Taking integration paths seriously

[pablosanjose]

We introduce submodule Paths that contain different possible integration paths for densitymatrix and josephson. The default densitymatrix(gs, ω0; kw...) corresponds to densitymatrix(gs, Paths.radial(ω0, π/4); kw...). This no longer an integral from -ω0 to ω0, but rather along a path as shown in the figure, where the angle in orange is set to ϕ = π/4, and the outer arcs are taken to infinity. (The angle can of course be changed by using the full densitymatrix(gs, Paths.radial(ω0, ϕ); kw...) syntax)

The integral over the arcs is analytic (thanks to spectral weight normalization), so only the radial parts need to be computed numerically. Along these radial parts the path is parametrized as ω = x * ω0 * cis(sign(x)*ϕ) with x from -Inf to Inf, so that ω0 has the meaning of a radial speed. This method is often faster than the old sawtooth path default, and does not require precomputing ω0 precisely as we did before (where it needed to be greater than the system half-bandwidth): the integral will converge to the same value regardless of ω0, only integration time will change, making this a much more robust default.

The old sawtooth path is available through Paths.sawtooth(reak_ω_points...; slope = 1, imshift = true) and we also have a general Paths.polygon(complex_ω_points...). Note that josephson(gs, ωs::Tuple) defaults to josephson(gs, Paths.sawtooth(ωs...)), and the same for densitymatrix.

The old path_override hack is no longer necessary, since we now also have Paths.polygon((µ, kBT; params...) -> ωpoints) as a possible path, which changes depending on chemical potential, temperature or system parameters.

As this is a breaking change, I summarize what needs to be done to old codebases to adapt:

• josephson(gs, ωmax) will now use a different (often faster) algorithm, and ωmax may be replaced with any estimated ω0 energy scale that dominates the contribution to the Josephson current (no need to compute a band range explicitly). This parameter only affects integration speed, not the result. To get the old behavior use josephson(gs, (-ωmax, ωmax)) instead. Same story for densitymatrix
• J(kBT, ωpoints; params...) as syntax to override ωpoints should now be replace with J = josephson(gs, Paths.polygon((µ, kBT; params...) -> ωpoints), and then simply J(kBT; params...). Note the use of µ in the polygon function signature, which is simply ignored in the josephson case, as µ = 0 there. Same story for densitymatrix, although in that case ωpoints can depend on µ too.

[codecov-commenter]

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Codecov Report

Attention: Patch coverage is 92.71255% with 18 lines in your changes missing coverage. Please review.

Project coverage is 92.45%. Comparing base (6197b8b) to head (4a828d2).
Report is 1 commits behind head on master.

Files with missing lines	Patch %	Lines
src/integrator.jl	90.27%	7 Missing ⚠️
src/observables.jl	92.72%	4 Missing ⚠️
src/solvers/green/internal.jl	88.88%	2 Missing ⚠️
src/types.jl	60.00%	2 Missing ⚠️
src/greenfunction.jl	97.87%	1 Missing ⚠️
src/specialmatrices.jl	97.05%	1 Missing ⚠️
src/tools.jl	90.00%	1 Missing ⚠️

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@@            Coverage Diff             @@
##           master     #325      +/-   ##
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- Coverage   92.47%   92.45%   -0.03%     
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  Files          39       41       +2     
  Lines        6907     7074     +167     
==========================================
+ Hits         6387     6540     +153     
- Misses        520      534      +14     

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