irbasis3 3.0a8

intermediate representation (IR) basis for electronic propagator

irbasis3 - 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

pip install irbasis3 xprec

Though not strictly required, we strongly recommend installing the xprec package alongside irbasis3 as it allows to compute the IR basis functions with greater accuracy.

Quick start

Here is some python code illustrating the API:

# Compute IR basis for fermions and β = 10, W <= 4.2
import irbasis3, numpy
K = irbasis3.KernelFFlat(lambda_=42)
basis = irbasis3.FiniteTempBasis(K, statistics='F', beta=10)

# Assume spectrum is a single pole at ω = 2.5, compute G(iw)
# on the first few Matsubara frequencies. (Fermionic/bosonic Matsubara
# frequencies are denoted by odd/even integers.)
gl = basis.s * basis.v(2.5)
giw = gl @ basis.uhat([1, 3, 5, 7])

# Reconstruct same coefficients from sparse sampling on the Matsubara axis:
smpl_iw = irbasis3.MatsubaraSampling(basis)
giw = -1/(1j * numpy.pi/basis.beta * smpl_iw.wn - 2.5)
gl_rec = smpl_iw.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_ >= β*W, where β 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

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