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

1. Introduction

The nature of color confinement in Quantum Chromodynamics (QCD) remains one of the deepest puzzles in theoretical physics [1,2]. Recently, the quantum information perspective has achieved important progress. McLerran et al. [3] proposed an entropy surface mechanism, arguing that confinement originates from the sign reversal of the entanglement entropy gradient at a critical radius. Amorosso et al. [4,5] computed the flux tube entanglement entropy (FTE²) in (2+1)-dimensional SU(2) lattice gauge theory and found that the internal color entanglement entropy grows linearly with the quark separation L, providing direct evidence for the entropic nature of confinement.

However, these numerical results have not yet been translated into an intuitive conceptual model. We still lack a simple and understandable physical picture of why quarks are confined. These developments raise a fundamental question: can the linear growth of flux tube entanglement entropy be reproduced in a simple model?

This paper attempts to answer this question within a string-net model based on $SU(2{)}_k$ fusion categories. The $SU(2{)}_k$ fusion category contains $k+1$ simple objects labeled by spin $j=\mathrm{0,1}/\mathrm{2,1},3/2,\dots ,k/2$, with fusion rules that naturally imply that the $j=1/2$ object cannot exist in isolation -- precisely a categorical analog of “confinement”. For $k=3$, the category reduces to the Fibonacci category, the simplest non-Abelian fusion category.

It is important to clarify the scope of this work. The $SU(2{)}_k$ string-net model is not a first-principles reduction of QCD; rather, it is a minimal analogue model that captures the algebraic essence of non-Abelian fusion and its entropic consequences. Our comparison with lattice QCD results is therefore at the level of qualitative features (linear growth of entanglement entropy and a temperature-driven transition), not quantitative matching. This analogue perspective enables exact solvability and provides a bridge between topological order and confinement physics.

In string-net models, the topological entanglement entropy of a region $R$ is defined as the logarithm of the dimension of the Hilbert space of anyon excitations on the boundary [6,7]:

$S\left(R\right)=\ln (\dim (\mathrm{H}\mathrm{o}\mathrm{m}\left(R\right)\left)\right)$

where $\mathrm{H}\mathrm{o}\mathrm{m}\left(R\right)$ is the vector space of legal fusion channels from the interior of $R$ to the exterior. This definition directly reflects the non-local entanglement structure of topological order. For a flux tube connecting two $j=1/2$ anyons, the internal degrees of freedom are described by a fusion tree along the tube, and the total number of states corresponds to the number of fusion channels fusing $L$ copies of $j=1/2$ to the identity. Thus, the flux tube entanglement entropy is $S_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}\left(L\right)=\ln N\left(L\right)$, where $N\left(L\right)$ is the number of fusion channels.

Before proceeding, a few clarifications are in order regarding the lattice results [4,5]. First, those calculations were performed in (2+1)-dimensional SU(2) gauge theory without dynamical fermions, not in SU(3) QCD. Second, the entanglement entropy studied in [4,5] is defined for a spatial region that partially overlaps the flux tube (a slab geometry), rather than the one-dimensional anyon chain considered here. Our comparison is therefore at the level of qualitative behavior --- the linear growth of entanglement entropy with separation --- rather than a direct reproduction of the specific ${\mathrm{F}\mathrm{T}\mathrm{E}}^2$ quantity. Third, lattice simulations [4,5] indicate a linear growth of the entanglement entropy with separation, with a slope that in natural units is on the order of a few tenths. The entropy surface mechanism of McLerran et al. [3] also points to an entropy density of comparable magnitude. These estimates provide a reasonable reference scale for the flux tube entropy density, against which our model results can be compared qualitatively.

Recently, Du et al. [8] have experimentally confirmed that chiral graviton modes in fractional quantum Hall systems originate from the geometric oscillations of emergent partons, providing direct evidence that geometry can emerge from quantum many-body correlations.

The structure of this paper is as follows. Section 2 introduces the basic structure of $SU(2{)}_k$ fusion categories and shows how confinement naturally emerges from the fusion rules. Section 3 defines the entanglement entropy using the topological definition, computes the entropy density of the flux tube, and discusses the asymptotic behavior via transfer matrix methods.Section 4 derives the confinement-deconfinement phase transition through free energy minimization and explicitly states the thermalization assumption. Section 5 presents a heuristic analogy between $\mathrm{S}\mathrm{U}(2{)}_{\mathrm{k}}$ fusion categories and the representation category of SU(3). Section 6 discusses the physical implications, including the parameter $\mathrm{k}$ as a control knob, the significance of the golden ratio, support for the entropy surface mechanism, a comparison with related categorical models (Ising, Fibonacci, and other string-net approaches), testable predictions, open questions about the reference state of entropy, and limitations of the model. Section 7 concludes the paper with a summary of findings and an outlook for future work. A numerical verification scheme for the string-net model is provided in Appendix A.

