# DMRG

ITensors.dmrgFunction
dmrg(H::MPO, psi0::MPS; kwargs...)
dmrg(H::MPO, psi0::MPS, sweeps::Sweeps; kwargs...)

Use the density matrix renormalization group (DMRG) algorithm to optimize a matrix product state (MPS) such that it is the eigenvector of lowest eigenvalue of a Hermitian matrix H, represented as a matrix product operator (MPO).

dmrg(Hs::Vector{MPO}, psi0::MPS; kwargs...)
dmrg(Hs::Vector{MPO}, psi0::MPS, sweeps::Sweeps; kwargs...)

Use the density matrix renormalization group (DMRG) algorithm to optimize a matrix product state (MPS) such that it is the eigenvector of lowest eigenvalue of a Hermitian matrix H. This version of dmrg accepts a representation of H as a Vector of MPOs, Hs = [H1, H2, H3, ...] such that H is defined as H = H1 + H2 + H3 + ... Note that this sum of MPOs is not actually computed; rather the set of MPOs [H1,H2,H3,..] is efficiently looped over at each step of the DMRG algorithm when optimizing the MPS.

dmrg(H::MPO, Ms::Vector{MPS}, psi0::MPS; weight=1.0, kwargs...)
dmrg(H::MPO, Ms::Vector{MPS}, psi0::MPS, sweeps::Sweeps; weight=1.0, kwargs...)

Use the density matrix renormalization group (DMRG) algorithm to optimize a matrix product state (MPS) such that it is the eigenvector of lowest eigenvalue of a Hermitian matrix H, subject to the constraint that the MPS is orthogonal to each of the MPS provided in the Vector Ms. The orthogonality constraint is approximately enforced by adding to H terms of the form w|M1><M1| + w|M2><M2| + ... where Ms=[M1, M2, ...] and w is the "weight" parameter, which can be adjusted through the optional weight keyword argument.

Note

dmrg will report the energy of the operator H + w|M1><M1| + w|M2><M2| + ..., not the operator H. If you want the expectation value of the MPS eigenstate with respect to just H, you can compute it yourself with an observer or after DMRG is run with inner(psi', H, psi).

The MPS psi0 is used to initialize the MPS to be optimized.

The number of sweeps of thd DMRG algorithm is controlled by passing the nsweeps keyword argument. The keyword arguments maxdim, cutoff, noise, and mindim can also be passed to control the cost versus accuracy of the algorithm - see below for details.

Alternatively the number of sweeps and accuracy parameters can be passed through a Sweeps object, though this interface is no longer preferred.

Returns:

• energy::Number - eigenvalue of the optimized MPS
• psi::MPS - optimized MPS

Keyword arguments:

• nsweeps::Int - number of "sweeps" of DMRG to perform

Optional keyword arguments:

• maxdim - integer or array of integers specifying the maximum size allowed for the bond dimension or rank of the MPS being optimized.
• cutoff - float or array of floats specifying the truncation error cutoff or threshold to use for truncating the bond dimension or rank of the MPS.
• eigsolve_krylovdim::Int = 3 - maximum dimension of Krylov space used to locally solve the eigenvalue problem. Try setting to a higher value if convergence is slow or the Hamiltonian is close to a critical point. [krylovkit]
• eigsolve_tol::Number = 1e-14 - Krylov eigensolver tolerance. [krylovkit]
• eigsolve_maxiter::Int = 1 - number of times the Krylov subspace can be rebuilt. [krylovkit]
• eigsolve_verbosity::Int = 0 - verbosity level of the Krylov solver. Warning: enabling this will lead to a lot of outputs to the terminal. [krylovkit]
• ishermitian=true - boolean specifying if dmrg should assume the MPO (or more general linear operator) represents a Hermitian matrix. [krylovkit]
• noise - float or array of floats specifying strength of the "noise term" to use to aid convergence.
• mindim - integer or array of integers specifying the minimum size of the bond dimension or rank, if possible.
• outputlevel::Int = 1 - larger outputlevel values make DMRG print more information and 0 means no output.
• observer - object implementing the Observer interface which can perform measurements and stop DMRG early.
• write_when_maxdim_exceeds::Int - when the allowed maxdim exceeds this value, begin saving tensors to disk to free RAM memory in large calculations
• write_path::String = tempdir() - path to use to save files to disk (to save RAM) when maxdim exceeds the write_when_maxdim_exceeds option, if set
source
• krylovkitThe dmrg function in ITensors.jl currently uses the eigsolve function in KrylovKit.jl as the internal the eigensolver. See the KrylovKit.jl documention on the eigsolve function for more details: KrylovKit.eigsolve.