w2dynamics: Local one- and two-particle quantities from dynamical mean field theory

• w2dynamics is licensed under GPLv3 and is implemented in Python, Fortran 90, and C++11, with listed required and optional dependencies.Source: PROGRAM SUMMARY · p. 1
• The package targets numerically unbiased finite-temperature one- and two-particle propagators for quantum impurity models, and approximate solutions for strongly correlated lattice models.Source: PROGRAM SUMMARY · p. 1
• It uses hybridization-expansion continuous-time quantum Monte Carlo, including worm sampling, for impurity problems and provides a DMFT solver for lattice problems.Source: PROGRAM SUMMARY · p. 1

1. Introduction

• CT-QMC is presented as the method of choice when considering more orbitals or calculating general two-particle Green’s functions with three independent frequencies.The paper notes that Fock-space scaling makes numerical solutions of correlated lattice models impractical beyond very few sites, except for a few special cases.Source: §1 · p. 2
• The CT-HYB solver uses worm sampling for general two-particle Green’s functions and improved self-energy estimators that help reduce high-frequency signal-to-noise problems.The method expands the partition function in bath hybridization and stochastically samples determinants.Source: §1 · p. 2
• w2dynamics can solve lattice Hamiltonians in DMFT, including material Hamiltonians derived from density functional theory or the GW approximation.It also interfaces with Wannier90-generated Hamiltonians through the wien2wannier interface.Source: §1 · p. 2
• For cluster extensions, treating the entire cluster as the impurity is possible in principle, but CT-HYB encounters a severe sign problem at moderate cluster sizes.Other CT-QMC methods fare better there but are restricted to single orbitals and/or density-density-like interactions.Source: §1 · p. 2
• w2dynamics calculates multi-orbital two-particle Green’s functions used to obtain local vertices for post-DMFT approaches and DMFT susceptibilities.These local vertices support diagrammatic extensions that determine non-local vertices and self-energies using ladder or parquet diagrams.Source: §1 · p. 3
• The paper outlines code and CT-HYB descriptions, followed by installation, usage parameters, and HDF5 output guidance.The user-oriented sections are introductory and refer readers to the online tutorial and wiki for further details.Source: §1 · p. 3

Implementation

• The DMFT modules are modularized for unit testing and reuse in other codes, with DMFTStep as the driver class.Source: §2 · p. 4

Lattice

• The lattice module computes the local Green’s function using a read-in k-grid, a semicircular Bethe-lattice density of states, or frequency-dependent hybridization to external leads.For DOS=ReadIn, correlated orbitals must precede uncorrelated orbitals for each atom in the input structure.Source: §2 · pp. 4–5, Figure 2 · p. 5

Filling

• Precomputing eigenvalues for fixed-filling chemical-potential searches trades O(NflNkNω) memory for reduced repeated computation; without sufficient memory, the code uses the direct strategy.The direct approach scales as O(Nfl^3NkNωNit), while the precomputed approach scales as O(Nfl^3NkNω) + O(NitNflNkNω).Source: §2 · p. 5

Atom

• Single-site DMFT uses atom-block-diagonal self-energies; atom-local projections map the full supercell problem to each impurity, while the user defines consistent atom groupings.The lattice Hamiltonian can retain inter-atom hopping, whereas the local interaction is assumed atom-local.Source: §2 · pp. 3, 5, 6
• Treating ligand orbitals as non-interacting spectators and separating atoms avoids putting their correlations into the impurity solver, reducing the number of impurity orbitals and computational burden.Alternatively, grouping all supercell atoms into one w2dynamics atom includes inter-atom correlations and self-energy elements, at increased computational cost.Source: §2 · p. 6

Local interaction

• Density interactions use orbital-occupation terms, while Kanamori adds spin-flip and pair-hopping terms to restore SU(2) symmetry; Coulomb and input-tensor parameterizations are also supported.The density-density scheme breaks SU(2) symmetry because it cannot represent spin-flip or pair-hopping terms.Source: §2 · pp. 6–7
• For density-density interactions, the Screening parameter adds a retarded interaction term to the effective impurity action.The code also supports finding a minimal block structure for general interactions by adding All to the QuantumNumber list.Source: §2 · pp. 7–8

ImpurityProblem

• Each DMFT iteration builds atom-specific Weiss fields, solves inequivalent impurity problems for Green’s functions, and returns the resulting self-energies blockwise to update the lattice Green’s function.Self-energy mixing is available to help stabilize the loop, while the user decides when it has converged.Source: §2 · p. 8

3. CT-HYB impurity solver

• CT-HYB expands the SIAM partition function in the hybridization and samples diagrams weighted by a local trace and a bath determinant.Source: §3 · p. 9

3.1. Steps

• Add/remove moves change expansion order; worm moves sample Green-function-like observables directly, while global moves target configurations separated by phase-space barriers.Source: §3.1 · pp. 10–12

