Public API

A quasi Monte Carlo inchworm impurity solver for multi-orbital fermionic models.

Strong coupling pseudo-particle expansion problem.

Exports

• Expansion
• InteractionPair
• add_corr_operators!

struct Expansion{ScalarGF<:Keldysh.AbstractTimeGF{ComplexF64, true}, PPGF_P0<:Union{Vector{Keldysh.GenericTimeGF{ComplexF64, false, Keldysh.FullTimeGrid}}, Vector{Keldysh.ImaginaryTimeGF{ComplexF64, false}}, Vector{QInchworm.exact_atomic_ppgf.ExactAtomicPPGF}, Vector{QInchworm.spline_gf.IncSplineImaginaryTimeGF{ComplexF64, false}}, Vector{QInchworm.spline_gf.SplineInterpolatedGF{Keldysh.GenericTimeGF{ComplexF64, false, Keldysh.FullTimeGrid}, ComplexF64, false}}, Vector{QInchworm.spline_gf.SplineInterpolatedGF{Keldysh.ImaginaryTimeGF{ComplexF64, false}, ComplexF64, false}}}, PPGF_P<:Union{Vector{Keldysh.GenericTimeGF{ComplexF64, false, Keldysh.FullTimeGrid}}, Vector{Keldysh.ImaginaryTimeGF{ComplexF64, false}}, Vector{QInchworm.exact_atomic_ppgf.ExactAtomicPPGF}, Vector{QInchworm.spline_gf.IncSplineImaginaryTimeGF{ComplexF64, false}}, Vector{QInchworm.spline_gf.SplineInterpolatedGF{Keldysh.GenericTimeGF{ComplexF64, false, Keldysh.FullTimeGrid}, ComplexF64, false}}, Vector{QInchworm.spline_gf.SplineInterpolatedGF{Keldysh.ImaginaryTimeGF{ComplexF64, false}, ComplexF64, false}}}}

The Expansion structure contains the components needed to define a strong coupling pseudo-particle expansion problem.

Fields

• ed: Exact diagonalization solver for the local degrees of freedom

• P0: Non-interacting propagator (pseudo-particle Green's function)

• P: Interacting propagator (pseudo-particle Green's function)

• pairs: List of pseudo-particle interactions

• determinants: List of hybridization function determinants (not implemented yet)

• corr_operators: List of operator pairs used in accumulation of two-point correlation functions

• identity_mat: Block matrix representation of the identity operator

• pair_operator_mat: Block matrix representation of paired operators (operator_i, operator_f)

• corr_operators_mat: Block matrix representation of corr_operators

• subspace_attachable_pairs: Given a subspace index s_i, lists indices of all interaction pairs with operator_i acting non-trivially on that subspace.

Expansion(
    ed::KeldyshED.EDCore,
    grid::Keldysh.AbstractTimeGrid,
    interaction_pairs::Array{QInchworm.expansion.InteractionPair{ScalarGF}, 1};
    corr_operators,
    interpolate_ppgf
) -> QInchworm.expansion.Expansion

Parameters

• ed: Exact diagonalization solution of the local problem.
• grid: Contour time grid to define the local propagators on.
• interaction_pairs: The list of pair interactions to expand in.
• corr_operators: The list of operator pairs used in accumulation of two-point correlation functions.
• interpolate_ppgf: Use a quadratic spline interpolation to represent and evaluate the local propagators. Currently works only with the imaginary time propagators.

Expansion(
    hamiltonian::KeldyshED.Operators.OperatorExpr,
    soi::KeldyshED.Hilbert.SetOfIndices,
    grid::Keldysh.ImaginaryTimeGrid;
    hybridization,
    nn_interaction,
    corr_operators,
    interpolate_ppgf
)

A higher-level constructor of Expansion that solves the local problem defined by a Hamiltonian and internally generates a list of pseudo-particle pair interactions from hybridization and $nn$-interaction functions.

