CGmapping_from_AL(AL::AbstractTensorMap, k0::Integer, n::Integer)

Constructs a CG mapping from an abstract tensor map AL.

Arguments

• AL::AbstractTensorMap: The abstract tensor map.
• k0::Integer: The value of k0.
• n::Integer: The value of n.

Returns

• If n == k0, returns a tuple (V0, V1, V2) where:
  • V0: A matrix of size D^2 x (d^k0) representing the finite part of the CG mapping.
  • V1: A sparse matrix of size D^2 x (D^2 * d) representing the left-infinite part of the CG mapping.
  • V2: A sparse matrix of size D^2 x (D^2 * d) representing the right-infinite part of the CG mapping.
• If n != k0, returns a tuple (V0, L, R) where:
  • V0: A matrix of size D^2 x (d^k0) representing the finite part of the CG mapping.
  • L: A matrix of size D^2 x (D^2 * d) representing the left-infinite part of the CG mapping.
  • R: A matrix of size D^2 x (D^2 * d) representing the right-infinite part of the CG mapping.

approxgroundstate(H::MPOHamiltonian{T}, ψ_good::InfiniteMPS, d::Integer, D::Integer) where {T}

Approximates the ground state of a given Hamiltonian using the variational uniform matrix product state (VUMPS) algorithm.

Arguments

• H::MPOHamiltonian{T}: The Hamiltonian for which the ground state is to be approximated.
• ψ_good::InfiniteMPS: An initial guess for the ground state.
• d::Integer: The local Hilbert space dimension.
• D::Integer: The bond dimension.

Returns

• ψ_approx::InfiniteMPS: The approximate ground state.

good_ground_state(H::MPOHamiltonian{T}, D::Integer) where {T}

Compute the ground state of a given MPOHamiltonian using the VUMPS algorithm.

Arguments

• H::MPOHamiltonian{T}: The MPOHamiltonian representing the Hamiltonian of the system.
• D::Integer: The bond dimension of the MPS.

Returns

• groundstate: The ground state of the system.

load_Hamiltonian(filename::String)

Load the Hamiltonian matrix and the exact eigenvalues from a file.

This function is only used when loading ground state MPS from MATLAB computation using VUMPS implemented here

Arguments

• filename::String: The name of the file containing the Hamiltonian matrix.

Returns

• H: The Hamiltonian matrix.
• Eexact: The exact eigenvalues.

load_MPS(filename::String, D::Integer)

Load an MPS (Matrix Product State) from a file.

Arguments

• filename::String: The path to the file containing the MPS data.
• D::Integer: The virtual bond dimension of the MPS.

Returns

• ψ: The loaded MPS as an InfiniteMPS object.
• upperBdFromMPS: The upper bound of ground state energy achieved by the MPS.

onestepapprox_dual(h::AbstractMatrix{V}, n::Integer, optimizer=SCS.Optimizer) where {V}

Compute the one-step approximation of the dual energy for a given Hamiltonian.

Arguments

• h::AbstractMatrix{V}: The Hamiltonian matrix.
• n::Integer: The number of spins.
• optimizer=SCS.Optimizer: The optimizer to use for solving the optimization problem.

Returns

• ElocTIRig::Float64: The one-step approximation of the dual energy with the lowest eigenvalue added.
• ElocTI::Float64: The one-step approximation of the dual energy.

Example

two_step_approx(h::AbstractMatrix{V}, D::Integer, n::Integer,
    W2::AbstractMatrix{T}, L2::AbstractMatrix{T}, R2::AbstractMatrix{T},
    optimizer=SCS.Optimizer) where {V,T}

Approximates a two-step Renormalization Group (RG) transformation for a given Hamiltonian h using the specified parameters.

Arguments

• h::AbstractMatrix{V}: The input Hamiltonian matrix.
• D::Integer: The bond dimension.
• n::Integer: The number of RG steps.
• W2::AbstractMatrix{T}: The second layer of the RG transformation.
• L2::AbstractMatrix{T}: The left transformation matrix for the second layer.
• R2::AbstractMatrix{T}: The right transformation matrix for the second layer.
• optimizer=SCS.Optimizer: The optimizer to use for solving the optimization problem.

Returns

The objective value of the optimization problem.

Example