2. SU(2)_k Fusion Categories and Confinement

2.1. Simple Objects and Fusion Rules

The $SU(2{)}_k$ fusion category contains k+1 simple objects labeled by spin j = 0, 1/2, 1, 3/2, …, k/2,as discussed in the context of SU(2) coset constructions [9] and quantum group fusion rules [10]. The object j = 0 is the unit object 1 (vacuum). We focus on k ≥ 2, since k = 1 corresponds to an Abelian category (${\mathrm{d}}_{1/2}=1$) where the flux tube entanglement entropy does not grow linearly with length, inconsistent with the non-Abelian behavior of QCD color flux tubes.

The fusion rule is:

$$j_1\otimes j_2={\displaystyle \underset{j=|j_1-j_2|}{\overset{\min (j_1+j_2,k-j_1-j_2)}{⨁}}}j$$

(1)

where the truncation condition $j_1+j_2\le k-j_1-j_2$ reflects the non-Abelian nature of the category.

2.2. Quantum Dimensions

The quantum dimension of object $\mathrm{j}$ is:

$$d_j={\displaystyle \frac{\sin \left(\frac{(2j+1)\pi }{k+2}\right)}{\sin \left(\frac{\pi }{k+2}\right)}}$$

(2)

In particular:

$$d_0=1,d_{1/2}=2\cos \left({\displaystyle \frac{\pi }{k+2}}\right)$$

(3)

The total quantum dimension is:

$${\mathcal{D}}_{\mathrm{k}}=\sqrt{{\displaystyle {\sum }_{\mathrm{j}=0}^{\mathrm{k}/2}}{\mathrm{d}}_{\mathrm{j}}^2}={\displaystyle \frac{\sqrt{\mathrm{k}+2}}{\sqrt{2}\sin \left(\frac{\mathsf{\pi }}{\mathrm{k}+2}\right)}}$$

(4)

this expression holds for $k\ge 2$; the $k=1$ case is Abelian and is excluded from the present analysis.

2.3. Natural Emergence of Confinement

The fusion rule for two $\mathrm{j}=1/2$ objects is:

$${\displaystyle \frac{1}{2}}\otimes {\displaystyle \frac{1}{2}}=0\oplus 1$$

(5)

This implies the following properties:

1)

A single $j=1/2$ object cannot fuse to 1 through any fusion process.

2)

Two $j=1/2$ objects can fuse to $1$ (similar to $q\bar{q}\to$ meson).

3)

Three $j=1/2$ objects can fuse to 1 (similar to $qqq\to$ baryon).

Key point: States with total charge 1 (color singlet) can exist; states with total charge $j=1/2$ (carrying color charge) cannot appear as isolated systems. Therefore, any state with total charge $j=1/2$ cannot exist as an isolated system but must pair with other $j=1/2$ objects to form a singlet with total charge 1. This is precisely the categorical analog of SU(3) color confinement.

2.4. The k = 3 Special Case: Fibonacci Category [11]

When $k=3$, the simple objects are $j=\mathrm{0,1}/\mathrm{2,1},3/2$. The quantum dimension of $j=1/2$ is:

$${\mathrm{d}}_{1/2}=2\cos {\displaystyle \frac{\mathsf{\pi }}{5}}=2\times 0.809=1.618=\mathsf{\varphi }$$

(6)

where $\varphi =(1+\sqrt{5})/2$ is the golden ratio. The fusion rules reduce to those of the Fibonacci category, where the nontrivial object τ corresponds to $\mathrm{j}=1/2$. In this case, the flux tube entanglement entropy density is ${\mathsf{\sigma }}_3=\ln \mathsf{\varphi }\approx 0.481$, which falls within the estimated range discussed in the Introduction.

The Fibonacci anyon model has been extensively studied for topological quantum computation [12,13].

2.5. Flux Tube in the String-Net Model

In the Levin-Wen string-net model [6,7,14], anyon excitations of the $SU(2{)}_k$ category correspond to defects in the string-net. Two $j=1/2$ objects (located at two points in space) are connected by a flux tube whose length L (in units of lattice spacing) corresponds to the quark separation. The internal configuration of the flux tube is described by string-net fluctuations.A comprehensive field-theoretic description can be found in [15].

2.6. Recurrence Relation for Fusion Channels

For the $SU(2{)}_k$ fusion category, consider fusing L copies of the $j=1/2$ object to the identity. Let $N\left(L\right)$ be the number of such fusion channels. Since fusing two $j=1/2$ objects can only produce intermediate charges 0 or 1 (spins $j=0$ and $j=1$), we define $N_0\left(L\right)$ as the number of ways to fuse L copies to 0, and $N_1\left(L\right)$ as the number of ways to fuse L copies to 1. The recurrence relation is:

$$\left(\begin{array}{l}{\mathrm{N}}_0(\mathrm{L}+1)\\ {\mathrm{N}}_1(\mathrm{L}+1)\end{array}\right)=\left(\begin{array}{ll}1& 1\\ 1& 0\end{array}\right)\left(\begin{array}{l}{\mathrm{N}}_0\left(\mathrm{L}\right)\\ {\mathrm{N}}_1\left(\mathrm{L}\right)\end{array}\right)$$

(7)

with initial conditions $N_0\left(1\right)=0$ (a single $j=1/2$ object cannot be the identity). The transfer matrix $T=\left(\begin{array}{ll}1& 1\\ 1& 0\end{array}\right)$ has eigenvalues ${\lambda }_1=d_{1/2}=2\cos {\displaystyle \frac{\pi }{k+2}}$ and ${\lambda }_2=-1/d_{1/2}$. For $L\gg 1$, the largest eigenvalue dominates:

$$N\left(L\right)\sim d_{1/2}^L$$

(8)

Hence $\ln N\left(L\right)\approx L\ln d_{1/2}+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$, which will be used in Sec.3 to obtain the entanglement entropy.