Parameters

• hamiltonian: Hamiltonian of the local problem.
• soi: An ordered set of indices carried by creation/annihilation operators of the local problem.
• grid: Imaginary time grid to define the local propagators on.
• hybridization: A matrix-valued hybridization function $\Delta_{ij}(\tau)$. A correspondence between the matrix elements $(i, j)$ and operators $c^\dagger_i, c_j$ is established by soi.
• nn_interaction: A matrix-valued $nn$-interaction function $U_{ij}(\tau)$. A correspondence between the matrix elements $(i, j)$ and operators $n_i, n_j$ is established by soi.
• corr_operators: The list of operator pairs used in accumulation of two-point correlation functions.
• interpolate_ppgf: Use a quadratic spline interpolation to represent and evaluate the local propagators.

struct InteractionPair{ScalarGF<:Keldysh.AbstractTimeGF{ComplexF64, true}}

Data type for pseudo-particle interactions containing two operators and one scalar propagator.

Indexed access to the operators stored in a pair::InteractionPair is supported: pair[1] and pair[2] are equivalent to pair.operator_i and pair.operator_f respectively.

Fields

• operator_f::KeldyshED.Operators.RealOperatorExpr: Final time operator

• operator_i::KeldyshED.Operators.RealOperatorExpr: Initial time operator

• propagator::Keldysh.AbstractTimeGF{ComplexF64, true}: Scalar propagator

add_corr_operators!(
    expansion::QInchworm.expansion.Expansion,
    ops::Tuple{KeldyshED.Operators.RealOperatorExpr, KeldyshED.Operators.RealOperatorExpr}
)

Add a pair of operators $(A, B)$ used to measure the two-point correlator $\langle A(t_1) B(t_2)\rangle$ to expansion.

Parameters

• expansion: Pseudo-particle expansion.
• ops: The pair of operators $(A, B)$.

High level functions implementing the quasi Monte Carlo inchworm algorithm.

Exports

• correlator_2p
• diff_inchworm!
• inchworm!

inchworm!(
    expansion::QInchworm.expansion.Expansion,
    grid::Keldysh.ImaginaryTimeGrid,
    orders,
    orders_bare,
    N_samples::Int64;
    n_pts_after_max,
    rand_params,
    seq_type
) -> Tuple{Dict, Dict}

Perform a complete qMC inchworm calculation of the bold propagators on the imaginary time segment. Results of the calculation are written into expansion.P.

Parameters

• expansion: Strong coupling expansion problem.
• grid: Imaginary time grid of the bold propagators.
• orders: List of expansion orders to be accounted for during a regular inchworm step.
• orders_bare: List of expansion orders to be accounted for during the initial inchworm step.
• N_samples: Number of samples to be used in qMC integration. Must be a power of 2.
• n_pts_after_max: Maximum number of points in the after-$\tau_w$ region to be taken into account. By default, diagrams with all valid numbers of the after-$\tau_w$ points are considered.
• rand_params: Parameters of the randomized qMC integration.
• seq_type: Type of the (quasi-)random sequence to be used for integration.

Returns

• Order-resolved contributions to the bold propagator as a dictionary Dict{Int, PPGF}.
• Estimated standard deviations of the order-resolved contributions as a dictionary Dict{Int, PPGF}.

diff_inchworm!(
    expansion::QInchworm.expansion.Expansion,
    grid::Keldysh.ImaginaryTimeGrid,
    orders,
    N_samples::Int64;
    rand_params,
    seq_type
) -> Tuple{Dict, Dict}

Perform a complete qMC inchworm calculation of the bold propagators on the imaginary time segment using the differential formulation of the method described in

"Inchworm Monte Carlo Method for Open Quantum Systems"
Z. Cai, J. Lu and S. Yang
Comm. Pure Appl. Math., 73: 2430-2472 (2020)

Results of the calculation are written into expansion.P.

Parameters

• expansion: Strong coupling expansion problem.
• grid: Imaginary time grid of the bold propagators.
• orders: List of expansion orders to be accounted for.
• N_samples: Number of samples to be used in qMC integration. Must be a power of 2.
• rand_params: Parameters of the randomized qMC integration.
• seq_type: Type of the (quasi-)random sequence to be used for integration.

