Add full update bond environment and gauge fixing

[Yue-Zhengyuan]

This PR adds the bond environment for nearest-neighbor full update, which is expressed using the CTM tensors. For the horizontal bond between sites at [r, c] and [r, c+1], the bond environment env is

    C1 --- T1 --------- T1 --- C2     r-1
    |      ‖             ‖     |
    T4 === XX ==     == YY === T2     r
    |      ‖             ‖     |
    C4 --- T3 --------- T3 --- C3     r+1
    c-1    c           c+1    c+2

where XX = X' X and YY = Y' Y, and X, Y are Q-tensors (PEPSOrth) obtained from _qr_bond.

However, this simple construction has two problems.

• Because of the approximate nature of CTMRG, env is not positive-definite, or even Hermitian.
• For physical (not random) iPEPS, its condition number (when solving equations to obtain the updated bond tensors) can be quite large (~1e+5 to 1e+7). The test test/bondenv/benv_fu.jl demonstrate this with an iPEPS obtained from Hubbard model simple update.

Thus this PR also implements a "gauge fixing" algorithm introduced in PhysRevB.90.064425 (and applied to FU in PhysRevB.92.035142; it can also be applied to NTU later), which does the following:

• Replace the bond environment env with (env + env') / 2; in addition, eigen-decomposition is used to remove possible small negative eigenvalues of env to make it semi-positive-definite.
• Use QR decomposition to "fix gauge" of env and greatly reduce its condition number (to ~1e+1).

[Yue-Zhengyuan]

@lkdvos I hope to get this merged before the NTU bondenv (#144) since I think FU may be a more widely acknowledged algorithm.

[codecov]

Codecov Report

All modified and coverable lines are covered by tests ✅

Files with missing lines	Coverage Δ
src/PEPSKit.jl	100.00% <ø> (ø)
src/algorithms/contractions/bondenv/benv_ctm.jl	100.00% <100.00%> (ø)
src/algorithms/contractions/bondenv/gaugefix.jl	100.00% <100.00%> (ø)

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[Yue-Zhengyuan]

The test failures are due to GitHub outage today. Tests relevant to the changes all pass.

[Yue-Zhengyuan]

Do you know the reason on why the bond environment condition number is quite large for a physical state, but small for a random state? And why can QR decomposition help?

[lkdvos]

I'm not sure I really have an explanation for the condition number of a physical state vs a random state. Speculation: It could be that for a random initial guess the gauge is just very bad, which affects the condition number more than the actual state it represents. Therefore it's not actually the state, but rather the gauging that makes this better, which is why the QR might help since that is a partial gauging procedure?

[Yue-Zhengyuan]

@lkdvos Besides the issue with inv of R, L which we decide to set aside, do you have other suggestions?

[lkdvos]

I think this looks good to me!