Clean up docstring for CTMRG contractions, particularly labeling of projectors

[Yue-Zhengyuan]

The docstrings (and @tensor index labels) for CTMRG contractions urgently need a careful revision, which is now causing me lots of headaches... I can easily spot several inconsistencies. To list a few:

• In renormalize_corner, the codomain index of P_right is called envlabel(:out):

PEPSKit.jl/src/algorithms/contractions/ctmrg_contractions.jl

Lines 584 to 589 in aa5875c

	|~~~~~~~~| -- |~~~~~~|
	|quadrant| |P_left| --
	|~~~~~~~~| -- |~~~~~~|
	| |
	[P_right]
	|

PEPSKit.jl/src/algorithms/contractions/ctmrg_contractions.jl

Lines 598 to 602 in aa5875c

	P_right_e = tensorexpr(
	:P_right,
	(envlabel(:out),),
	(envlabel(:L), ntuple(i -> virtuallabel(:L, i), N - 1)...),
	)

However, immediately below in renormalize_northwest_corner, it is called χ_in. It is also against the intuition that codomain indices are "out" indices.

PEPSKit.jl/src/algorithms/contractions/ctmrg_contractions.jl

Lines 658 to 666 in aa5875c

	function renormalize_northwest_corner(
	E_west, C_northwest, E_north, P_left, P_right, A::PEPSSandwich
	)
	return @autoopt @tensor corner[χ_in; χ_out] :=
	P_right[χ_in; χ1 D1 D2] *
	E_west[χ1 D3 D4; χ2] * C_northwest[χ2; χ3] * E_north[χ3 D5 D6; χ4] *
	ket(A)[d; D5 D7 D1 D3] * conj(bra(A)[d; D6 D8 D2 D4]) *
	P_left[χ4 D7 D8; χ_out]
	end

• Suppose we accept the convention for P_left, P_right in renormalize_north_edge:

PEPSKit.jl/src/algorithms/contractions/ctmrg_contractions.jl

Lines 940 to 956 in aa5875c

	```
	|~~~~~~| -- E_north -- |~~~~~~~|
	-- |P_left| | |P_right| --
	|~~~~~~| -- A -- |~~~~~~~|
	|
	```
	"""
	function renormalize_north_edge(
	(row, col), env::CTMRGEnv, P_right, P_left, network::InfiniteSquareNetwork
	)
	return renormalize_north_edge(
	env.edges[NORTH, _prev(row, end), col],
	P_right[NORTH, row, col],
	P_left[NORTH, row, _prev(col, end)],
	network[row, col], # so here it's fine
	)
	end

It looks fine that P_left is one column before the others. But then in renormalize_south_edge:

PEPSKit.jl/src/algorithms/contractions/ctmrg_contractions.jl

Lines 1020 to 1037 in aa5875c

	```
	|
	|~~~~~~~| -- A -- |~~~~~~|
	-- |P_right| | |P_left| --
	|~~~~~~~| -- E_south -- |~~~~~~|
	|
	```
	"""
	function renormalize_south_edge(
	(row, col), env::CTMRGEnv, P_left, P_right, network::InfiniteSquareNetwork
	)
	return renormalize_south_edge(
	env.edges[SOUTH, _next(row, end), col],
	P_left[SOUTH, row, col],
	P_right[SOUTH, row, _next(col, end)],
	network[row, col],
	)
	end

First, line 1025 shouldn't be there. Besides, suddenly P_right is at the next column (instead of previous), despite being on the left. I don't quite get it... It may be due to the convention used by SimultaneousCTMRG, but it is really a weird convention.

Besides, I also want to stop calling the projectors P_left, P_right, P_bottom, P_top. It looks kind of unnatural in the above example that P_right is appearing to the left of the south edge. Are they currently named along the "canonical" environment space arrow direction?

    ┌-←-P_right-←-P_left-←-┐
    ↓                      ↑
    P_top                  P_bot
    ↓                      ↑
    P_bot                  P_top
    ↓                      ↑
    └-→-P_left-→-P_right-→-┘

• Finally, one more example in which P_top appears at the bottom of the east edge (possibly of the convention drawn above?):

PEPSKit.jl/src/algorithms/contractions/ctmrg_contractions.jl

Lines 1061 to 1080 in aa5875c

	```
	|
	[~P_bottom~]
	| |
	E_west -- A --
	| |
	[~~P_top~~~]
	|
	```
	"""
	function renormalize_west_edge( # For simultaneous CTMRG scheme
	(row, col), env::CTMRGEnv, P_top::Array{Pt, 3}, P_bottom::Array{Pb, 3}, network::InfiniteSquareNetwork,
	) where {Pt, Pb}
	return renormalize_west_edge(
	env.edges[WEST, row, _prev(col, end)],
	P_top[WEST, row, col],
	P_bottom[WEST, _next(row, end), col],
	network[row, col],
	)
	end

Anyway, may I ask @pbrehmer to summarize how the projectors are labeled in the docstring?

[pbrehmer]

Sorry for causing confusion and indeed there are several inconsistencies. In fact, @leburgel noticed some of them a while ago and he also got very confused.

• Regarding the "in" and "out" labels: they are indeed inconsistent with the usual TensorKit convention of labeling the codomain with "out". This should definitely be changed.
• Regarding P_left and P_right: I called them P_left and P_right since these projectors correspond to left and right isometries. I usually remember this by looking at the north edge where the left and right isometries have their larger spaces to the left and right, respectively. At the bottom, this sequence is of course reversed since the environment is rotated by 180 degrees. If you turn your head by 180 degrees as well, P_left again has the larger spaces to the left. I do think that this makes sense and would tend to keep the names unless there is a good alternative suggestion.
• Regarding P_top and P_bottom: This naming convention I initially used in the context of the :sequential CTMRG scheme since there you only consider left moves (the west edge). I do agree that this is inconsistent with calling the other projectors P_left and P_right, so I would also like to change that.
• My suggestion: We relabel all indices with "in" and "out" labels that are consistent with TensorKit and we call all projectors P_left or P_right based on the fact that they are left and right isometries (so that means we get rid of P_top and P_bottom).

I won't be able to work on this immediately since I have other things to do but maybe we can discuss further details on Friday.