Excitations

Reference for the excitation interface, its algorithms, and the quasiparticle state types it produces. For a task-oriented walkthrough see the how-to guides. The full, canonical docstrings for the whole package live in the Library index.

Interface

MPSKit.excitations Function

julia

excitations(
        H, algorithm::QuasiparticleAnsatz, ψ::FiniteQP, [left_environments],
        [right_environments]; num = 1
    ) -> (energies, states)
excitations(
        H, algorithm::QuasiparticleAnsatz, ψ::InfiniteQP, [left_environments],
        [right_environments]; num = 1
    ) -> (energies, states)
excitations(
        H, algorithm::FiniteExcited, ψs::NTuple{<:Any, <:FiniteMPS};
        num = 1, init
    ) -> (energies, states)
excitations(
        H, algorithm::ChepigaAnsatz, ψ::FiniteMPS, [envs];
        num = 1, pos = length(ψ) ÷ 2
    ) -> (energies, states)
excitations(
        H, algorithm::ChepigaAnsatz2, ψ::FiniteMPS, [envs];
        num = 1, pos = length(ψ) ÷ 2
    ) -> (energies, states)

Compute the first excited states and their energy gap above a ground state.

Arguments

H::AbstractMPO: operator for which to find the excitations
algorithm: optimization algorithm
ψ::QP: initial quasiparticle guess
ψs::NTuple{N, <:FiniteMPS}: N first excited states
[left_environments]: left ground state environment
[right_environments]: right ground state environment

Keyword Arguments

num::Int: number of excited states to compute
solver: algorithm for the linear solver of the quasiparticle environments
init: initial excited state guess; defaults to a copy of the first state in ψs
pos: position of perturbation

source

Algorithms

MPSKit.QuasiparticleAnsatz Type

julia

struct QuasiparticleAnsatz{A, E} <: MPSKit.Algorithm

Optimization algorithm for quasi-particle excitations on top of MPS groundstates.

Constructors

julia

QuasiparticleAnsatz()
QuasiparticleAnsatz(; kwargs...)
QuasiparticleAnsatz(alg)

Create a QuasiparticleAnsatz algorithm with the given eigensolver, or by passing the keyword arguments to Arnoldi.

Fields

alg::Any: algorithm used for the eigenvalue solvers
alg_environments::Any: algorithm used for the quasiparticle environments

See also

Used as the algorithm argument of excitations.

References

Haegeman et al. Phys. Rev. Let. 111 (2013)

source

MPSKit.FiniteExcited Type

julia

struct FiniteExcited{A} <: MPSKit.Algorithm

Variational optimization algorithm for excitations of finite MPS by minimizing the energy of

ᵢ ᵢ ᵢ

Fields

gsalg::Any: optimization algorithm
weight::Float64: energy penalty for enforcing orthogonality with previous states

See also

Used as the algorithm argument of excitations.

source

MPSKit.ChepigaAnsatz Type

julia

struct ChepigaAnsatz{A<:KrylovKit.KrylovAlgorithm} <: MPSKit.Algorithm

Single-site optimization algorithm for excitations on top of MPS groundstates.

Constructors

julia

ChepigaAnsatz()
ChepigaAnsatz(; kwargs...)
ChepigaAnsatz(alg)

Create a ChepigaAnsatz algorithm with the given eigensolver, or by passing the keyword arguments to Arnoldi.

Fields

alg::KrylovKit.KrylovAlgorithm: algorithm used for the eigenvalue solvers

See also

Used as the algorithm argument of excitations.

References

Chepiga et al. Phys. Rev. B 96 (2017)

source

MPSKit.ChepigaAnsatz2 Type

julia

struct ChepigaAnsatz2{A<:KrylovKit.KrylovAlgorithm} <: MPSKit.Algorithm

Two-site optimization algorithm for excitations on top of MPS groundstates.

Constructors

julia

ChepigaAnsatz2()
ChepigaAnsatz2(; kwargs...)
ChepigaAnsatz2(alg, trunc)

Create a ChepigaAnsatz2 algorithm with the given eigensolver and truncation, or by passing the keyword arguments to Arnoldi.

Fields

alg::KrylovKit.KrylovAlgorithm: algorithm used for the eigenvalue solvers, defaults to Arnoldi(; krylovdim = 30, tol = 1.0e-10, eager = true)
trunc::Any: truncation strategy used when splitting the optimized two-site tensor, defaults to notrunc()

See also

Used as the algorithm argument of excitations.

References

Chepiga et al. Phys. Rev. B 96 (2017)

source

Quasiparticle states

These are the ansatz states produced by, and passed to, excitations on top of a ground state.

MPSKit.QP Type

julia

QP{S, T1, T2}

Union of the quasiparticle excitation ansätze LeftGaugedQP and RightGaugedQP. It is used for dispatch and to share their gauge-independent interface; it is not a concrete type and cannot be constructed on its own. The internal aliases FiniteQP and InfiniteQP further restrict the ground-state type to FiniteMPS or InfiniteMPS respectively.

source

MPSKit.LeftGaugedQP Type

julia

struct LeftGaugedQP{S, T1, T2, E<:Number}

Left-gauged quasiparticle excitation ansatz on top of a matrix product state ground state. The excitation is parametrized through the left-gauge nullspace of the ground-state tensors, and the object behaves as a vector so it can be handed directly to the iterative eigensolvers used by excitations.

For a FiniteMPS ground state this represents a finite (localized) quasiparticle; for an InfiniteMPS ground state it represents a momentum eigenstate with the given momentum. When left_gs !== right_gs the ansatz describes a domain wall between the two ground states.

Constructors

julia

LeftGaugedQP(datfun, left_gs, right_gs = left_gs; sector, momentum = 0.0)

These states are normally produced by excitations with a QuasiparticleAnsatz rather than constructed directly. When constructing manually, datfun initializes the variational tensors (e.g. rand/randn), sector selects the charge sector of the excitation, and momentum sets the momentum for infinite ground states.

Fields

left_gs, right_gs: the ground state(s) the excitation lives on; distinct values yield a domain wall.
VLs: left-nullspace tensors of the ground-state AL (satisfying AL' * VL == 0).
Xs: the variational parameters of the ansatz.
momentum: the excitation momentum (used for infinite ground states).

See also

RightGaugedQP, QP

source

MPSKit.RightGaugedQP Type

julia

struct RightGaugedQP{S, T1, T2, E<:Number}

Right-gauged counterpart of LeftGaugedQP: the same quasiparticle excitation ansatz, but parametrized through the right-gauge nullspace of the ground-state tensors. It is most often obtained via convert(RightGaugedQP, ϕ) from a LeftGaugedQP rather than constructed directly.

Constructors

julia

RightGaugedQP(datfun, left_gs, right_gs = left_gs; sector, momentum = 0.0)

Fields

left_gs, right_gs: the ground state(s) the excitation lives on; distinct values yield a domain wall.
Xs: the variational parameters of the ansatz.
VRs: right-nullspace tensors of the ground-state AR.
momentum: the excitation momentum (used for infinite ground states).

See also

LeftGaugedQP, QP

source

Excitations ​

Interface ​

Algorithms ​

Quasiparticle states ​

Excitations

Interface

Algorithms

Quasiparticle states