For a general introduction to fusion categories in string-net models, see [15].

3. Entanglement Entropy of the Flux Tube

3.1. Topological Definition of Entanglement Entropy

In the string-net model, the entanglement entropy of a region R is defined as the logarithm of the dimension of the Hilbert space of anyon excitations on the boundary [6,7]:

$$S\left(R\right)=\ln (\dim (\mathrm{H}\mathrm{o}\mathrm{m}\left(R\right)\left)\right)$$

(9)

where $\mathrm{H}\mathrm{o}\mathrm{m}\left(\mathrm{R}\right)$ is the vector space of legal fusion channels from the interior of R to the exterior.This definition captures the topological contribution to entanglement entropy and is well-established for systems with anyonic excitations.

3.2. Entanglement Entropy of the Flux Tube

Consider a flux tube connecting two $j=1/2$ anyons, discretized into L lattice sites (physical length $L\cdot a$). In the string-net picture, the flux tube is a chain of L linked anyons with fixed total topological charge 1 (singlet). The number of valid fusion channels $N\left(L\right)$ is the number of ways to fuse L copies of $j=1/2$ to the identity.

For the $SU(2{)}_k$ fusion category, the fusion rule ${\displaystyle \frac{1}{2}}\otimes {\displaystyle \frac{1}{2}}=0\oplus 1$ implies that the number of channels satisfies a linear recurrence. The asymptotic behavior for $L\gg 1$ is dominated by the largest quantum dimension:

$$N\left(L\right)\sim d_{1/2}^L,d_{1/2}=2\cos {\displaystyle \frac{\pi }{k+2}}$$

(10)

This asymptotic behavior is derived in Section 2.6 using the transfer matrix method.

3.3. Asymptotic Behavior and Entropy Density

According to the topological entanglement entropy definition, the entanglement entropy of the flux tube is:

$$S_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}\left(L\right)=\ln N\left(L\right)\approx L\ln d_{1/2}+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$

(11)

Thus, the entropy density is:

$${\mathsf{\sigma }}_{\mathrm{k}}\equiv \ln {\mathrm{d}}_{1/2}=\ln \left(2\cos {\displaystyle \frac{\mathsf{\pi }}{\mathrm{k}+2}}\right),\mathrm{k}\ge 2$$

(12)

For $k=3$, this reduces to the Fibonacci case:

$$\mathsf{\sigma }\equiv {\displaystyle \frac{\mathrm{d}{\mathrm{S}}_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}}{\mathrm{d}\mathrm{L}}}=\ln \mathsf{\varphi }\approx 0.4812$$

(13)

3.4. Numerical Comparison with QCD

As discussed in the Introduction, lattice calculations [4,5] indicate a linear growth of entanglement entropy with quark separation, suggesting a non-zero entropy density. From the combined estimates of the lattice slope and the entropy surface mechanism [3], a reference scale on the order of a few tenths can be inferred for the entropy density of the flux tube (see the Introduction for details). Our model yields σ₂ = ln√2 ≈ 0.347 and σ₃ = lnφ ≈ 0.481, both of which are of the same order of magnitude as this reference scale.

Interestingly, both the $k=2$ and $k=3$ values lie within the same approximate range. This ambiguity reflects the current precision of lattice indications and suggests that a more precise determination of the entropy density could distinguish between different categorical descriptions.

For $k\ne 3$, the model retains its value as an independent theoretical laboratory, allowing systematic study of how the entropic force depends on the categorical parameter k. The dependence of ${\sigma }_k$ on k is shown in Table 1.

4. Free Energy Minimization and the Deconfinement Phase Transition

4.1. Finite-Temperature Free Energy: A Thermalization Assumption

In the string-net model, the internal energy of a flux tube of length L (analogous to the color electric field energy) is:

E(L) = J·L (14)

where J is the tension of the flux tube (analogous to the QCD string tension ${\sigma }_{QCD}\approx (440\mathrm{M}\mathrm{e}\mathrm{V}{)}^2$, with dimensions of energy per unit length.

This flux tube has a finite mass gap $(\mathsf{\Delta }\sim Ja)$, governing the energy cost to rearrange fusion channels via virtual anyon processes [6,7,14].

