# DMRG

`ITensors.ITensorMPS.dmrg`

— Function```
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.

`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

- 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.