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

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.