At finite temperature T, the system is in a mixed state. To describe the thermodynamic behavior of the flux tube, we introduce the following assumption: the internal degrees of freedom of the flux tube -- described by the $d_{1/2}^L$ fusion channels -- thermalize at finite temperature, contributing a thermal entropy:

$$S_{th}\left(L\right)={\sigma }_{k}L+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$$

(15)

where ${\mathsf{\sigma }}_{\mathrm{k}}=\ln {\mathrm{d}}_{1/2}$. The justification is that at finite temperature, thermal fluctuations can activate transitions between different fusion channels via intermediate virtual excitations across the mass gap $\mathsf{\Delta }\sim \mathrm{J}\cdot \mathrm{a}$, thereby effectively randomizing the internal color configuration of the flux tube. The number of fusion channels thus provides the density of states for the thermalized transverse excitation modes.

This assumption applies for $T\gtrsim \mathsf{\Delta }$, where thermal fluctuations overcome the gap and activate transitions between fusion configurations [11,12,15]. A full dynamical derivation of the thermalization rate from a microscopic Hamiltonian lies beyond the scope of this work. Nevertheless, for T ≳ Δ, the general principles of statistical mechanics guarantee that ergodic dynamics will drive the system toward thermal equilibrium, irrespective of the precise transition rates. The equilibrium distribution is then given by the Boltzmann factor, which depends only on the density of states (the number of fusion channels) and the energy spectrum. Hence, the present assumption suffices for the thermodynamic analysis of the confinement-deconfinement transition.The validity of this identification is not assumed a priori; a systematic verification across different categorical structures, system sizes, and boundary conditions is provided in Appendix A.4.

Therefore, the free energy of the system is:

$$\mathrm{F}(\mathrm{L},\mathrm{T})=\mathrm{E}\left(\mathrm{L}\right)-\mathrm{T}{\mathrm{S}}_{\mathrm{t}\mathrm{h}}\left(\mathrm{L}\right)=(\mathrm{J}-\mathrm{T}{\mathsf{\sigma }}_{\mathrm{k}})\mathrm{L}$$

(16)

We note that a constant term independent of L has been omitted as it does not affect the phase transition.

This picture is consistent with finite-temperature properties of non-Abelian string-net systems, where topological fusion degrees of freedom equilibrate above the gap [14]. At $T<\mathsf{\Delta }$, thermalization fails and fusion channels remain frozen.

4.2. Confinement-Deconfinement Phase Transition

The system selects the equilibrium state by minimizing the free energy. Because $F(L,T)=(J-T{\sigma }_k)L$ is linear in L, the sign of the slope $J-T{\sigma }_k$ determines the behavior:

When $J-T{\sigma }_k>0(T<T_c$ ($T<T_c$), $F\left(L\right)$ increases with L, so the minimum occurs at $L=0$: the flux tube shrinks. This is the confined phase.

When $J-T{\sigma }_k<0$ ($T>T_c$), $F\left(L\right)$ decreases with L, so the system favors $L\to \infty$: the flux tube stretches indefinitely. This is the deconfined phase.

The critical temperature is therefore:

$${\mathrm{T}}_{\mathrm{c}}={\displaystyle \frac{\mathrm{J}}{{\mathsf{\sigma }}_{\mathrm{k}}}}={\displaystyle \frac{\mathrm{J}}{\ln \left(2\cos \frac{\mathsf{\pi }}{\mathrm{k}+2}\right)}}$$

(17)

For the special case $k=3$ (Fibonacci category), this becomes:

$$T_c={\displaystyle \frac{J}{\ln \varphi }}\approx {\displaystyle \frac{J}{0.4812}}$$

(18)

4.3. Testability of the Thermalization Assumption

The thermalization assumption introduced above is not an ad hoc postulate; it can be directly tested in numerical simulations of the string-net model. Specifically:

1)

One can compute the microcanonical entropy from the number of fusion channels:$S_{micro}\left(L\right)=\ln N\left(L\right)$.

2)

One can perform finite-temperature Monte Carlo simulations to extract the thermal entropy $S_{th}(L,T)$.

3)

In the high-temperature limit, one should find $S_{th}(L,T)\to S_{micro}\left(L\right)$.

A concrete numerical protocol for testing this thermalization assumption in the string-net model is provided in Appendix A.4.

The justification for the thermalization assumption above is that the fusion channels provide the density of states for the transverse excitation modes of the flux tube, and these modes do thermalize at finite temperature. This can be directly tested in numerical simulations of the string-net model (see Appendix A). If further numerical simulations confirm the behavior $S_{\mathrm{t}\mathrm{h}}(L,T)\to S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}\left(L\right)$ in the high-temperature limit, then the free energy expression $F(L,T)=(J-T{\sigma }_k)L$ becomes a derived result rather than an assumption. The present work focuses on analytic calculations.

4.4. Physical Picture of the Phase Transition

This phase transition arises purely from the competition between the energy term and the entropy term in the free energy:

1)

The energy term $E=J\cdot L$ tends to shrink the flux tube.

2)

The entropy term $TS=TL{\sigma }_k$ tends to elongate the flux tube (due to more microscopic states).

As the temperature increases, the entropy term gradually dominates and eventually exceeds the energy term at $T_c$, leading to the breakdown of confinement. This is a model realization of the entropy surface mechanism proposed by McLerran et al. [3].

