yomichi / SpinMonteCarlo.jl

Markov chain Monte Carlo solver for lattice spin systems implemented in Julialang

SpinMonteCarlo.jl

Markov chain Monte Carlo solver for finite temperature problem of lattie spin system implemented by Julia language.

Online manual

Install

Pkg> add SpinMonteCarlo

Simple example

The following program calculates temperature v.s. specific heat of the ferromagnetic Ising model on a $16\times 16$ square lattice by Swendsen-Wang algorithm.

using SpinMonteCarlo
using Printf

const model = Ising
const lat = "square lattice"
const L = 16
const update = SW_update!

const Tc = 2.0/log1p(sqrt(2))
const Ts = Tc*range(0.85, stop=1.15, length=31)
const MCS = 8192
const Therm = MCS >> 3

for T in Ts
    params = Dict{String,Any}("Model"=>model, "Lattice"=>lat,
                              "L"=>L, "T"=>T, "J"=>1.0,
                              "Update Method"=>update,
                              "MCS"=>MCS, "Thermalization"=>Therm,
                             )
    result = runMC(params)
    @printf("%f %.15f %.15f\n",
            T, mean(result["Specific Heat"]), stderror(result["Specific Heat"]))
end

Implemented

Model

• Classical spin model
  • Ising model
  • Q state Potts model
    • order parameter defined as $M = n_1(Q-1)/Q - (1-n_1)/Q$, where $n_1$ is the number density of $q=1$ spins.
  • XY model
  • Q state Clock model
  • AshkinTeller model
• Quantum spin model
  • spin-S QuantumXXZ model
    • $\mathcal{H} = \sum_{ij} [ Jz_{ij} S_i^z S_j^z + \frac{Jxy_{ij}}{2} (S_i^+ S_j^- + S_i^-S_j^+) ] - \sum_i \Gamma_i S_i^x$

Lattice

• chain lattice
  • L
• bond-alternating chain lattice
  • L
• square lattice
  • L * W
• J1J2 square lattice
  • L * W
• triangular lattice
  • L * W
• cubic lattice
  • L * W * H
• fully connected graph
  • N

Update algorithm

• Classical spin
  • local_update!
  • SW_update!
  • Wolff_update!
• Quantum spin
  • loop_update!

Physical quantities

• Ising, Potts
  • Magnetization
    • $\braket{m} := \braket{ M_\text{total}/N_\text{site} }$
  • |Magnetization|
    • $\braket{|m|} := \braket{|M_\text{total}/N_\text{site}|}$
  • Magnetization^2
    • $\braket{m^2} := \braket{(M_\text{total}/N_\text{site})^2}$
  • Magnetization^4
    • $\braket{m^4} := \braket{(M_\text{total}/N_\text{site})^4 }$
  • Binder Ratio
    • $U_{4,2} := \braket{m^4}/\braket{m^2}^2$
  • Susceptibility
    • $\chi := \partial_h \braket{m} = (N/T)(\braket{m^2} - \braket{m}^2)$
  • Connected Susceptibility
    • $\chi := (N/T)(\braket{m^2} - \braket{|m|}^2)$
  • Energy
    • $E := \braket{\mathcal{H}} = \braket{E_\text{total}}/N_\text{site}$
  • Energy^2
    • $E^2 := \braket{\mathcal{H}^2}$
  • Specific Heat
    • $C := \partial_\beta \braket{\mathcal{H}} = (N/T^2)(\braket{\mathcal{H}^2} - \braket{\mathcal{H}}^2)$
• XY, Clock
  • |Magnetization|
  • |Magnetization|^2
  • |Magnetization|^4
  • Binder Ratio
  • Susceptibility
  • Connected Susceptibility
  • Magnetization x
  • |Magnetization x|
  • Magnetization x^2
  • Magnetization x^4
  • Binder Ratio x
  • Susceptibility x
  • Connected Susceptibility x
  • Magnetization y
  • |Magnetization y|
  • Magnetization y^2
  • Magnetization y^4
  • Binder Ratio y
  • Susceptibility y
  • Connected Susceptibility y
  • Helicity Modulus x
  • Helicity Modulus y
  • Energy
  • Energy^2
  • Specific Heat
• QuantumXXZ
  • Magnetization
    • $\braket{m} := \braket{\sum_i S_i^z } / N_\text{site}$
  • Magnetization^2
    • $\braket{m^2}:= \braket{(\sum_i S_i^z)^2 } / N_\text{site}^2$
  • Magnetization^4
    • $\braket{m^4}:= \braket{(\sum_i S_i^z)^4 } / N_\text{site}^4$
  • Binder Ratio
    • $U_{4,2} := \braket{m^4}/\braket{m^2}^2$
  • Susceptibility
    • $\chi := \partial_h \braket{m} = (N/T)(\braket{m^2} - \braket{m}^2)$
  • Energy
    • $E := \braket{\mathcal{H}}$
  • Energy^2
    • $E^2 := \braket{\mathcal{H}^2}$
  • Specific Heat
    • $C := \partial_\beta \braket{\mathcal{H}} = (N/T^2)(\braket{\mathcal{H}^2} - \braket{\mathcal{H}}^2)$

Future work

• Model
  • Classical model
    • Heisenberg model
    • antiferro interaction
    • magnetic field
  • Quantum model
    • SU(N) model
• Update Method
  • worm algorithm
• Others
  • random number parallelization
    • NOTE: parameter parallelization can be realized simply by using @parallel for or pmap.
  • write algorithmic note
    • especially, Foutuin-Kasteleyn representaion and improved estimators

Author

Yuichi Motoyama, the University of Tokyo, 2016-

This package is distributed under the MIT license.