sparse-ir

sparse-ir - A library for the intermediate representation of propagators

This library provides routines for constructing and working with the intermediate representation of correlation functions. It provides:

• on-the-fly computation of basis functions for arbitrary cutoff Λ
• basis functions and singular values are accurate to full precision
• routines for sparse sampling

Installation

Install via pip <https://pypi.org/project/sparse-ir>_::

pip install sparse-ir

Install via conda <https://anaconda.org/spm-lab/sparse-ir>_::

conda install -c spm-lab sparse-ir

sparse-ir requires numpy <https://numpy.org/>, scipy <https://scipy.org/>, and pylibsparseir <https://pypi.org/project/pylibsparseir>_ (a thin Python wrapper for the libsparseir C API from sparse-ir-rs <https://github.com/SpM-lab/sparse-ir-rs>_).

To manually install the current development version, you can use the following::

Only recommended for developers - no automatic updates!

git clone https://github.com/SpM-lab/sparse-ir cd sparse-ir uv sync

Note: uv is a fast Python package manager. If you don't have it installed, you can install it with pip install uv or use pip install -e . instead.

Building documentation

To build the documentation locally, first install the development dependencies::

uv sync --group doc

Then build the documentation::

uv run sphinx-build -M html doc _build/html

The documentation will be available in _build/html/html/index.html.

Documentation and tutorial

Check out our comprehensive tutorial_, where we self-contained notebooks for several many-body methods - GF(2), GW, Eliashberg equations, Lichtenstein formula, FLEX, ... - are presented.

Refer to the API documentation_ for more details on how to work with the python library.

The symbols and conventions are defined on the notation page_, shared with the tutorial and the Julia and C libraries. In short: G(τ) = -⟨T c(τ) c†(0)⟩ for both statistics; Matsubara frequencies iν are passed as the reduced frequency n, an integer with ν = nπ/β (odd for fermions, even for bosons); G(iν) = ∫₀^β dτ e^{iντ} G(τ); and G(τ) is evaluated for τ in [-β, β], where -0.0 stands for 0⁻.

There is also a Julia library_ and (currently somewhat restricted) C library with Fortran bindings_ available for the IR basis and sparse sampling.

.. _comprehensive tutorial: https://spm-lab.github.io/sparse-ir-tutorial-v2/ .. _API documentation: https://sparse-ir.readthedocs.io .. _notation page: https://spm-lab.github.io/sparse-ir-doc/src/notation.html .. _Julia library: https://github.com/SpM-lab/SparseIR.jl .. _C library with Fortran bindings: https://github.com/SpM-lab/sparse-ir-rs

Getting started

Here is a full second-order perturbation theory solver (GF(2)) in a few lines of Python code::

# Construct the IR basis and sparse sampling for fermionic propagators
import sparse_ir, numpy as np
basis = sparse_ir.FiniteTempBasis('F', beta=10, wmax=8, eps=1e-6)
stau = sparse_ir.TauSampling(basis)
siv = sparse_ir.MatsubaraSampling(basis, positive_only=True)

# Solve the single impurity Anderson model coupled to a bath with a
# semicircular density of states with unit half bandwidth.
U = 1.2
def rho0w(w):
    return np.sqrt(1-w.clip(-1,1)**2) * 2/np.pi

# Compute the IR basis coefficients G0_l = -S_l rho0_l for the
# non-interacting propagator
rho0l = basis.v.overlap(rho0w)
G0l = -basis.s * rho0l

# Self-consistency loop: alternate between second-order expression for the
# self-energy and the Dyson equation until convergence.
Gl = G0l
Gl_prev = 0
while np.linalg.norm(Gl - Gl_prev) > 1e-6:
    Gl_prev = Gl
    Gtau = stau.evaluate(Gl)
    Sigmatau = U**2 * Gtau**3
    Sigmal = stau.fit(Sigmatau)
    Sigmaiv = siv.evaluate(Sigmal)
    G0iv = siv.evaluate(G0l)
    Giv = 1/(1/G0iv - Sigmaiv)
    Gl = siv.fit(Giv)

You may want to start with reading up on the intermediate representation. It is tied to the analytic continuation of bosonic/fermionic spectral functions from (real) frequencies to imaginary time, a transformation mediated by the kernel K(τ, ω) = e^{-τω}/(1 + e^{-βω}). The basis depends on the inverse temperature beta and on the frequency cutoff wmax, which you should choose so that the spectral function vanishes outside [-wmax, wmax]. In the dimensionless variables x = 2τ/β - 1 and y = ω/wmax, the kernel, and hence its singular value expansion, depends on them only through the cutoff Λ = beta * wmax (``basis.lambda``).

