[test] imtime gf interp

[HugoStrand]

Dear Joseph,

I think there might be a problem with the interpolation for imaginary time Green's functions. The interpolation result seems not to be time-translational invariant. I was expecting that

$$ G(\tau_1, \tau_2) = G(\tau_1 - \tau_2, 0) $$

In this pull request is a test that checks for this by calling the interpolation routine for the two cases above, comparing the result.

For me the test fails with:

% j time_gf.jl 
Test Summary: | Pass  Total
time_gf       |    8      8
t0  = BranchPoint(0.0 + 0.0im, 0.0, imaginary_branch)
t12 = BranchPoint(0.0 - 0.4im, 0.4, imaginary_branch)
t1  = BranchPoint(0.0 - 0.5im, 0.5, imaginary_branch)
t2  = BranchPoint(0.0 - 0.1im, 0.1, imaginary_branch)
d_12  = 0.0 - 0.16222222222222224im
d_120 = 0.0 - 0.16000000000000003im
imtime interpolation: Test Failed at /.../dev/Keldysh.jl/test/time_gf.jl:127
  Expression: diff < tol
   Evaluated: 0.0022222222222222088 < 1.0e-12
Stacktrace:
 [1] macro expansion
   @ /.../julia/share/julia/stdlib/v1.7/Test/src/Test.jl:445 [inlined]
 [2] macro expansion
   @ ~/dev/Keldysh.jl/test/time_gf.jl:127 [inlined]
 [3] macro expansion
   @ /.../julia/share/julia/stdlib/v1.7/Test/src/Test.jl:1283 [inlined]
 [4] top-level scope
   @ ~/dev/Keldysh.jl/test/time_gf.jl:93
Test Summary:        | Fail  Total
imtime interpolation |    1      1
ERROR: LoadError: Some tests did not pass: 0 passed, 1 failed, 0 errored, 0 broken.
in expression starting at /.../dev/Keldysh.jl/test/time_gf.jl:91

Could this be a bug?

Best, Hugo

[kleinhenz]

I believe this is because the interpolation function is doing bilinear interpolation on $G(\tau_1, \tau_2)$ rather than linear interpolation on $G(\tau_1 - \tau_2, 0)$. The reason for this is that the interpolation function is generic so it is doing the same thing in imaginary time as in real time and doesn't know anything about the symmetries of the problem. All of the symmetries are implemented at the level of discrete indexing. I believe the invariance should hold up to the same accuracy as the interpolation scheme overall.

In #13 I implement the imaginary time invariance symmetry exactly.

[HugoStrand]

Dear Joseph,

Ok, I understand. I think you are right that the error difference is only linear in $\Delta \tau$. Thank you for the fix in #13 it is good to exploit the simplification in imaginary time here.

I found this behaviour while trying to debug some other code and was a bit puzzled. I think the difference is not a big deal in the end. It turned out that I had bigger problems like time differences that should be $0^+$ but that were rounded to $0^-$ and wrapped to $\beta$ with a fermi sign.

Cheers, Hugo