Returns

• Order-resolved contributions to the pseudo-particle self-energy as a dictionary Dict{Int, PPGF}.
• Estimated standard deviations of the order-resolved contributions as a dictionary Dict{Int, PPGF}.

correlator_2p(
    expansion::QInchworm.expansion.Expansion,
    grid::Keldysh.ImaginaryTimeGrid,
    A_B_pair_idx::Int64,
    τ::Keldysh.TimeGridPoint,
    top_data::Vector{QInchworm.inchworm.TopologiesInputData};
    seq_type,
    tmr
) -> Tuple{ComplexF64, ComplexF64}

Calculate value of a two-point correlator $\langle A(\tau) B(0)\rangle$ for one value of the imaginary time argument $\tau$. The pair of operators $(A, B)$ used in the calculation is taken from expansion.corr_operators[A_B_pair_idx].

Parameters

• expansion: Strong coupling expansion problem. expansion.P must contain precomputed bold propagators.
• grid: Imaginary time grid of the correlator to be computed.
• A_B_pair_idx: Index of the $(A, B)$ pair within expansion.corr_operators.
• τ: The imaginary time argument $\tau$.
• top_data: Accumulation input data.
• seq_type: Type of the (quasi-)random sequence to be used for integration.
• tmr: A TimerOutput object used for profiling.

Returns

• Accumulated value of the two-point correlator.
• Estimated standard deviations of the computed correlator.

correlator_2p(
    expansion::QInchworm.expansion.Expansion,
    grid::Keldysh.ImaginaryTimeGrid,
    orders,
    N_samples::Int64,
    ::QInchworm.randomization.RequestStdDev;
    rand_params,
    seq_type
) -> Tuple{Any, Any}

Calculate a two-point correlator $\langle A(\tau) B(0)\rangle$ on the imaginary time segment. Accumulation is performed for each pair of operators $(A, B)$ in expansion.corr_operators. Only the operators that are a single monomial in $c/c^\dagger$ are supported.

This method is selected by the flag argument of type RequestStdDev and returns randomized qMC estimates of both mean and standard deviation of the correlators.

Parameters

• expansion: Strong coupling expansion problem. expansion.P must contain precomputed bold propagators.
• grid: Imaginary time grid of the correlator to be computed.
• orders: List of expansion orders to be accounted for.
• N_samples: Number of samples to be used in qMC integration. Must be a power of 2.
• rand_params: Parameters of the randomized qMC integration.
• seq_type: Type of the (quasi-)random sequence to be used for integration.

Returns

• A list of scalar-valued GF objects containing the computed correlators, one element per a pair in expansion.corr_operators.
• A list of scalar-valued GF objects containing estimated standard deviations of the computed correlators, one element per a pair in expansion.corr_operators.

correlator_2p(
    expansion::QInchworm.expansion.Expansion,
    grid::Keldysh.ImaginaryTimeGrid,
    orders,
    N_samples::Int64;
    rand_params,
    seq_type
) -> Any

Calculate a two-point correlator $\langle A(\tau) B(0)\rangle$ on the imaginary time segment. Accumulation is performed for each pair of operators $(A, B)$ in expansion.corr_operators. Only the operators that are a single monomial in $c/c^\dagger$ are supported.

Parameters

• expansion: Strong coupling expansion problem. expansion.P must contain precomputed bold propagators.
• grid: Imaginary time grid of the correlator to be computed.
• orders: List of expansion orders to be accounted for.
• N_samples: Number of samples to be used in qMC integration. Must be a power of 2.
• rand_params: Parameters of the randomized qMC integration.
• seq_type: Type of the (quasi-)random sequence to be used for integration.

Returns

A list of scalar-valued GF objects containing the computed correlators, one element per a pair in expansion.corr_operators.

Randomized quasi Monte Carlo.

Exports

• RandomizationParams
• RequestStdDev

struct RandomizationParams

Parameters of the randomized qMC integration.

Fields

• rng::Union{Nothing, Random.AbstractRNG}: Random Number Generator used to scramble Sobol sequences or nothing to disable scrambling

• N_seqs::Int64: Maximal number of scrambled Sobol sequences to be used

• target_std::Float64: Target standard deviation of the accumulated quantity (∞-norm for matrices)

struct RequestStdDev

Singleton type used to select methods that return an estimate of the standard deviation in addition to the mean.

Pseudo-particle Green's functions (propagators) of finite fermionic systems and related tools.

