Time evolution

Schwinger.jl supports time-evolving states using all three backends:

ED: Uses Krylov methods from KrylovKit.jl
ITensors: Uses the time-dependent variational principle (TDVP) from ITensorMPS.jl
MPSKit: Uses TDVP from MPSKit.jl

The evolve function evolves a state forwards in time, and monitors any given observables. It returns a final state along with a DataFrame of the observables. For example, here is a simulation of flux unwinding.

using Schwinger, Plots

lat = Lattice(10; F = 1, L = 2, periodic = true)
gs = groundstate(Hamiltonian(lat; backend=:ED))

_, df = evolve(WilsonLoop(lat; backend=:ED) * gs, 15;
    nsteps = 30,
    observable = (ψ, t) -> sum(electricfields(ψ))/10
)

scatter(df.time, df.observable, xlabel = "gt", ylabel = "Average electric field", label = "Schwinger.jl")
plot!(0:.1:15, [cos(t/√(π)) for t in 0:.1:15], label = "Exact")

Schwinger.evolve — Function

evolve(state::EDState, t::Real; nsteps::Int = 1, tol = 1E-12, observable = nothing, kwargs...)

Evolve an exact diagonalization state by time t using matrix exponentiation.

Arguments

state::EDState: The state to evolve.
t::Real: The time to evolve by.
nsteps::Int = 1: Number of time steps to divide the evolution into.
tol = 1E-12: Tolerance for the matrix exponentiation.
observable::Union{Nothing,Function,Dict} = nothing: Observable(s) to monitor during evolution. Can be a single function (state, t) -> value or a dictionary of name => function pairs.

Returns

EDState: The evolved state.
obs: Observer object containing the history of observables and metadata.

Examples

evolved_state, obs = evolve(state, 1.0; nsteps=10, observable=s->energy(s))

source

evolve(state::ITensorState, t::Real; nsteps::Int = 1, observable = nothing, kwargs...)

Evolve an ITensor MPS state by imaginary time t using TDVP.

Arguments

state::ITensorState: The state to evolve.
t::Real: The time to evolve by (will be multiplied by -im for imaginary time evolution).
nsteps::Int = 1: Number of time steps for the TDVP algorithm.
observable::Union{Nothing,Function,Dict} = nothing: Observable(s) to monitor during evolution. Can be a single function (state, t) -> value or a dictionary of name => function pairs.
kwargs...: Additional keyword arguments passed to the TDVP algorithm.

Returns

ITensorState: The evolved state.
obs: Observer object containing the history of observables and metadata.

Notes

Uses ITensors.jl TDVP algorithm.
Time is tracked as real values (divided by -im) in the observer.

source

evolve(state::MPSKitState, t::Real; nsteps::Int = 1, dt = nothing, observable = nothing, kwargs...)

Evolve an MPSKit MPS state by time t using MPSKit's TDVP algorithm.

Arguments

state::MPSKitState: The state to evolve.
t::Real: The time to evolve by.
nsteps::Int = 1: Number of time steps to divide the evolution into.
dt::Union{Nothing,Real} = nothing: Time step size (currently unused, evolution uses t/nsteps).
observable::Union{Nothing,Function,Dict} = nothing: Observable(s) to monitor during evolution. Can be a single function (state, t) -> value or a dictionary of name => function pairs.
kwargs...: Additional keyword arguments.

Returns

MPSKitState: The evolved state.
obs: Observer object containing the history of observables and metadata.

Notes

Uses MPSKit.jl TDVP algorithm.
Evolves with imaginary time (-im*t/nsteps).

source