One can now perform a singular value expansion_ on this kernel, which generates two sets of orthonormal basis functions, one set V_l(ω), basis.v[l](w), on the real-frequency axis, orthonormal on [-wmax, wmax], and one set U_l(τ), basis.u[l](tau), for the same object in imaginary (Euclidean) time, orthonormal on [0, beta], together with the singular values S_l, basis.s[l], which couple the two sides.

By this construction, the imaginary time basis can be shown to be optimal in terms of compactness.

.. _intermediate representation: https://arxiv.org/abs/2106.12685 .. _singular value expansion: https://w.wiki/3poQ

License and citation

This software is released under the MIT License. See LICENSE.txt for details.

If you find the intermediate representation, sparse sampling, or this software useful in your research, please consider citing the following papers:

• Hiroshi Shinaoka et al., Phys. Rev. B 96, 035147_ (2017)
• Jia Li et al., Phys. Rev. B 101, 035144_ (2020)
• Markus Wallerberger et al., SoftwareX 21, 101266_ (2023)

If you are discussing sparse sampling in your research specifically, please also consider citing an independently discovered, closely related approach, the MINIMAX isometry method (Merzuk Kaltak and Georg Kresse, Phys. Rev. B 101, 205145_, 2020).

.. _Phys. Rev. B 96, 035147: https://doi.org/10.1103/PhysRevB.96.035147 .. _Phys. Rev. B 101, 035144: https://doi.org/10.1103/PhysRevB.101.035144 .. _SoftwareX 21, 101266: https://doi.org/10.1016/j.softx.2022.101266 .. _Phys. Rev. B 101, 205145: https://doi.org/10.1103/PhysRevB.101.205145

Development

Updating pylibsparseir Dependency

When updating the ``pylibsparseir`` dependency version, you must update it in
**both** ``pyproject.toml`` and ``.conda/meta.yaml`` to maintain consistency:

1. **Update pyproject.toml**::

       # Edit dependencies in pyproject.toml
       dependencies = [
           "pylibsparseir>=0.8.0,<0.9.0",  # Update version range
       ]

2. **Update .conda/meta.yaml**::

       # Edit both host and run requirements in .conda/meta.yaml
       requirements:
         host:
           - spm-lab::pylibsparseir >=0.8.0,<0.9.0
         run:
           - spm-lab::pylibsparseir >=0.8.0,<0.9.0

3. **Verify consistency**::

       python check_libsparseir_version_consistency.py

   This should output ``✅ Version specifications are consistent!``

4. **Commit changes**::

       git add pyproject.toml .conda/meta.yaml
       git commit -m "chore: update pylibsparseir dependency to >=0.8.0,<0.9.0"

Version Consistency Check
~~~~~~~~~~~~~~~~~~~~~~~~~~
This repository includes a tool to ensure consistency between different package managers:

- **Version Consistency Check**: Ensures that ``pylibsparseir`` version
  specifications in ``pyproject.toml`` and ``.conda/meta.yaml`` are consistent.

  Run the check manually::

      python check_libsparseir_version_consistency.py

  Or install as a pre-commit hook::

      pip install pre-commit
      pre-commit install

Release Process
~~~~~~~~~~~~~~~
To release a new version (e.g., ``2.0.0a10``):

1. **Create a working branch for version bump**::

       git checkout mainline
       git pull origin mainline
       git checkout -b bump-to-2.0.0a10

2. **Update version in pyproject.toml**::

       # Edit pyproject.toml: version = "2.0.0a10"

3. **Commit and push**::

       git add pyproject.toml
       git commit -m "Bump to v2.0.0a10"
       git push --set-upstream origin bump-to-2.0.0a10

4. **Create Pull Request and merge to mainline**

5. **Create and push tag**::

       git checkout mainline
       git pull origin mainline
       git tag v2.0.0a10
       git push origin v2.0.0a10

6. **Automated builds** (triggered by tag push):

   - PyPI: ``wheel.yml`` workflow builds and uploads to PyPI
   - conda: ``conda.yml`` workflow builds and uploads to SpM-lab channel

Both workflows are automatically triggered when a tag starting with ``v`` is pushed.