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Does the Gribov horizon change the magic of the confining vacuum?

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Statement

Benedetti–Dabholkar–Dalmonte prove that a local QFT vacuum is necessarily magical because it is cyclic-separating (Reeh–Schlieder), has the continuous Bisognano–Wichmann modular spectrum, and lives on a type III₁ algebra. The Refined Gribov–Zwanziger vacuum satisfies none of these hypotheses in an obvious way: the functional integral is restricted to the first Gribov region, the propagators have complex conjugate poles (no Källén–Lehmann positivity), and Reeh–Schlieder is not known to hold. Does the Gribov-horizon restriction change the magic content of the vacuum — and can magic (anti-flatness) therefore serve as an order parameter for confinement?

Why It Matters

This is the natural place where the paper’s result collides with the group’s signature framework, and only this group is positioned to push it. Two outcomes are both interesting:

  • If the RGZ vacuum is less magical (flatter entanglement spectrum) than the standard vacuum, magic becomes a new confinement diagnostic — a departure from the universal “all physical states are magical” statement, signalling that the confining vacuum sits closer to the stabilizer / topological sector. The paper already isolates the stabilizer-compatible sector as TQFT (flat spectrum, not type III₁); a confining IR that flows toward a topological description (higher-form symmetry realization) would be expected to shed magic, tying to entanglement-as-confinement-probe.
  • If magic is unchanged, that is evidence the type III₁ + BW structure survives the Gribov restriction — a foundational algebraic fact the whole program tacitly assumes (Gap 4 of type-iii-algebras-across-areas) and which is currently open.

Either way it sharpens the boundary of validity of the magic universality theorem and connects the gribov-zwanziger and quantum-information lines.

What We Know

  • RGZ modifies the vacuum non-trivially. Complex-pole propagators, a soft BRST breaking, and the horizon function change the two-point structure that the modular operator is built from.
  • Reeh–Schlieder / BW status is open in RGZ. The algebraic structure of the RGZ vacuum is itself unresolved — entanglement-as-confinement-probe flags that the Haag–Kastler axioms and Reeh–Schlieder are not straightforwardly applicable when the functional integral is restricted by the non-local horizon condition.
  • Anti-flatness is computable from a two-point function. As in magic-of-coherent-squeezed-cat-states, can be evaluated once the (modified) wedge reduced density matrix is known, so the RGZ computation is a modification of a computation the group can already do.
  • TQFT is the flat-spectrum exception. The paper’s own boundary case (topological theories can be stabilizer) is the conceptual anchor for expecting confinement — if it is a flow toward a topological IR — to reduce magic.

Possible Approaches

  1. RGZ Rindler reduced state. Build the wedge reduced density matrix from the RGZ (complex-pole) propagator and evaluate anti-flatness ; compare to the standard-vacuum value at matched cutoff.
  2. Modular operator under the horizon restriction. Ask whether the restricted algebra still yields a continuous modular spectrum (type III₁) or acquires a gap / discrete component (a signature of a different algebraic type), which would show up as reduced anti-flatness.
  3. Confinement order-parameter test. Track across a deconfinement transition (finite , or as the Gribov parameter ) and check whether magic behaves as an order parameter, alongside the Polyakov loop / Wilson loop and higher-form symmetry realization.
  4. Consistency with positivity violation. Relate any anti-flatness change to the spectral positivity violation of the RGZ propagator — is lost magic tied to lost positivity?

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