For a system defined by a time-independent Hamiltonian $\hat H$, the pseudo-particle Green's function (PPGF) is

\[P(z, z') = \left\{ \begin{array}{ll} -i (-1)^{\hat N} e^{-i \hat H(z-z')},& z \succ -i\beta \cap -i\beta \succeq z',\\ -i e^{-i \hat H(z-z')},& \text{otherwise}. \end{array} \right.\]

In particular, on the imaginary time segment alone one has $P(\tau) = -i e^{-\hat H \tau}$.

This operator has a block-diagonal structure determined by the symmetry sectors of $\hat H$, and is stored as a vector of GF containers corresponding to the individual diagonal blocks (FullTimePPGF, ImaginaryTimePPGF).

Exports

• FullTimePPGF
• ImaginaryTimePPGF
• atomic_ppgf
• density_matrix
• normalize!
• partition_function

An atomic propagator (PPGF) defined on a full Keldysh contour

An atomic propagator (PPGF) defined on an imaginary time segment

atomic_ppgf(
    β::Float64,
    ed::KeldyshED.EDCore
) -> Vector{QInchworm.exact_atomic_ppgf.ExactAtomicPPGF}

Construct the exact atomic pseudo-particle Green's function.

Parameters

• β: Inverse temperature.
• ed: Exact diagonalization structure describing the atomic problem.

atomic_ppgf(
    grid::Keldysh.FullTimeGrid,
    ed::KeldyshED.EDCore
) -> Vector{Keldysh.GenericTimeGF{ComplexF64, false, Keldysh.FullTimeGrid}}

Compute atomic pseudo-particle Green's function on a full contour time grid for a time-independent exact diagonalization problem ed.

As the resulting PPGF $P(z, z')$ is defined up to a multiplier $e^{-i\lambda (z-z')}$, we choose the energy shift $\lambda$ to fulfil the normalization property $\mathrm{Tr}[i P(-i\beta, 0)] = 1$.

atomic_ppgf(
    grid::Keldysh.ImaginaryTimeGrid,
    ed::KeldyshED.EDCore
) -> Vector{Keldysh.ImaginaryTimeGF{ComplexF64, false}}

Compute atomic pseudo-particle Green's function on an imaginary time grid for a time-independent exact diagonalization problem ed.

As the resulting PPGF $P(\tau)$ is defined up to a multiplier $e^{-\lambda\tau}$, we choose the energy shift $\lambda$ to fulfil the normalization property $\mathrm{Tr}[i P(\beta)] = 1$.

partition_function(
    P_0::Vector{QInchworm.exact_atomic_ppgf.ExactAtomicPPGF}
) -> ComplexF64

Extract the partition function $Z = \mathrm{Tr}[i P_0(-i\beta, 0)]$ from a un-normalized pseudo-particle Green's function P_0.

partition_function(
    P::Vector{<:Keldysh.AbstractTimeGF}
) -> ComplexF64

Extract the partition function $Z = \mathrm{Tr}[i P(-i\beta, 0)]$ from a un-normalized pseudo-particle Green's function P.

density_matrix(
    P::Vector{QInchworm.exact_atomic_ppgf.ExactAtomicPPGF}
) -> Vector{Matrix{ComplexF64}}

Extract the equilibrium density matrix $\rho = i P(-i\beta, 0)$ from a normalized pseudo-particle Green's function P. The density matrix is block-diagonal and is returned as a vector of blocks.

density_matrix(
    P::Vector{<:Keldysh.AbstractTimeGF}
) -> Vector{Matrix{ComplexF64}}

Extract the equilibrium density matrix $\rho = i P(-i\beta, 0)$ from a normalized pseudo-particle Green's function P. The density matrix is block-diagonal and is returned as a vector of blocks.

normalize!(
    P::Vector{<:Keldysh.AbstractTimeGF},
    β::Float64
) -> ComplexF64

Normalize a pseudo-particle Green's function P by multiplying it by $e^{-i\lambda (z-z')}$ with $\lambda$ chosen such that $\mathrm{Tr}[i P(-i\beta, 0)] = 1$.

