Fuzzy manifolds

GetTorusLz2QNDiag(nm :: Int64, nf :: Int64) :: QNDiag

Return the QNDiag of twice the angular momentum $2L_z\mod 2N_m$, with an offset such that one fully filled LLL has angular momentum 0, implemented as

QNDiag("Lz", [2 * m - 1 + mod(nm, 2) for m = 1 : nm for f = 1 : nf], 2 * nm)

GetTorusTranslQNOffd(nm :: Int64, nf :: Int64, Δ :: Int64 = 1) :: QNOffd

Return the QNOffd of translation

\[ c_m↦c_{m+Δ\mod N_m}\]

on the torus, implemented as

    QNOffd([((m - 1 + Δ) % nm) * nf + f for m = 1 : nm for f = 1 : nf], nm)

GetTorusIntMatrix(nm :: Int64, lx :: Number, ps_pot :: Vector{<:Number}) :: Array{ComplexF64, 3}
GetTorusIntMatrix(nm :: Int64, ps_pot :: Vector{<:Number} ; aspect_ratio :: Number = 1.0) :: Array{ComplexF64, 3}

Gives the interaction matrix $U_{m_1m_2m_3m_4}$ from the pseudopotentials.

\[ U_{m_1m_2m_3m_4}=δ'_{m_1+m_2,m_3+m_4}\frac{1}{N_m}∑_{l,𝐪}δ'_{m_1-m_4,t}U_lL_l(q^2)e^{-q^2/2}e^{2πis(m_1-m_3)/N_m}\]

where $𝐪=(2πs/L_x,2πt/L_y), s,t∈ℤ$, $L_xL_y=2πN_m$ and the Kronecker $δ$ is defined in a sense of mod $N_m$

Argument

• nm :: Int64 is the number of orbitals.
• lx :: Number is the length along $x$-direction. Facultative.
• ps_pot :: Vector{<:Number} is the vector of non-zero pseudopotentials.
• aspect_ratio :: Number is the ratio $L_y/L_x$. Facultative, at most one of lx and aspect_ratio is given. If both are omitted, $L_x=L_y$ is taken.

Output

• A nm×nm×nm array giving the interaction matrix $U_{m_1m_2m_3m_4}$ where $m_4=m_1+m_2-m_3\mod N_m$.

GetTorusDenIntTerms(nm :: Int64, nf :: Int64, lx :: Number, ps_pot :: Vector{<:Number}[, mat_a :: Matrix{<:Number}[, mat_b :: Matrix{<:Number}]]) :: Terms
GetTorusDenIntTerms(nm :: Int64, nf :: Int64, ps_pot :: Vector{<:Number}[, mat_a :: Matrix{<:Number}[, mat_b :: Matrix{<:Number}]] ; aspect_ratio :: Number = 1.0) :: Terms

Return the normal-ordered density-density term in the Hamiltonian

\[∑_{\{m_i,f_i\}}U_{m_1m_2m_3m_4}M^A_{f_1f_4}M^B_{f_2f_3}c^{†}_{m_1f_1}c^{†}_{m_2f_2}c_{m_3f_3}c_{m_4f_4}.\]

Arguments

• nm :: Int64 is the number of orbitals.
• nf :: Int64 is the number of flavours.
• lx :: Number is the length along $x$-direction. Facultative.
• ps_pot :: Vector{<:Number} is a list of numbers specifying the pseudopotentials for the interacting matrix $U_{m_1m_2m_3m_4}$.
• mat_a :: Matrix{<:Number} is a nf×nf matrix specifying $M^A_{ff'}$. Facultative, $I_{N_f}$ by default.
• mat_b :: Matrix{<:Number} is a nf×nf matrix specifying $M^B_{ff'}$. Facultative, the Hermitian conjugate of mat_a by default.
• aspect_ratio :: Number is the ratio $L_y/L_x$. Facultative, at most one of lx and aspect_ratio is given. If both are omitted, $L_x=L_y$ is taken.

GetTorusPairIntTerms(nm :: Int64, nf :: Int64, lx :: Number, ps_pot :: Vector{<:Number}, mat_a :: Matrix{<:Number}[, mat_b :: Matrix{<:Number}]) :: Terms
GetTorusPairIntTerms(nm :: Int64, nf :: Int64, ps_pot :: Vector{<:Number}, mat_a :: Matrix{<:Number}[, mat_b :: Matrix{<:Number}] ; aspect_ratio :: Number = 1.0) :: Terms

Return the normal-ordered pair-pair interaction term in the Hamiltonian

\[∑_{\{m_i,f_i\}}U_{m_1m_2m_3m_4}M^A_{f_1f_2}M^B_{f_3f_4}c^{†}_{m_1f_1}c^{†}_{m_2f_2}c_{m_3f_3}c_{m_4f_4}.\]

Arguments

• nm :: Int64 is the number of orbitals.
• nf :: Int64 is the number of flavours.
• lx :: Number is the length along $x$-direction. Facultative.
• ps_pot :: Vector{<:Number} is a list of numbers specifying the pseudopotentials for the interacting matrix $U_{m_1m_2m_3m_4}$.
• mat_a :: Matrix{<:Number} is a nf×nf matrix specifying $M^A_{ff'}$. Facultative, $I_{N_f}$ by default.
• mat_b :: Matrix{<:Number} is a nf×nf matrix specifying $M^B_{ff'}$. Facultative, the Hermitian conjugate of mat_a by default.
• m_kept :: Vector{Int64} is a list of orbitals that range from 1 to nm. Facultative, if specified, only terms for which all $m_i$ are in the list are kept.
• aspect_ratio :: Number is the ratio $L_y/L_x$. Facultative, at most one of lx and aspect_ratio is given. If both are omitted, $L_x=L_y$ is taken.