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https://github.com/numericaleft/lehmann.jl

Compact Spectral Representation for Imaginary-time/Matsubara-frequency Green's Functions
https://github.com/numericaleft/lehmann.jl

feynman-diagrams greens-functions julia many-body-physics quantum-statistics

Last synced: 27 days ago
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Compact Spectral Representation for Imaginary-time/Matsubara-frequency Green's Functions

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README 

        
# Lehmann



[![Stable](https://img.shields.io/badge/docs-stable-blue.svg)](https://numericaleft.github.io/Lehmann.jl/dev)

[![Dev](https://img.shields.io/badge/docs-dev-blue.svg)](https://numericaleft.github.io/Lehmann.jl/dev)

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This package provides subroutines representing and manipulating Green's functions in the imaginary time or the Matsubara-frequency domain. 



Imaginary-time Green's functions encode the thermodynamic properties of quantum many-body systems. They are typically singular and hard to deal with at low temperatures in numerical calculations. 



## Features

We provide the following components to ease the numerical manipulation of the Green's functions:



- Algorithms to generate the discrete Lehamnn representation (DLR), a generic and  compact representation of Green's functions proposed in Ref. [1]. DLR generally only requires $\\sim \\log(1/T)\\log(1/\\epsilon)$ numbers to represent a Green's function at a temperature T up to a given accuracy ϵ. This package provides two algorithms: one algorithm is based on a conventional QR algorithm, and another is based on a functional QR algorithm. The latter extends DLR to extremely low temperatures.



- Dedicated DLR for Green's functions with the particle-hole symmetry (e.g., phonon propagator) or particle-hole antisymmetry (e.g., superconductor gap function).



- Fast and accurate Fourier transform between the imaginary-time domain and the Matsubara-frequency domain with a cost $O(\\log(1/T)\\log(1/\\epsilon))$ and an accuracy $\\sim 100\epsilon$.



- Fast and accurate Green's function interpolation with a cost $O(\\log(1/T)\\log(1/\\epsilon))$ and accuracy $\\sim 100\\epsilon$.



- Fit a Green's function with noise.



## Installation



This package has been registered. So, type `import Pkg; Pkg.add("Lehmann")` in the Julia REPL to install.



## Basic Usage



In the following [demo](example/demo.jl), we will show how to compress a Green's function of ~10000 data points into ~20 DLR coefficients and perform fast interpolation and Fourier transform up to the accuracy ~1e-10.



```julia

using Lehmann

β = 100.0 # inverse temperature

Euv = 1.0 # ultraviolt energy cutoff of the Green's function

rtol = 1e-8 # accuracy of the representation

isFermi = false

symmetry = :none # :ph if particle-hole symmetric, :pha is antisymmetric, :none if there is no symmetry



diff(a, b) = maximum(abs.(a - b)) # return the maximum deviation between a and b



dlr = DLRGrid(Euv, β, rtol, isFermi, symmetry) #initialize the DLR parameters and basis

# A set of most representative grid points are generated:

# dlr.ω gives the real-frequency grids

# dlr.τ gives the imaginary-time grids

# dlr.ωn and dlr.n gives the Matsubara-frequency grids. The latter is the integer version.



println("Prepare the Green's function sample ...")

Nτ, Nωn = 10000, 10000 # many τ and n points are needed because Gτ is quite singular near the boundary

τgrid = collect(LinRange(0.0, β, Nτ))  # create a τ grid

Gτ = Sample.SemiCircle(dlr, :τ, τgrid) # Use semicircle spectral density to generate the sample Green's function in τ

ngrid = collect(-Nωn:Nωn)  # create a set of Matsubara-frequency points

Gn = Sample.SemiCircle(dlr, :n, ngrid) # Use semicircle spectral density to generate the sample Green's function in ωn



println("Compress Green's function into ~20 coefficients ...")

spectral_from_Gτ = tau2dlr(dlr, Gτ, τgrid)

spectral_from_Gω = matfreq2dlr(dlr, Gn, ngrid)

# You can use the above functions to fit noisy data by providing the named parameter ``error``



println("Prepare the target Green's functions to benchmark with ...")

τ = collect(LinRange(0.0, β, Nτ * 2))  # create a dense τ grid to interpolate

Gτ_target = Sample.SemiCircle(dlr, :τ, τ)

n = collect(-2Nωn:2Nωn)  # create a set of Matsubara-frequency points

Gn_target = Sample.SemiCircle(dlr, :n, n)



println("Interpolation benchmark ...")

Gτ_interp = dlr2tau(dlr, spectral_from_Gτ, τ)

println("τ → τ accuracy: ", diff(Gτ_interp, Gτ_target))

Gn_interp = dlr2matfreq(dlr, spectral_from_Gω, n)

println("iω → iω accuracy: ", diff(Gn_interp, Gn_target))



println("Fourier transform benchmark...")

Gτ_to_n = dlr2matfreq(dlr, spectral_from_Gτ, n)

println("τ → iω accuracy: ", diff(Gτ_to_n, Gn_target))

Gn_to_τ = dlr2tau(dlr, spectral_from_Gω, τ)

println("iω → τ accuracy: ", diff(Gn_to_τ, Gτ_target))

```



## Build DLR basis file

 A set of basis files have been precalculated and stored in the folder [basis](basis/). They cover most of the use cases. For edge cases, you may generate your own basis file use this [script](build.jl).



 In the above script, user can choose the folder to store the generated basis file. To use the new basis file, pass the folder as an argument when creating ``DLRGrid`` struct. More information can be found in the [documentation](https://numericaleft.github.io/Lehmann.jl/dev/lib/dlr/)



## Citation



If this library helps you to create software or publications, please let us know and cite



[1] ["Discrete Lehmann representation of imaginary time Green's functions", Jason Kaye, Kun Chen, and Olivier Parcollet, arXiv:2107.13094](https://arxiv.org/abs/2107.13094)



[2] ["libdlr: Efficient imaginary time calculations using the discrete Lehmann representation", Jason Kaye, Kun Chen and Hugo U.R. Strand, arXiv:2110.06765](https://arxiv.org/abs/2110.06765)



## Related Package

[__libdlr__](https://github.com/jasonkaye/libdlr) by Jason Kaye and Hugo U.R. Strand.



## Questions and Contributions



Contributions are very welcome, as are feature requests and suggestions. Please open an issue if you encounter any problems.