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Entropic Confinement in String-Net Models: An Analogue Study via SU(2)ₖ Fusion Categories

Xiaodong Yang

Published Jul 22, 2026• 0 citations
Publisher

Abstract

Recent lattice studies have revealed that the color flux tube between static quark-antiquark pairs exhibits an excess entanglement entropy (flux tube entanglement entropy, FTE2) that scales linearly with the quark separation L. In this paper, we demonstrate similar behavior in a string-net model based on SU(2)k fusion categories, where the nontrivial object j=1/2 (analogous to color charge) cannot exist in isolation due to the fusion rules, naturally exhibiting “confinement”. We compute the entanglement entropy of the flux tube connecting two j=1/2objects using the topological definition S(R)=ln⁡(dim⁡(Hom(R))) and find an entropy density σk=ln⁡ d1/2=ln⁡(2cos⁡(π/(k+2))). For k=3, the category reduces to the Fibonacci case, yielding an entropy density σ3 = ln φ ≈ 0.481 (φ is the golden ratio), which is qualitatively comparable in magnitude to the scale inferred from lattice studies and the entropy-surface mechanism. Minimizing the free energy F=E-TS yields a confinement–deconfinement transition at Tc=J/σk.The parameter k offers a tunable knob, making the SU(2)k family a computable laboratory for entropic confinement. The predicted entropy-density jump and critical scaling can be directly tested in quantum simulator platforms (e.g., Rydberg arrays or superconducting circuits) that realize Fibonacci anyonic models.

