tau"
  • magnetization density (tau spin)
• "|Magnetization tau|"
• "Magnetization tau^2"
• "Magnetization tau^4"

improved_estimator(model::Ising, T::Real, Js::AbstractArray, sw::SWInfo)

Returns the following observables as Dict{String, Any} using cluster information sw

Observables

• "Energy"
  • energy density
• "Energy^2"
  • square of energy density
• "Magnetization"
  • magnetization density
• "|Magnetization|"
  • absolute value of magnetization density
• "Magnetization^2"
  • square of magnetization density
• "Magnetization^4"
  • quadruple of magnetization density
• "Clustersize^2"
  • $\sum_c r_c^2$, where $r_c$ is the size density of $c$-th cluster
• "Clustersize^4"
  • $\sum_c r_c^4$
• "Clustersize^2 Clustersize^2"
  • $\sum_{c\ne c'} r_c^2 r_{c'}^2$

improved_estimator(model::Potts, T::Real, Js::AbstractArray, sw::SWInfo)

Returns the following observables as Dict{String, Any} using cluster information sw

Observables

• "Energy"
  • energy density
• "Energy^2"
  • square of energy density
• "Magnetization"
  • magnetization density
• "|Magnetization|"
  • absolute value of magnetization density
• "|Magnetization|^2"
  • square of magnetization density
• "|Magnetization|^4"
  • quadruple of magnetization density
• "Clustersize^2"
  • $\sum_c r_c^2$, where $r_c$ is the size density of $c$-th cluster
• "Clustersize^4"
  • $\sum_c r_c^4$
• "Clustersize^2 Clustersize^2"
  • $\sum_{c\ne c'} r_c^2 r_{c'}^2$

improved_estimator(model::QuantumXXZ, T::Real, Js::AbstractArray, uf::UnionFind)

Returns the following observables as Dict{String, Any} using loop information uf

Observables

• "Sign"
  • Sign of the weight function
• "Sign * Energy"
  • Energy per spin (site)
• "Sign * Energy^2"
• "Sign * Magnetization"
  • Total magnetization (Sz) per spin (site)
• "Sign * |Magnetization|"
• "Sign * Magnetization^2"
• "Sign * Magnetization^4"

gen_snapshot!(model, T, [N=1])

generate and return N snapshots (spin configuration).

Arguments

• model::Model : model
• T::Real : temperature
• N::Integer=1 : the number of snapshots to be generated
• MCS::Integer=8192 : the number of Monte Carlo steps followed by snapshot generation

gensave_snapshot!(io, model, T, [N=1])
gensave_snapshot!(filename, model, T, [N=1])

generate and write N snapshots into io or filename.

Arguments

• io::IO : output stream where snapshots will be written
• filename::String : the name of file where snapshots will be written
• model::Model : model to be simulated
• T::Real : temperature
• N::Integer=1 : the number of snapshots to be generated
• MCS::Integer=1 : the number of Monte Carlo steps followed by snapshot generation
• sep::String=" " : field separator of snapshot

load_snapshot(io, splitter)
load_snapshot(filename, splitter)

load snapshot from io or filename. splitter is the set of field splitter and will be passed Base.split as the second argument (see the doc of Base.split).

Input parameter of simulation

convert_parameter(model, param)

Generates arguments of updater and estimator.

Example

julia> model = Ising(chain_lattice(4));

julia> p = convert_parameter(model, Parameter("J"=>1.0))
(1.0, [1.0, 1.0]) # T and Js

julia> p = convert_parameter(model, Parameter("J"=>[1.5, 0.5]))
(1.0, [1.5, 0.5]) # J can take a vector whose size is `numbondtypes(model)`

julia> model.spins
4-element Array{Int64,1}:
  1
  1
  1
 -1

julia> local_update!(model, p...);

julia> model.spins
4-element Array{Int64,1}:
  1
  1
 -1
 -1

convert_parameter(model::Ising, param::Parameter)

Keynames:

• "T": a scalar (default: 1.0).
• "J": a vector with numbondtypes(model) elements (default: 1.0).

convert_parameter(model::Potts, param::Parameter)

Keynames:

• "T": a scalar (default: 1.0).
• "J": a vector with numbondtypes(model) elements (default: 1.0).

convert_parameter(model::Clock, param::Parameter)

Keynames:

• "T": a scalar (default: 1.0).
• "J": a vector with numbondtypes(model) elements (default: 1.0).

convert_parameter(model::XY, param::Parameter)

Keynames:

• "T": a scalar (default: 1.0).
• "J": a vector with numbondtypes(model) elements (default: 1.0).

convert_parameter(model::AshkinTeller, param::Parameter)

Keynames:

• "T": a scalar (default: 1.0).
• "Jsigma": a vector with numbondtypes(model) elements (default: 1.0).
• "Jtau": a vector with numbondtypes(model) elements (default: 1.0).
• "K": a vector with numbondtypes(model) elements (default: 0.0).

convert_parameter(model::QuantumXXZ, param::Parameter)

Keynames:

• "T": a scalar (default: 1.0).
• "Jz": a vector with numbondtypes(model) elements (default: 1.0).
• "Jxy": a vector with numbondtypes(model) elements (default: 1.0).
• "Gamma": a vector with numsitetypes(model) elements (default: 0.0).