lsdensities is a Python library for the calculation of smeared spectral densities from lattice correlators.
Solutions can be obtained with the Hansen Lupo Tantalo method and Bayesian inference with Gaussian Processes, or combinations of the two.
This library is based on mpmath for performing the high-precision arithmetic operations that are necessary for the solution of the inverse problem.
Niccolò Forzano, Alessandro Lupo.
One can download, build and install the package
pip install https://github.com/LupoA/lsdensitiesPreliminary tests can be found in the tests folder, and tested using the pytest command.
Usage examples can be found in the examples folder.
The most basic workflow is illustrated in examples/runExact.py,
which generates a high-precision correlator, and computes the corresponding spectral density smeared with one of the
available kernels.
A realistic example is shown in examples/runInverseProblem.py, where input data for the correlator
needs to be provided.
The most complete class is src/lsdensities/InverseProblemWrapper.py, which
provides utilities for estimating errors and treating
the bias both in the HLT and in the Bayesian framework.
Function call example:
from lsdensities.utils.rhoUtils import (
init_precision,
Inputs,
)
from mpmath import mp, mpf
from lsdensities.core import hlt_matrix
from lsdensities.transform import coefficients_ssd, get_ssd_scalar
from lsdensities.utils.rhoMath import gauss_fp
# compute the smeared spectral density at some energy,
# from a lattice correlator
init_precision(128)
parameters = Inputs()
parameters.time_extent = 32
parameters.kerneltype = 'FULLNORMGAUSS' # Kernel smearing spectral density
parameters.periodicity = 'EXP' # EXP / COSH for open / periodic boundary conditions
parameters.sigma = 0.25 # smearing radius in given energy units
peak = 1 # energy level in the correlator
energy = 0.5 # energy at which the smeared spectral density
# is evaluated in given energy units
parameters.assign_values() # assigns internal variables
# based on given inputs
# such as tmax = number of data points,
# which is inferred from time_extent and periodicity,
# if not specified
lattice_correlator = mp.matrix(parameters.tmax, 1) # vector; fill with lattice data
lattice_covariance = mp.matrix(parameters.tmax) # matrix; fill with data covariance
for t in range(parameters.tmax): # mock data
lattice_correlator[t] = mp.exp(-mpf(t + 1) * mpf(str(peak)))
lattice_covariance[t,t] = lattice_correlator[t] * 0.02
regularising_parameter = mpf(str(1e-6)) # regularising parameters; must be tuned.
# Automatic tuning is provided in InverseProblemWrapper.py
# this example has exact data, so the parameters
# can be made as small as zero,
# in which case the result will be exact in
# the limit of infinite tmax
regularised_matrix = hlt_matrix(parameters.tmax, alpha=0) + (regularising_parameter * lattice_covariance)
matrix_inverse = regularised_matrix**(-1)
coeff = coefficients_ssd(matrix_inverse, # linear coefficients
parameters,
energy,
alpha=0)
result = get_ssd_scalar(coeff, # linear combination of data
lattice_correlator,
parameters)
true_value = gauss_fp(peak, energy, parameters.sigma, norm="full")
print("Result: ", float(result)) # reconstructed smeared spectral density at E = energy
print("Exact results :", true_value) # exact smeared spectral density at E = energyPull requests are welcome. For major changes, please open an issue first to discuss what you would like to change.
Please make sure to update tests as appropriate.
Development requirements can be installed by using pip install -r requirements.txt, and they are listed in requirements.txt.
For the main ideas: https://arxiv.org/pdf/1903.06476.pdf
For the Bayesian setup and the general treatment of the bias: https://arxiv.org/pdf/2311.18125.pdf