3.2. Estimators

• Partition-function sampling measures one- and two-particle Green's functions and susceptibilities, while worm sampling accesses diagrams absent from the partition-function expansion.Source: §3.2 · pp. 12–13

3.3. Statistical uncertainties

• Estimator variances differ; runs discard warmups and estimate errors across cores, while calculating autocorrelation times across sampling schemes remains cumbersome.Source: §3.3 · pp. 13, 15

3.4. Local trace

• The local trace is an exact-diagonalization calculation over a Fock space of size 2^Nfl; superstate or state sampling avoids explicitly summing all impurity eigenstates.Source: §3.4 · p. 15

3.5. Quantum numbers

• Choosing conserved quantum numbers block-diagonalizes Hloc to reduce the largest block, while truncating Hloc is intended for sensitivity checks or cheaper DMFT pre-convergence.Source: §3.5 · pp. 15–16

3.6. Bath trace

• Offdiagonal hybridizations can be minimized by rotating the single-particle basis but not generally removed; combining them with Kanamori or full-Hubbard interactions requires special moves.Source: §3.6 · p. 16
• Four-operator moves are the default for generating spin-flip and pair-hopping configurations; flavor-change moves have much higher acceptance but produce more noise.Source: §3.6 · p. 16

4.1. Where to find the project and the code

• The GPLv3 code is publicly available through a GitHub repository, with issue tracking, wiki pages, mailing lists, and planned community workshops.The authors aim to build a community around a flexible DMFT code that can be embedded in other tools through its Python interface.Source: §4.1 · pp. 16–17

4.2. Installation

• Installation requires CMake, Fortran and C++ compilers, BLAS/LAPACK, Python, FFTW, and MPI; CMake can install several optional dependencies when absent.The text lists NFFT, numpy, HDF5 with Fortran bindings, mpi4py, h5py, scipy, and python-configobj among dependencies it can install.Source: §4.2 · p. 17
• With CMake installed and source files downloaded, the documented commands build CT-HYB and Maxent binaries; compilation is routinely tested on several Linux distributions.The authors recommend following tutorials covering Green’s functions, self-energy, occupations, Wannier Hamiltonians, basis-set benchmarks, and worm-sampled two-particle Green’s functions.Source: §4.2 · p. 17
• The repository’s benchmarks use problem-specific directories and parameter files; users run DMFT.py to produce an HDF5 archive of simulation data.The described workflow begins by inspecting Parameters.in before executing the script.Source: §4.2 · p. 18

Input parameters

• Configuration files use [General], [Atoms], and [QMC] sections, with numbered [Atoms] subsections for each unit-cell atom; users can override predefined defaults.A sample configuration is distributed, and Table 4 lists mandatory user-defined parameters.Source: §5 · p. 18, Table 4 · p. 19

HDF5 output

• HDF5 stores measured or calculated quantities alongside run metadata, including parameters, the shell environment, scheduler settings, shared axes, and quantity-to-axis links.The file also has iteration groups: “start” holds non-interacting data, while “dmft-”, “stat-”, and “finish” hold other quantities grouped by inequivalent atoms.Source: §5 · p. 18, Table 5 · p. 20

hgrep utility

• The hgrep command-line utility extracts subsets from one or more HDF5 files and can tabulate or plot them, including selected self-energy components and frequency ranges.The example selects the imaginary impurity self-energy for specified iterations, impurity, orbitals, spin, and frequencies.Source: §5 · p. 20

Conclusions

• The package calculates local two- and four-point fermionic Green’s functions for multi-orbital impurities, including impurities attached to leads or auxiliary impurities in DMFT and DFT+DMFT.The stated approximation level is DMFT; the package adds no further approximations to the numerically exact CT-QMC algorithm.Source: §6 · p. 20
• Arbitrary one- and two-particle Green’s functions and local susceptibilities provide inputs for physical-susceptibility calculations and diagrammatic extensions of DMFT.The package directly interfaces with AbinitioDGA and LadderDGA.Source: §6 · p. 20
• Worm sampling measures general n-particle Green’s-function components and extends CT-HYB to strongly insulating multi-orbital systems; superstate sampling speeds local-weight computation.Worm sampling also supports equal-time dynamic susceptibilities, improved self-energy estimators, and four-point vertex asymptotics.Source: §6 · p. 21
• Supported four-fermion interactions range from density-density to rotationally invariant Kanamori and full Coulomb tensors, and can include retardation effects.Examples of retardation sources include electron-phonon interactions and cRPA screening.Source: §6 · p. 21
• The DFT interface accommodates large unit cells with inequivalent atoms and supports Wannier90 Hamiltonians; magnetic phases and off-diagonal hybridization are supported, while full complex inter-orbital and inter-spin support is planned.The planned complex support is intended for calculations with spin-orbit coupling.Source: §6 · p. 21