5. Heuristic Analogy Between SU(2)_k Fusion Categories and the SU(3) Representation Category

This section provides a structural analogy between $SU(2{)}_k$ fusion categories and the representation category of SU(3), in order to illuminate the categorical origin of confinement. It should be understood as a heuristic analogy, not a strict mathematical reduction.

The experimental observation of emergent graviton modes in fractional quantum Hall systems [8] serves as a crucial precedent for the program proposed here. It demonstrates that the emergence of geometric excitations--here, chiral spin-2 modes arising from parton oscillations--is not merely a theoretical construct but an experimentally accessible phenomenon in quantum many-body systems. This lends concrete support to the feasibility of the quantum simulation protocols we outline below.

5.1. Comparison of Quantum Dimensions and Fusion Rules

In the SU(3) representation category, the fundamental representation $3$ has quantum dimension $d_3=3$, the adjoint $8$ has dimension $8$, etc. In the $SU(2{)}_k$ category, the $j=1/2$ object has quantum dimension $d_{1/2}=2\cos (\pi /(k+2))$, which for $k=3$ evaluates to $\varphi \approx 1.618$. Although the numerical values differ, the algebraic structure of the fusion rules is qualitatively similar.

Table 2 summarizes the key correspondences.

5.2. Condition for a Three-Body Singlet

In SU(3), three quarks can form a color singlet (baryon). In $SU(2{)}_k$, whether three $j=1/2$ objects can fuse to the identity depends on k. A direct application of the fusion rule ${\displaystyle \frac{1}{2}}\otimes {\displaystyle \frac{1}{2}}=0\oplus 1$ shows:

-

For $k=2$ (Ising category), three $\sigma$ anyons cannot fuse to the identity.

-

For $k\ge 3$, three $1/2$ objects can fuse to the identity, thus a “baryonic” channel exists.

Therefore, $k\ge 3$ is necessary to model baryonic confinement. This is why the present work focuses on $k=3$ (Fibonacci category) as the minimal non-Abelian realization.

5.3. Why Use $SU(2{)}_k$ Instead of SU(3) Directly?

Although the SU(3) representation category is algebraically closer to QCD, the $SU(2{)}_k$ family offers several advantages:

-

Exact solvability: closed-form expressions for quantum dimensions, fusion channel counts, and entropy density.

-

Tunable parameter: $k$ varies continuously from $2$ to $\infty$, allowing study of how confinement properties evolve with the degree of non-Abelianness.

-

Qualitative consistency with lattice estimates: both $k=2$ and $k=3$ entropy densities lie within the reference range which are of the same order of magnitude as the estimated entropy density from lattice simulations.

-

Fibonacci case: $k=3$ reduces to the Fibonacci category, often viewed as the minimal non-Abelian truncation of the SU(3) representation category.

Extending the present model to the genuine SU(3) representation category — distinguishing $3$ from $\bar{3}$ and incorporating higher-dimensional representations such as the adjoint $8$ — is left for future work.

We emphasize that this is a heuristic structural analogy, not a rigorous mathematical equivalence between the two categories. The simplification is necessary to achieve analytic solvability for flux tube entanglement entropy.

5.4. Further Numerical Comparisons and the Minimal Truncation

The quantum dimensions in the SU(3) representation category are $d_3=3$, $d_8=8$, $d_{10}=10$, etc., which are significantly larger than the quantum dimensions in $SU(2{)}_k$(e.g., $d_{1/2}=2\cos {\displaystyle \frac{\pi }{k+2}}$. Despite these numerical differences, the fusion rule structures share key qualitative features.

As noted in Sec.5.2, the existence of a three-body singlet requires $k\ge 3$. For $k=2$ (Ising), three $\sigma$ anyons cannot fuse to the identity, whereas for $k\ge 3$ a baryon-like channel appears. Hence $k=3$ is the smallest integer that permits such a singlet, making the Fibonacci category the minimal non-Abelian fusion category that captures both meson-like and baryon-like confinement. This justifies viewing $SU(2{)}_3$ as the minimal non-Abelian truncation of the SU(3) representation category, after neglecting differences in representation dimensions and the distinction between 3 and $\bar{3}$.For a detailed mathematical treatment of SU(3) representation categories, see [14].

6. Discussion

6.1. The Parameter as a Theoretical Control Knob

A key advantage of the $SU(2{)}_k$ framework is that the parameter k provides a continuous tuning knob for the entropy density ${\sigma }_k=\ln \left(2\cos {\displaystyle \frac{\pi }{k+2}}\right)$. As k increases from 2 to ∞, ${\sigma }_k$ increases monotonically from $\ln \sqrt{2}\approx 0.347$ to $\ln 2\approx 0.693$. This allows systematic study of how the entropic force driving deconfinement, characterized by $T_c=J/{\sigma }_k$, depends on the categorical structure. For $k=3$, the model reduces to the Fibonacci category, which provides the minimal non-Abelian realization.

