Is magic an order parameter for confinement and higher-form symmetry realization?

Statement

Benedetti–Dabholkar–Dalmonte show that magic sharply separates local QFT (necessarily magical, a net of type III₁ factors) from TQFT (can be a flat-spectrum stabilizer state — the toric code is the paradigm, $Z[{\mathcal{M}}_n]=(Z[{\mathcal{M}}_1]{)}^n{D}^{n-1}$). Since a deconfined gauge theory flows in the deep IR to a topological description while a confining one does not, does the magic of the ground state behave as an order parameter across the confinement/deconfinement transition — and does it track the realization vs. breaking of the relevant higher-form symmetry?

Why It Matters

This lifts the paper’s TQFT-vs-local-QFT boundary into the group’s confinement and generalized-symmetries programs, and connects three of its threads at once. Confinement is now understood as the (un)realization of a 1-form symmetry (Wilson-line area vs. perimeter law); topological order is a flat-spectrum / low-magic regime; and entanglement has already been proposed as a confinement diagnostic. If magic drops in the deconfined/topological phase and is non-zero (or peaks) across the transition, it would be a genuinely new, complexity-flavored order parameter — distinct from the Wilson/‘t Hooft loops and from entanglement entropy — with a clean many-body implementation. It also sharpens why topological phases are classically simulable (low magic) while genuine QFT phases are not.

What We Know

• Magic distinguishes the phases algebraically. The flat-spectrum (stabilizer) sector is exactly the topological one in the paper’s analysis, because TQFT is not a net of type III₁ factors. See magic-nonstabilizerness, topological-order, string-net-condensation.
• Lattice magic diagnostics exist. Stabilizer Rényi entropy and anti-flatness have been computed across criticality and in gauge theories on the lattice (Tarabunga–Tirrito–Dalmonte 2023; Dalmonte is a co-author of the QFT paper) — the condensed-matter bridge makes these directly usable.
• Higher-form-symmetry framing is in place. The wiki already tracks confinement as higher-form symmetry realization and the Wilson loop as the charged line; magic would be a state-property complement to these operator diagnostics.
• Gribov-side sibling. The closely-related question of whether the specific RGZ vacuum’s magic differs is magic-and-the-gribov-horizon; this question is the broader phase-diagram / symmetry-realization framing.

Possible Approaches

1. Lattice gauge theory scan. Compute the stabilizer Rényi entropy / anti-flatness of the ground state of a ${\mathbb{Z}}_N$ or $SU(N)$ lattice gauge theory across the confinement/deconfinement transition (Dalmonte-style Pauli–Markov sampling), and correlate with the Wilson-loop law.
2. Toric code as the deconfined anchor. Use the exactly-solvable toric code / deconfined ${\mathbb{Z}}_2$ phase (zero magic, flat spectrum) as the reference, and perturb toward the confined phase to watch magic turn on.
3. Higher-form order-parameter correspondence. Ask whether the onset of magic coincides with the restoration of the 1-form symmetry (perimeter→area law), making the correspondence “magic ↔ 1-form symmetry realization” precise.
4. Continuum reading. Interpret the lattice result through the paper’s algebraic lens: does approaching the topological IR correspond to the local algebra degenerating away from type III₁?

Related Questions

• entanglement-as-confinement-probe — the parent question on entanglement/Bell as confinement order parameters.
• magic-and-the-gribov-horizon — the Gribov-vacuum-specific version.
• julia-toulouse-higher-form-symmetries, non-invertible-symmetries-mcs — the generalized-symmetries context.
• magic-of-coherent-squeezed-cat-states — the free-field magic computation on the QFT side.