API Documentation

The TRIQS library provides various useful tools for studying interacting quantum systems.

The following provides a detailed reference documentation grouped into logical units.

If you are looking for a specific function, class, etc., try using the search bar in the top left corner.

Atomic problem diagonalization

Atomic problem diagonalization provides a lightweight exact diagonalization solver for finite fermionic Hamiltonians, together with helpers that build derived quantities from a solved eigensystem:

• triqs::atom_diag::atom_diag holds the block-diagonal Hamiltonian, its eigensystem, and the matrix representations of the fundamental creation/annihilation operators in the eigenbasis. The Hilbert space is partitioned into invariant subspaces by auto-partitioning, by user-supplied quantum numbers, or by a particle-number window, and each block is then diagonalized with the QR algorithm.
• Thermodynamic averages of an observable are computed via triqs::atom_diag::partition_function, triqs::atom_diag::atomic_density_matrix and triqs::atom_diag::trace_rho_op, and the action of an operator on a state vector is provided by triqs::atom_diag::act.
• Eigenvalues of conserved quantities can be tabulated with triqs::atom_diag::quantum_number_eigenvalues and triqs::atom_diag::quantum_number_eigenvalues_checked.
• The atomic Green's function on an imaginary-time, Matsubara, Legendre or real-frequency mesh is built by triqs::atom_diag::atomic_g_tau, triqs::atom_diag::atomic_g_iw, triqs::atom_diag::atomic_g_l and triqs::atom_diag::atomic_g_w, all of which derive from the Lehmann representation exposed by triqs::atom_diag::atomic_g_lehmann.

Determinant manipulation

Determinant manipulation provides tools to efficiently compute determinants and ratios of determinants, as needed for the acceptance probabilities of CTQMC solvers. By caching the inverse matrix and the determinant of an internally permuted matrix, rows and columns can be inserted, removed or changed at low cost, with most operations split into a cheap try step that only evaluates the determinant ratio and a complete step that applies the accepted update.

Experimental tools

Experimental tools collect functionality that is still under active development and whose API may change between releases. They live in the triqs::experimental namespace and are organized into two submodules:

• Lattice tools (experimental) provides advanced lattice tools, such as Fourier-series representations of lattice functions, tight-binding Hamiltonians, superlattice folding, Wannier90 file loaders and Brillouin-zone integration.
• Utilities (experimental) provides numerical helpers, such as adaptive integration and root finding.

Green's functions

Green's functions are generic containers for functions that are not known analytically, such as quantum many-body Green's functions, self energies and other correlation functions. A container is parameterised by a mesh (see Meshes), which fixes the domain and representation of the function, and a target, which fixes the value stored at each mesh point.

• GF concepts specify the requirements for Green's function types.
• GF containers are the central containers and their views, including their block-diagonal variants.
• GF functions provide common physical operations on Green's functions.
• GF input/output provides HDF5 serialisation.
• GF transforms convert between different representations.
• GF utilities provide factory functions, expression templates, MPI support and other utilities.

Hilbert space

Hilbert space provides the fermionic Fock space and related tools for performing many-body calculations:

• triqs::hilbert_space::fundamental_operator_set enumerates single-particle indices.
• triqs::hilbert_space::hilbert_space represents the fermionic Fock space generated by a fundamental operator set, and triqs::hilbert_space::sub_hilbert_space represents one of its invariant subspaces.
• Many-body states within a (sub-)Fock space are stored either as a dense vector (triqs::hilbert_space::state<HS, T, false>) or as a sparse map (triqs::hilbert_space::state<HS, T, true>).
• triqs::hilbert_space::imperative_operator applies a many-body operator directly to a state.
• triqs::hilbert_space::space_partition partitions a Fock space into the invariant subspaces of a Hermitian operator.

Lattice tools

Lattice tools provide classes to describe crystal lattices and tight-binding Hamiltonians:

• triqs::lattice::bravais_lattice represents a Bravais lattice in 1, 2 or 3 dimensions, defined by its basis vectors and the positions of atomic orbitals within the unit cell.
• triqs::lattice::brillouin_zone represents the first Brillouin zone of a given Bravais lattice via its reciprocal-space basis vectors.
• triqs::lattice::tight_binding represents a tight-binding Hamiltonian on a Bravais lattice. It supports Fourier transforms to reciprocal space, dispersion (band-structure) calculations on Brillouin zone meshes and produces matrix-valued Green's functions on those meshes.
• triqs::lattice::grid_generator generates a regular grid of points inside the d-dimensional unit cube, useful for Brillouin zone sampling.