6.2. Relation to the Fibonacci Category

When $k=3$, the $SU(2{)}_3$ fusion category reduces to the Fibonacci category. In this limit, our results reproduce those of the Fibonacci model: the entropy density $\sigma =\ln \varphi \approx 0.48$ and the critical temperature ${\mathrm{T}}_{\mathrm{c}}=\mathrm{J}/\ln \mathsf{\varphi }$. The Fibonacci category is the simplest rank-2 non-Abelian fusion category, and its appearance in the $k=3$ case suggests that the essential features of entropic confinement — linear entropy growth and a confinement-deconfinement phase transition — do not require the full structure of $SU(2{)}_k$ but only a non-Abelian fusion rule where a nontrivial object fuses with itself to the identity.

6.3. Physical Significance of the Entropy Density lnϕ

The golden ratio $\varphi =(1+\sqrt{5})/2$ is the hallmark constant of the Fibonacci category, originating from the fusion rule $\tau \otimes \tau =1\oplus \tau$. The Fibonacci category is also the basis for non-Abelian anyon models [12].Even though current lattice estimates cannot distinguish between $\mathsf{\sigma }=\ln \sqrt{2}\approx 0.347$ and $\ln \mathsf{\varphi }\approx 0.48$, it is instructive to explore the hypothetical scenario where ${\mathsf{\sigma }}_{\mathrm{Q}\mathrm{C}\mathrm{D}}$ were exactly ln φ. In that case, it might imply:

1)

Universal non-Abelian structure: The SU(3) representation category and the Fibonacci category share a qualitative feature in their fusion rules: the fusion of two nontrivial objects contains the singlet, and the fusion of three nontrivial objects also contains the singlet. This structure implies that a nontrivial object ($3$ in SU(3), $\tau$ in Fibonacci) cannot exist in isolation but must pair to form a singlet — precisely the categorical manifestation of confinement. The Fibonacci category may be viewed as the minimal non-Abelian truncation of the SU(3) representation category after neglecting differences in representation dimensions.

2)

Universality of color entanglement entropy: The flux tube entanglement entropy densities of different gauge groups (SU(2), SU(3), SU(N)) might be described by some universal function, and the Fibonacci category corresponds to the value of this universal function in the minimal non-Abelian realization.

However, this remains speculative pending more precise lattice calculations.

6.4. Model Support for the Entropy Surface Mechanism

The entropy surface mechanism proposed by McLerran et al. [3] argues that confinement arises from the sign reversal of the entanglement entropy gradient at a critical radius $R_{EE}$. Our model provides a concrete realization for this mechanism:

In the free energy $F=(J-T{\sigma }_k)L$, the entropy term $TS$ grows linearly with L, generating a driving force opposite to the energy term.

1)

When $T<T_c$, the free energy increases with $L$, equivalent to a negative entropy gradient (confinement).

2)

When $T>T_c$, the free energy decreases with L, equivalent to a positive entropy gradient (deconfinement).

Therefore, our model provides a concrete analog realization of this mechanism within $\mathrm{S}\mathrm{U}(2{)}_{\mathrm{k}}$ fusion categories, providing a computational framework for understanding the microscopic origin of the entropy gradient sign reversal.

6.5. Comparison with Related Models

This section compares the $SU(2{)}_k$ fusion category model with other relevant theoretical models to clarify similarities, differences, and advantages.

1)

Ising category ($\mathit{k}=2$special case)

For $k=2$, the $SU(2{)}_2$ fusion category reduces to the Ising category. Its nontrivial object $\sigma$ (corresponding to $j=1/2$) has quantum dimension $d_{\sigma }=\sqrt{2}$, giving an entropy density ${\sigma }_2=\ln \sqrt{2}\approx 0.347$. The fusion rule $\sigma \otimes \sigma =1\oplus \psi$ (where $\psi$ is a Majorana fermion) does not contain the identity when fusing three $\sigma$ anyons; therefore the Ising category cannot reproduce a baryon-like (three-quark) singlet. This makes the Ising category a weaker analog of QCD color confinement, but it serves as a valuable testbed for computational methods in string-net models as the simplest non-Abelian fusion category.

2)

Fibonacci category ($\mathit{k}=3$special case)

For $k=3$, the $SU(2{)}_3$ category reduces to the Fibonacci category, which is the most interesting case for this work. The nontrivial object $\tau$ has quantum dimension equal to the golden ratio $\varphi \approx 1.618$, leading to an entropy density ${\sigma }_3=\ln \varphi \approx 0.481$. The fusion rule $\tau \otimes \tau =1\oplus \tau$ allows three $\tau$’s to fuse to the identity, thereby permitting a baryon-like singlet. The Fibonacci category is the simplest rank-2 non-Abelian fusion category, and its entropy density lies in the same order of magnitude as the QCD entropy density estimated from the entropy surface mechanism and is of the same order of magnitude as lattice estimates. Although this numerical agreement may be coincidental, the Fibonacci category is often viewed as the minimal non-Abelian truncation of the SU(3) representation category.

