sparse-ir 1.1.2

intermediate representation (IR) basis for electronic propagator

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:

pip install sparse-ir[xprec]

The above line is the recommended way to install sparse-ir. It automatically installs the xprec package, which allows one to compute the IR basis functions with greater accuracy. If you do not want to do this, simply remove the string [xprec] from the above command.

Install via conda:

conda install -c spm-lab sparse-ir xprec

Other than the optional xprec dependency, sparse-ir requires only numpy and scipy.

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
pip install -e sparse-ir/[xprec]

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.

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

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)
siw = sparse_ir.MatsubaraSampling(basis, positive_only=True)

# Solve the single impurity Anderson model coupled to a bath with a
# semicircular 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 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)
    Sigmaiw = siw.evaluate(Sigmal)
    G0iw = siw.evaluate(G0l)
    Giw = 1/(1/G0iw - Sigmaiw)
    Gl = siw.fit(Giw)

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 a kernel K. The kernel depends on a cutoff, which you should choose to be lambda_ >= beta * W, where beta is the inverse temperature and W is the bandwidth.

One can now perform a singular value expansion on this kernel, which generates two sets of orthonormal basis functions, one set v[l](w) for real frequency side w, and one set u[l](tau) for the same obejct in imaginary (Euclidean) time tau, together with a “coupling” strength s[l] between the two sides.

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

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).