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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.