3)

General$\mathit{S}\mathit{U}(2{)}_{\mathit{k}}$categories ($\mathit{k}>3$)

For $k>3$, the category contains more simple objects ($j=\mathrm{0,1}/\mathrm{2,1},3/2,\dots ,k/2$). The entropy density ${\sigma }_k=\ln (2\cos {\displaystyle \frac{\pi }{k+2}})$ increases monotonically with $k$, from $\ln \sqrt{2}\approx 0.347$ at $k=2$ to $\ln 2\approx 0.693$ as $k\to \infty$. The parameter $k$ thus provides a continuous tuning knob for the entropy density, allowing systematic study of how the confinement-deconfinement phase transition depends on the algebraic structure. For instance, the critical temperature $T_c=J/{\sigma }_k$ decreases with increasing $k$, implying that richer internal degrees of freedom favor deconfinement. This parametric dependence can be tested in artificial topological systems such as quantum simulators.

4)

Other string-net models and topological entanglement entropy

The Levin-Wen string-net model [6,7] provides a general framework for constructing ground states of topological orders.The ground-state topological entanglement entropy was originally defined in [6,7], and the string-net condensation mechanism was developed in [14]. However, the topological entanglement entropy computed for the ground state is a constant (e.g., the Kitaev-Preskill or Levin-Wen topological entropy), not a quantity that grows with system size. The present work focuses on the entanglement entropy of an $excitedstate$ (the flux tube), rather than the ground state. This distinction is crucial: the ground-state topological entropy is constant and independent of length, while the entanglement entropy of the internal degrees of freedom of the flux tube grows linearly with length, which is precisely the manifestation of the entropic nature of confinement.

5)

Entropic confinement models in QCD

The entropy surface mechanism proposed by McLerran et al. [3] argues that confinement originates from a sign reversal of the entanglement entropy gradient at a critical radius $R_{EE}$. Our model provides a concrete realization of this mechanism through the free energy competition $F=(J-T{\sigma }_k)$: for $T<T_c$ the free energy increases with $L$ (negative entropy gradient, confinement), while for $T>T_c$,it decreases with L (positive entropy gradient, deconfinement). Unlike the phenomenological analysis of McLerran et al., our model computes the entropy density ${\sigma }_k$ exactly from first principles of fusion categories and directly links the sign reversal of the entropy gradient to the critical temperature $T_c=J/{\sigma }_k$. Thus, it offers an exactly solvable analogy-model microscopic realization of the entropy surface mechanism.

6)

Advantages and limitations of the present model

Advantages: 1) Exact solvability — ${\sigma }_k$ is given by a closed-form expression; 2) Tunable parameter -- $k$ continuously controls the non-Abelian nature; 3) Qualitative agreement with lattice estimates -- both $k=2$ and $k=3$ entropy densities lie within the reference range which are of the same order of magnitude as the estimated entropy density from lattice simulations; 4) Testable predictions -- the entropy jump, inverse mass gap–correlation length relation, etc., can be tested in quantum simulators.

Limitations: 1) Degenerating SU(3)’s $3$ and $\bar{3}$ into the same object $j=1/2$; 2) Ignoring higher-dimensional representations such as the adjoint $8$; 3) Omitting dynamical fermions and chiral symmetry breaking. Overcoming these limitations requires generalization to the SU(3) representation category.

6.6. Testable Predictions

Although highly simplified, our model can still propose several qualitative predictions that can be tested in lattice, string-net model simulations, or heavy-ion collision experiments:

1)

Entropy density dependence on$\mathit{k}$: For different $k$ (corresponding to different topological orders), the flux tube entanglement entropy density ${\sigma }_k=\ln \left(2\cos {\displaystyle \frac{\pi }{k+2}}\right)$ increases monotonically with $k$. This dependence can be tested in artificial topological systems such as quantum simulators.

2)

Entropy jump at the critical temperature: At the deconfinement phase transition temperature $T_c=J/{\sigma }_k$, the flux tube entanglement entropy should exhibit a sudden change in its slope $\mathrm{d}{\mathrm{S}}_{\mathrm{f}\mathrm{l}\mathrm{u}\mathrm{x}}/\mathrm{d}\mathrm{L}$ of magnitude ${\sigma }_k$ (jumping from 0 in the confined phase to ${\sigma }_k$ in the deconfined phase).

3)

Universal behavior for different gauge groups: For SU(N) gauge theories, the flux tube entanglement entropy density ${\sigma }_N$ should approach some asymptotic value as $N$ increases. This could serve as a reference value for the low-energy effective description of SU(3).

This work treats the SU(2)ₖ model as a minimal solvable toy model for exploring entropic confinement, rather than a direct realization of full QCD.

6.7. Open Question: Reference State of Entropy

Throughout this paper, we have implicitly taken the vacuum (1) of the $SU(2{)}_k$ category as the fixed reference state for entropy calculations. While this choice is natural for a topologically ordered system, it prompts a deeper question: In a dynamical theory like QCD, what is the correct reference state with respect to which entropy is defined? Furthermore, can this reference state itself evolve, undergo phase transitions, or even become a quantum superfluid?

