This note records the current design thinking for real nonsymmetric Arnoldi solves in iDMRG transfer-matrix workflows. It is intentionally exploratory.

Context

For real tensor-network problems, the transfer matrix is often real but non-Hermitian. At convergence it may be close to normal if the fixed point does not break time-reversal or reflection symmetry. During early iDMRG iterations, however, the transfer operator can be substantially non-normal, and in some physical cases the dominant fixed-point structure may genuinely require complex arithmetic.

The Krylov layer should therefore not silently assume that a real nonsymmetric operator has real wanted eigenvectors. It should make the real-versus-complex decision visible to the caller.

Policy Meaning

RealNonsymmetricPolicy is intended to answer one question:

‍When a real nonsymmetric Arnoldi solve finds that a wanted Ritz value is not numerically real, what should the real-arithmetic solver do?

RequireRealEigenpairs

Use this when the caller expects a real fixed point.

If a wanted Ritz value is classified as complex, or possibly ambiguous, the solver should stop and report the condition rather than silently continue. This is the conservative default for real iDMRG calculations where the physical fixed point is expected to remain real.

PromoteToComplexSuggested

Use this when the caller is willing to restart the larger calculation in complex arithmetic, but the Krylov solver itself should not perform that promotion implicitly.

For iDMRG this distinction matters because promotion affects tensors, environments, caches, and possibly symmetry handling. It should be an outer algorithm decision, not a hidden Krylov-side conversion.

AllowRealSchurPairs

Use this when the caller can consume a real two-dimensional invariant subspace for a complex conjugate pair.

For a real operator, complex eigenvectors u +/- i v correspond to a real invariant plane spanned by u and v. This is mathematically valid and may be useful for diagnostics or specialized algorithms. It is not automatically a drop-in replacement for an iDMRG fixed-point vector unless the outer algorithm knows how to use that plane.

This policy is currently a placeholder until real Schur two-plane output is implemented.

Status Semantics

The status set distinguishes the policy-specific outcomes for complex wanted Ritz values:

enum class NonsymmetricStatus {
  Converged,
  NotConverged,
  ComplexPairEncountered,
  ComplexPromotionRecommended,
  RealSchurPairRequired,
  AmbiguousReality,
  Breakdown,
  InvalidInput
};

Then the policy behavior would be:

Policy	Complex wanted Ritz value
`RequireRealEigenpairs`	`ComplexPairEncountered`
`PromoteToComplexSuggested`	`ComplexPromotionRecommended`
`AllowRealSchurPairs`	return real Schur/two-plane output, or `RealSchurPairRequired` until implemented

Ambiguous Ritz values should likely remain a separate AmbiguousReality status, because they indicate that the imaginary part is near the classification threshold and the caller may want to tighten tolerances, continue iterating, or inspect diagnostics before making a physical symmetry-breaking decision.

Non-Normality Is Separate

Complex wanted Ritz values are not the same thing as non-normality.

A real nonsymmetric transfer matrix can be highly non-normal while all wanted eigenvalues are real. Conversely, a nearly normal real operator can have a dominant complex conjugate pair. The policy above only handles the real-versus-complex output decision. It does not fully diagnose non-normal behavior.

Useful future diagnostics include:

conditioning or angle diagnostics for selected Ritz vectors,
residual history and stagnation information,
detection of near-degenerate real Ritz values that may become a complex pair under perturbation,
Schur residuals for nonnormal cases rather than relying only on individual eigenvector residuals.

For iDMRG, these diagnostics should help distinguish:

harmless early-iteration non-normality,
numerical ambiguity near a real fixed point,
genuine complex fixed-point structure,
cases where real arithmetic should stop and ask the outer algorithm to promote the calculation to complex arithmetic.

Current Implementation State

The current Phase 4 real nonsymmetric path keeps the Arnoldi basis and matrix-free operations real. It extracts complex Ritz values from the small projected Hessenberg matrix, classifies wanted Ritz values as real, ambiguous, or complex, reports the condition through NonsymmetricStatus, and records the projected departure from normality ||H^T H - H H^T||_F / max(1, ||H||_F^2) in diagnostics.

The current implementation does not yet:

return real Schur two-plane output,
compute non-normality diagnostics for the original matrix-free operator beyond the projected Hessenberg diagnostic,
restart the nonsymmetric Arnoldi factorization.

Those are future design and implementation steps.