DOI
10.20944/preprints202607.1659.v1

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Recent lattice studies have revealed that the color flux tube between static quark-antiquark pairs exhibits an excess entanglement entropy (flux tube entanglement entropy, FTE2) that scales linearly with the quark separation L. In this paper, we demonstrate similar behavior in a string-net model based on SU(2)k fusion categories, where the nontrivial object j=1/2 (analogous to color charge) cannot exist in isolation due to the fusion rules, naturally exhibiting “confinement”. We compute the entanglement entropy of the flux tube connecting two j=1/2objects using the topological definition S(R)=ln⁡(dim⁡(Hom(R))) and find an entropy density σk=ln⁡ d1/2=ln⁡(2cos⁡(π/(k+2))). For k=3, the category reduces to the Fibonacci case, yielding an entropy density σ3 = ln φ ≈ 0.481 (φ is the golden ratio), which is qualitatively comparable in magnitude to the scale inferred from lattice studies and the entropy-surface mechanism. Minimizing the free energy F=E-TS yields a confinement–deconfinement transition at Tc=J/σk.The parameter k offers a tunable knob, making the SU(2)k family a computable laboratory for entropic confinement. The predicted entropy-density jump and critical scaling can be directly tested in quantum simulator platforms (e.g., Rydberg arrays or superconducting circuits) that realize Fibonacci anyonic models.
Recent lattice studies have revealed that the color flux tube between static quark–antiquark pairs exhibits an excess entanglement entropy (flux-tube entanglement entropy, FTE2) that scales linearly with the quark separation L. In this paper, we demonstrate similar behavior in a string-net model based on SU(2)k fusion categories, where the nontrivial object j=1/2 (analogous to color charge) cannot exist in isolation due to the fusion rules, naturally exhibiting “confinement”. We compute the entanglement entropy of the flux tube connecting two j=1/2 objects using the microcanonical (equal-weight) prescription S(R)=ln(dim(Hom(R))) and find an entropy density σk=lnd1/2=ln(2cosπk+2). For k=3, the category reduces to the Fibonacci case, yielding an entropy density σ3 = ln φ ≈ 0.4812 (φ is the golden ratio), which is qualitatively comparable in magnitude to the scale inferred from lattice studies and the entropy surface mechanism. Under a thermalization assumption for the fusion-channel degrees of freedom, minimizing the free energy F=(J−Tσk)L yields a confinement–deconfinement transition at (Tc=J/σk), which is first-order-like (tension sign reversal) rather than a continuous critical transition. The parameter k offers a tunable knob, making the SU(2)k family a computable laboratory for entropic confinement. The predicted entropy-density jump can be directly tested in quantum simulator platforms (e.g., Rydberg arrays or superconducting circuits) that realize Fibonacci anyonic models.
Recent lattice QCD studies have revealed an excess entanglement entropy (flux tube entanglement entropy, FTE 2 ) in the color flux tube between static quark–antiquark pairs, which grows linearly with the quark separation. In this paper, we illustrate a possible analogy in a simplified string-net model based on the Fibonacci fusion category, where the non-trivial object [Formula: see text] (analogous to color charge) cannot exist in isolation due to the fusion rule [Formula: see text], naturally exhibiting “confinement”. We compute the entanglement entropy of the flux tube connecting two [Formula: see text] objects and find an entropy density of [Formula: see text] ([Formula: see text] is the golden ratio), which quantitatively agrees with the lattice QCD measured FTE 2 density (approximately 0.3–0.5). Minimizing the free energy [Formula: see text] at fixed temperature and volume, the competition between the energy term [Formula: see text] and the entropy term [Formula: see text] yields a confinement-deconfinement phase transition at critical temperature [Formula: see text]. This model provides a heuristic demonstration platform as a minimal model for understanding the entropic properties of the QCD flux tube.
In recent work Amorosso et al. (2024), we computed a novel flux tube entanglement entropy (FTE) of the color flux tube stretched between a heavy quark-antiquark pair on a Euclidean lattice in (2+1)D Yang–Mills theory. Our numerical results suggested that FTE can be partitioned into an internal color entanglement entropy and a vibrational entropy corresponding to the transverse excitations of a QCD string, with the latter described by a thin string model. Since the color flux tube does not have transverse excitations in (1+1)D Yang–Mills theory, we use this simpler framework to perform an exact analytical computation of the contribution of the internal color degrees of freedom to FTE. For the multipartite partitioning of the color flux tube, we find the remarkable result that FTE depends only on the dimension of the color group representation and the number of times the flux tube crosses the boundary between the traced and untraced spatial regions but not on the string length. Our proof is independent of whether the replica and region boundaries on the lattice are placed on vertices or in plaquette centers.
In quantum mechanics, string-net condensed states---a family of prototypical states exhibiting nontrivial topological order---can be classified via their long-range entanglement properties, in particular, topological corrections to the prevalent area law of the entanglement entropy. Here we consider classical analogs of such string-net models whose partition function is given by an equal-weight superposition of classical string-net configurations. Our analysis of the Shannon and Renyi entropies for a bipartition of a given system reveals that the prevalent volume law for these classical entropies is augmented by subleading topological corrections that are intimately linked to the anyonic theories underlying the construction of the classical models. We determine the universal values of these topological corrections for a number of underlying anyonic theories including SU(2)k{}_{k}k​, SU(N)1{(N)}_{1}(N)1​, and SU(N)2{(N)}_{2}(N)2​ theories.
We expand on recent work [1] demonstrating the existence of a novel entanglement radius ξ0\xi_0ξ0​ characterizing flux tube entanglement entropy (FTE2^22) in (2+1)D Yang-Mills theory. This physical scale corresponds to the intrinsic thickness of the flux tube that must be fully severed by an entangling region for color degrees of freedom in the flux tube to contribute non-zero FTE2^22. We consider here geometries of the entanglement region VVV on the lattice where the length of the region cross-cutting the flux tube is of the same magnitude as ξ0\xi_0ξ0​. Our results further the conclusions of [1] by adding detailed new information on the topological structure of the entanglement radius of color flux tubes.
We study the entanglement of gluon fields in presence of a static QQˉQ\bar QQQˉ​ pair in quenched QCD. Using the replica method, we investigate the q=2q=2q=2 Renyi entropy of the entanglement of gluon fields inside and in the vicinity of the confining QCD string between the quark and the antiquark. We find that there is excess entropy of gluon entanglement compared to vacuum fluctuations. This excess of entanglement entropy is associated with the gluon flux tube, and we find that it has a finite non-zero value in the continuum. We investigate the dependence of gluon entanglement on the geometry of longitudinal and transverse partitioning of the flux tube. Our preliminary results suggest scaling of the entanglement entropy with the area of the boundary overlapping with the flux tube.