BenchmarkFreeFermions.jl

 SingleParticleSpectrum(Tij::AbstractMatrix) -> ϵ::Vector{Float64}

Numerically diagonalize the hopping matrix Tij to get the single particle spectrum ϵ. Note the convention is H = - \sum_ij Tij c_i^dag c_j.

 EigenModes(Tij::AbstractMatrix{F}) -> ϵ::Vector{Float64}, V::Matrix{F}

Numerically diagonalize the hopping matrix Tij = - V diagm(ϵ) V' to get the single particle spectrum ϵ and the eigenvectors V. Note the convention is H = - \sum_ij Tij c_i^dag c_j and the decoupled Hamiltonian reads H = \sum_k ϵ_k f_k^dag f_k where c_i = \sum_k V[i,k] f_k.

 n_fermion(x::Real, β::Real) -> ::Float64
 n_fermion(x::Real, lsβ::AbstractVector{<:Real}) -> ::Vector{Float64}
  n_fermion(lsx::AbstractVector{<:Real}, β::Real) -> ::Vector{Float64}

The Fermi-Dirac distribution x -> 1 / (e^{βx} + 1).

 ParticleNumber(ϵ::Vector{Float64}, β::Real, μ::Real) -> Ntot::Float64

Return the total particle number Ntot = \sum_k nk with given chemical potential μ.

 LogPartition(ϵ::Vector{Float64}, β::Real, μ::Real) -> lnZ::Float64

Return the logarithm of the partition function.

 Energy(ϵ::Vector{Float64}, β::Real, μ::Real) -> E::Float64

Return the total inner energy E = \sum_k ϵk nk.

 FreeEnergy(ϵ::Vector{Float64}, β::Real, μ::Real) -> F::Float64

Return the free energy F = - lnZ/β + μN.

 Entropy(ϵ::Vector{Float64}, β::Real, μ::Real)) -> S::Float64

Return the thermal entropy S = β(E - F).

 SpecificHeat_μ(ϵ::Vector{Float64}, β::Real, μ::Real) -> C_μ::Float64

Return the fixed-chemical-potential specific heat C_μ = (∂(E - μN) / ∂T)_μ.

 SpecificHeat_N(ϵ::Vector{Float64}, β::Real, μ::Real) -> C_N::Float64

Return the fixed-particle-number specific heat C_N = (∂E / ∂T)_N.

 SolveChemicalPotential(ϵ::Vector{Float64},   
      β::Real,
      n::Real) -> μ::Float64

Return the chemical potential μ that leads to the given average particle number n ∈ (0, 1).

 GreenFunction(ξ::Vector{Float64}, V::Matrix{F}, β::Real; τ::Number=0.0) -> G::Matrix{F}
 GreenFunction(ξ::Vector{Float64}, V::Matrix{F}, β::Real,
 i::Int64, j::Int64;
 τ::Number=0.0,
 reverse::Bool = false) -> Gij::F

Compute the Green's funtion Gij(τ) = ⟨c_i(τ) c_j^dag⟩ (up to a coefficient) with the shifted single particle spectrum ξ = ϵ - μ and the eigenvectors V obtained from Tij = - V diagm(ϵ) V'.

Kwargs

 reverse::Bool = false

Return ⟨c_i^dag(τ) c_j⟩ instead if reverse = true.

 ExpectationValue(G::Matrix{F},
      si::Vector{Int64},
      dagidx::Vector{Int64}) -> ::F

Return the expectation value of severial fermionic operators. si denotes the sites and dagidx tells whether the operator is daggered or not. For example, si = [i, j] and dagidx = [2] means the single particle correlation c_i c_j^dag and si = [i, j, k, l] and dagidx = [3, 4] means the pairing correlation c_i c_j c_k^dag c_l^dag.

 TimeCorrelation(ξ::Vector{Float64},
      V::Matrix,
      β::Real,
      si::Vector{Int64},
      dagidx::Vector{Int64},
      lsτ::AbstractVector{<:Number}) -> ::Float64(::ComplexF64)

Return the time correlation function with the shifted single particle spectrum ξ = ϵ - μ and the eigenvectors V obtained from Tij = - V diagm(ϵ) V'. si denotes the sites and dagidx tells whether the operator is daggered or not. lsτ has the same length as si and tells the time of each operator. For example, si = [i, j, k, l], dagidx = [1, 3] and lsτ = [τ1, τ2, 0, 0] means c_i^dag(τ1) c_j(τ2) c_k^dag c_l.

 Density(G::Matrix, si::Int64) -> ::Float64
 Density(G::Matrix, lssi::AbstractVector{Int64}) -> ::Vector{Float64}

Return the local density n_i with the green function G.