Returns

The energy shift $\lambda$.

normalize!(
    P::Vector{<:Keldysh.AbstractTimeGF},
    τ::Keldysh.TimeGridPoint
)

Normalize a pseudo-particle Green's function P by multiplying it by $e^{-i\lambda (z-z')}$ with $\lambda$ chosen such that $\mathrm{max}[i P(-i\tau, 0)] = 1$.

normalize!(P_s::Keldysh.AbstractTimeGF, λ)

Multiply a given diagonal block of a pseudo-particle Green's function P_s by $e^{-i\lambda (z-z')}$.

normalize!(
    P_s::QInchworm.spline_gf.IncSplineImaginaryTimeGF{ComplexF64, false},
    λ
)

Multiply a given diagonal block of a pseudo-particle Green's function P_s by $e^{-i\lambda (z-z')}$. This method is defined for the spline-interpolated imaginary-time propagators.

Spline-interpolated Green's function containers.

Exports

• SplineInterpolatedGF
• ph_conj

struct SplineInterpolatedGF{GFType, T, scalar} <: Keldysh.AbstractTimeGF{T, scalar}

Wrapper around a Green's function object that allows for fast cubic spline interpolation on the time grid.

The wrapper supports square bracket access to the wrapped object, direct access to the grid property, eltype(), Keldysh.norbitals() and Keldysh.TimeDomain(). Evaluation at an arbitrary contour time point (via operator ()) is carried out by a stored set of pre-computed B-spline interpolants.

Fields

• GF::Any: Wrapped Green's function

• interpolants::Any: B-spline interpolants, one object per matrix element of G

SplineInterpolatedGF(
    GF::Keldysh.AbstractTimeGF{T<:Number, scalar};
    τ_max
) -> QInchworm.spline_gf.SplineInterpolatedGF{_A, T} where {_A, T<:Number}

Make a SplineInterpolatedGF wrapper around GF and compute interpolants of its data from the start of the grid up to τ_max. By default, the entire data array is used.

Extension of Keldysh.jl defining imaginary time Green's functions represented using the Discrete Lehmann Representation as implemented in Lehmann.jl.

Exports

• DLRImaginaryTimeGF
• DLRImaginaryTimeGrid
• ph_conj

struct DLRImaginaryTimeGrid <: Keldysh.AbstractTimeGrid

Wrapper around Lehmann.jl describing a Discrete Lehmann Representation imaginary time grid conforming to the interface of Keldysh.jl TimeGrids.

Fields

• contour::Keldysh.ImaginaryContour

• points::Vector{Keldysh.TimeGridPoint}

• branch_bounds::Tuple{Pair{Keldysh.TimeGridPoint, Keldysh.TimeGridPoint}}

• ntau::Int64

• dlr::Lehmann.DLRGrid

struct DLRImaginaryTimeGF{T, scalar} <: Keldysh.AbstractTimeGF{T, scalar}

Wrapper around Lehmann.jl describing a Discrete Lehmann Representation imaginary time Green's function conforming to the interface of Keldysh.jl AbstractTimeGF.

Fields

• grid::QInchworm.keldysh_dlr.DLRImaginaryTimeGrid

• mat::Keldysh.PeriodicStorage

• ξ::Keldysh.GFSignEnum

An assorted collection of utility types and functions.

Exports

• ph_conj

ph_conj(
    G_int::QInchworm.spline_gf.SplineInterpolatedGF{GFType<:Keldysh.ImaginaryTimeGF{T<:Number, true}, T<:Number, true}
)

Given a spline-interpolated scalar-valued imaginary time Green's function $g(\tau)$, return its particle-hole conjugate $g(\beta-\tau)$.

ph_conj(g::Keldysh.ImaginaryTimeGF{T, true}) -> Any

Given a scalar-valued imaginary time Green's function $g(\tau)$, return its particle-hole conjugate $g(\beta-\tau)$.

ph_conj(
    g::QInchworm.keldysh_dlr.DLRImaginaryTimeGF{T, true}
) -> QInchworm.keldysh_dlr.DLRImaginaryTimeGF{T, true} where T

Given a scalar-valued imaginary time Green's function $g(\tau)$, return its particle-hole conjugate $g(\beta-\tau)$.