MC tools

The MC tools module provides a generic, MPI-aware Markov-chain Monte Carlo framework. The user supplies moves (see MC moves), measurements (see MC measurements) and a sign type satisfying the concepts in MC concepts. These are registered with a triqs::mc_tools::mc_generic instance (see MC simulation) which drives the warmup and accumulation phases. Random number generation and other building blocks are documented under MC utilities.

Meshes

Meshes form the basis of the Green's functions containers in the TRIQS library. They define the mathematical domain of the underlying function, how it is represented in memory, and how it is evaluated.

• Mesh concepts specify the requirements for mesh types and mesh points (triqs::mesh::Mesh, triqs::mesh::MeshWithValues, triqs::mesh::MeshPoint).
• Mesh types are the concrete mesh classes provided by TRIQS:
  • Imaginary time and frequency meshes for imaginary time and Matsubara frequency domains, including Matsubara frequencies arithmetic.
  • Real time and frequency meshes for real time and frequency domains.
  • Lattice meshes for Bravais lattices and Brillouin zones, with k-vector expressions for lazy k-vector arithmetic.
  • Function space meshes for function-space representations such as Legendre polynomials and DLR.
  • Product meshes combine several meshes into a single product mesh.
  • Other mesh types contain base classes and rarely used mesh types.
• Mesh utilities provide common utilities used across mesh types (iteration, evaluation, helper functions).
• Tail fitting tools fits the high- and low-frequency tails of functions on real or imaginary frequency meshes.

Operators

Operators provides a symbolic algebra for second-quantized many-body operators that act on the fermionic Fock space documented under Hilbert space. It is split into two parts:

• Many-body operators is the many-body operator algebra. triqs::operators::many_body_operator (with its real- and complex-valued aliases) is a linear combination of monomials of canonical creation/annihilation operators, assembled from the factory functions triqs::operators::c_dag, triqs::operators::c and triqs::operators::n and combined with ordinary +, - and * arithmetic.
• Operator utilities decomposes an operator of known structure into coefficient dictionaries and dense arrays

Statistical analysis tools

Statistical analysis tools is a header-mostly toolkit for the statistical analysis of Monte Carlo data and other sampled observables:

• Accumulators provide streaming binning accumulators: triqs::stat::lin_binning groups consecutive samples into equally sized bins and keeps the bin data for downstream resampling, while triqs::stat::log_binning maintains one bin per power-of-two bin size and reports the standard error and integrated autocorrelation time at each binning level.
• Mean and error estimation offers standalone functions triqs::stat::mean, triqs::stat::mean_and_err and their MPI counterparts. The flavour of mean (sum vs arithmetic mean) and error (variance, standard error of data, standard error of mean, jackknife error) is selected through the triqs::stat::mean_tag and triqs::stat::error_tag enums.
• Resampling techniques provides triqs::stat::jackknife for bias-corrected estimates and jackknife errors of an arbitrary function of the averages of one or more sampled observables.
• Utilities contains triqs::stat::histogram for counting samples into equally spaced bins on a closed interval, plus small helpers shared by the rest of the module.
• Concepts specify the value and range types accepted by the API (triqs::stat::StatCompatible, triqs::stat::AccCompatible, triqs::stat::StatCompatibleRange).

Every component has an MPI-aware variant so the same client code runs serially or in parallel.

Test tools

Test tools is a header-only collection of GoogleTest-based helpers used throughout the TRIQS C++ test suite.

It provides various macros and helper functions to simplify writing unit tests involving TRIQS components.

Utilities

Utilities collects the foundational building blocks shared across the TRIQS library:

• I/O streams provide verbosity-controlled and auto-indenting output streams.
• Macros and exceptions provide the exception hierarchy, various macros, and the compiler / platform glue used by every other component.
• Math helpers expose numerical helpers used across the library, such as interpolation points and weights, analytic continuation and runtime-typed numbers.
• Other utilities collects general-purpose helpers that do not fit the other categories.
• Runtime utilities contains process-level utilities such as timers and signal handling tools.
• Tuple and variant tools contain various useful functions when working with std::tuple and std::variant types.
• Type traits and metaprogramming collect template-metaprogramming helpers, type traits and RAII helpers.