Time evolution

Reference for the time-evolution drivers and algorithms. For a task-oriented walkthrough see the how-to guides. The full, canonical docstrings for the whole package live in the Library index.

Drivers

MPSKit.time_evolve Function

time_evolve(ψ₀, H, t_span, alg, [envs]; kwargs...) -> (ψ, envs)
time_evolve!(ψ₀, H, t_span, alg, [envs]; kwargs...) -> (ψ₀, envs)

Time-evolve the initial state ψ₀ with Hamiltonian H over a given time span by stepping through each of the time points obtained by iterating t_span.

Arguments

• ψ₀::AbstractMPS: initial state

• H::AbstractMPO: operator that generates the time evolution (can be time-dependent).

• t_span::AbstractVector{<:Number}: time points over which the time evolution is stepped

• alg: algorithm to use for the time evolution, e.g. TDVP or TDVP2.

• envs: MPS environment manager

Keyword Arguments

• verbosity::Int = 0: verbosity level for logging

• imaginary_evolution::Bool = false: if true, the time evolution is done with an imaginary time step instead, (i.e.   instead of  ). This can be useful for using this function to compute the ground state of a Hamiltonian, or to compute finite-temperature properties of a system.

Returns

• ψ: the time-evolved state

• envs: the updated environment manager

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MPSKit.timestep Function

timestep(ψ₀, H, t, dt, alg, [envs]; kwargs...) -> (ψ, envs)
timestep!(ψ₀, H, t, dt, alg, [envs]; kwargs...) -> (ψ₀, envs)

Time-step the state ψ₀ with Hamiltonian H over a given time step dt at time t, solving the Schroedinger equation:  .

Arguments

• ψ₀::AbstractMPS: initial state

• H::AbstractMPO: operator that generates the time evolution (can be time-dependent).

• t::Number: starting time of time-step

• dt::Number: time-step magnitude

• alg: algorithm to use for the time evolution, e.g. TDVP or TDVP2.

• envs: MPS environment manager

Keyword Arguments

• imaginary_evolution::Bool = false: if true, the time evolution is done with an imaginary time step instead, (i.e.   instead of  ). This can be useful for using this function to compute the ground state of a Hamiltonian, or to compute finite-temperature properties of a system.

Returns

• ψ: the time-stepped state

• envs: the updated environment manager

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MPSKit.timestep! Function

timestep(ψ₀, H, t, dt, alg, [envs]; kwargs...) -> (ψ, envs)
timestep!(ψ₀, H, t, dt, alg, [envs]; kwargs...) -> (ψ₀, envs)

Time-step the state ψ₀ with Hamiltonian H over a given time step dt at time t, solving the Schroedinger equation:  .

Arguments

• ψ₀::AbstractMPS: initial state

• H::AbstractMPO: operator that generates the time evolution (can be time-dependent).

• t::Number: starting time of time-step

• dt::Number: time-step magnitude

• alg: algorithm to use for the time evolution, e.g. TDVP or TDVP2.

• envs: MPS environment manager

Keyword Arguments

• imaginary_evolution::Bool = false: if true, the time evolution is done with an imaginary time step instead, (i.e.   instead of  ). This can be useful for using this function to compute the ground state of a Hamiltonian, or to compute finite-temperature properties of a system.

Returns

• ψ: the time-stepped state

• envs: the updated environment manager

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MPS time-evolution algorithms

MPSKit.TDVP Type

struct TDVP{A, E, G, F} <: MPSKit.Algorithm

Single site MPS time-evolution algorithm based on the Time-Dependent Variational Principle.

For finite MPS, setting alg_expand to a bond-expansion algorithm (e.g. OptimalExpand, SketchedExpand) expands the bond with directions orthogonal to the current state ahead of each local integration, recovering Controlled Bond Expansion (CBE) TDVP and lifting the fixed-bond limitation of plain single-site TDVP. A truncating trscheme is then required to cut the enlarged bond back down (selecting the truncated-SVD gauge). The expansion is state-preserving, as required for a consistent time evolution.

Note

Real-time evolution preserves the norm: neither the bond expansion nor the truncation renormalizes, so the state norm reflects the accumulated truncation error. Imaginary-time evolution instead renormalizes at every step, like a ground-state search. CBE is only available for finite MPS.

Fields

• integrator::Any: algorithm used in the exponential solvers

• tolgauge::Float64: tolerance for gauging algorithm

• gaugemaxiter::Int64: maximal amount of iterations for gauging algorithm

• alg_expand::Any: algorithm used to expand the bond ahead of each local update, or nothing for none (finite CBE-TDVP)

• alg_gauge::Any: factorization used for the post-update gauge: a QR algorithm (no truncation) or a truncated SVD

• finalize::Any: callback function applied after each iteration, of signature finalize(iter, ψ, H, envs) -> ψ, envs

See also

Used as the algorithm argument of timestep, timestep! and time_evolve.

References

• Haegeman et al. Phys. Rev. Lett. 107 (2011)

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MPSKit.TDVP2 Type

struct TDVP2{A, S, F} <: MPSKit.Algorithm

Two-site MPS time-evolution algorithm based on the Time-Dependent Variational Principle.

Fields

• integrator::Any: algorithm used in the exponential solvers

• tolgauge::Float64: tolerance for gauging algorithm

• gaugemaxiter::Int64: maximal amount of iterations for gauging algorithm

• alg_svd::Any: algorithm used for the singular value decomposition

• trscheme::MatrixAlgebraKit.TruncationStrategy: algorithm used for truncation of the two-site update

• finalize::Any: callback function applied after each iteration, of signature finalize(iter, ψ, H, envs) -> ψ, envs

See also

Used as the algorithm argument of timestep, timestep! and time_evolve.

References

• Haegeman et al. Phys. Rev. Lett. 107 (2011)

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Time-evolution MPOs

For evolving with an explicitly constructed propagator MPO, e.g. for an InfiniteMPS, use make_time_mpo with one of the expansion algorithms below.

MPSKit.make_time_mpo Function

make_time_mpo(H::MPOHamiltonian, dt::Number, alg; kwargs...) -> O::MPO

Construct an MPO that approximates  .

Keyword Arguments

• imaginary_evolution::Bool = false: if true, the time evolution operator is constructed with an imaginary time step instead, (i.e.   instead of  ). This can be useful for using this function to compute the ground state of a Hamiltonian, or to compute finite-temperature properties of a system.

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MPSKit.TaylorCluster Type

struct TaylorCluster <: MPSKit.Algorithm

Algorithm for constructing the Nth order time evolution MPO using the Taylor cluster expansion.

Fields

• N::Int64: order of the Taylor expansion

• extension::Bool: include higher-order corrections

• compression::Bool: approximate compression of corrections, accurate up to order N

See also

Used as the algorithm argument of make_time_mpo.

References

• Van Damme et al. SciPost Phys. 17 (2024)

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MPSKit.WI Constant

const WI = TaylorCluster(; N = 1, extension = false, compression = false)

First order Taylor expansion for a time-evolution MPO.

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MPSKit.WII Type

struct WII <: MPSKit.Algorithm

Generalization of the Euler approximation of the operator exponential for MPOs.

Fields

• tol::Float64: tolerance for convergence criterium

• maxiter::Int64: maximal number of iterations

See also

Used as the algorithm argument of make_time_mpo.

References

• Zaletel et al. Phys. Rev. B 91 (2015)

• Paeckel et al. Ann. of Phys. 411 (2019)

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• time_evolve! exists in src but is NOT exported, so it is intentionally omitted from this page.

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