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.