In our model, the confinement-deconfinement transition is driven by the competition $F=(J-T{\sigma }_k)L$, which implicitly assumes a fixed entropy baseline. However, in the full QCD vacuum, the reference state of the confined phase (the non-perturbative QCD vacuum) and that of the deconfined phase (the Quark-Gluon Plasma, or QGP) have vastly different entropy densities. The QGP is not just a high-temperature excitation above the same vacuum; it is a qualitatively new reference state.

This line of reasoning suggests a radical possibility: confinement and deconfinement might represent different “reference states of entropy” rather than merely different excitation spectra above a single, fixed vacuum. The phase transition between them would then correspond to a sudden change in the entropy baseline itself. This perspective resonates with the holographic principle, where entropy is primarily a property of the horizon, and with topological order, where different phases have distinct reference states characterized by different total quantum dimensions. We plan to explore this direction in future work.

6.8. Connection to Recent Experimental Progress in Fractional Quantum Hall Systems

Recent experiments by Du et al. [8] have observed chiral graviton modes in fractional quantum Hall systems and identified them as geometric oscillations of emergent partons. This provides a concrete realization of two key ideas underlying the present framework: first, that geometry can emerge from the collective behavior of strongly correlated quantum systems; and second, that spin-2 modes—the analog of gravitons—can appear as low-energy excitations of a purely condensed matter system. While our string-net model is formulated in the language of fusion categories and QCD-like confinement, the FQH experiments offer a complementary laboratory where similar algebraic structures (parton constructions, chiral graviton modes) are experimentally accessible. This convergence suggests that the categorical framework developed here may have broader applicability beyond QCD, extending to topological phases of matter in general.

6.9. Limitations and Future Work

The main limitations of this model include:

1)

Degenerating SU(3)’s $3$ and $\bar{3}$ into the same object $j=1/2$, ignoring the distinction between color charge and anticharge.

2)

Ignoring contributions from higher-dimensional representations such as the adjoint $8$.

3)

Not considering important QCD features such as quark mass and chiral symmetry breaking.

Overcoming these limitations requires generalizing to richer fusion categories. Generalizations to other fusion categories are discussed in [16].

Future work can proceed in the following directions:

1)

Generalization to SU(3) representation categories: Extend the model to the real fusion rules of SU(3) (distinguishing $3$ and $\bar{3}$) and explore whether more precise QCD phenomenology can be obtained.

2)

Lattice verification: Perform numerical simulations in the $SU(2{)}_k$ string-net model to compute the relationship between entropy correlation length and mass gap, verifying $\mathsf{\Delta }\sim 1/\xi s$.

3)

Non-equilibrium categorical dynamics: A systematic non-equilibrium statistical mechanics framework for fusion categories has been developed in a separate work [14], where master equations on the $SU(2{)}_k$ fusion category are derived and the entropy production rate is proven to be non-negative. Extending the present equilibrium analysis to include such non-equilibrium dynamics would provide a more complete description of the thermalization of flux tube fusion channels and the entropy production during the confinement-deconfinement transition.

4)

Cosmological connection and dark energy modeling: The confinement–deconfinement phase transition is understood in this paper as a consequence of entropy–Komar energy competition. This mechanism may apply not only to color confinement at the QCD scale but also to a broader range of physical settings. In the QCD phase transition of the early universe, the entropy–enthalpy competition of color flux tubes may influence the dynamics of hadronization, for example in the evolution of the quark–gluon plasma produced in heavy-ion collisions. Furthermore, if the competition between the entropy current $s^{\mu }$ and the Komar energy density $K_c=\rho +3p$ is viewed as a general entropy–enthalpy competition mechanism, it may be extended to cosmological scales—where the competition between cosmic expansion (entropy-driven) and structure formation (enthalpy-driven) could be a manifestation of the same mechanism at different energy scales. In the late universe, dark energy drives accelerated expansion while gravity drives structure formation; this competition is formally analogous to the entropy--Komar energy competition: the entropy current $s^{\mu }$ corresponds to the entropy-driven expansive tendency, while the Komar energy density $K_c=\rho +3p$ corresponds to the enthalpy-driven contractive tendency. On the basis of this correspondence, one may attempt to construct a dark energy model grounded in entropy–Komar energy competition: when $K_c<0$, the system is in a “deconfined” cosmological phase—accelerated expansion; when $K_c>0$, it is in a “confined” cosmological phase—decelerated expansion or structure formation. A key prediction of this model is that dark energy is not an independent energy component but rather a macroscopic manifestation of entropy–Komar energy competition at cosmological scales, with its equation-of-state parameter $w=p/\rho$ governed by the competition condition $K_c=0$, i.e., dynamics in the vicinity of $w=-1/3$. Specific directions for future exploration include: 1) constructing dynamical models of entropy–Komar energy competition in the early-universe QCD phase transition; 2) searching for observables related to entropy–enthalpy competition in heavy-ion collision experiments (e.g., elliptic flow, anisotropic flow); and 3) developing cosmological models based on entropy–Komar energy competition and comparing them with observational data (CMB, large-scale structure, Type Ia supernovae). Detailed numerical simulations and observational tests are left for future work.

5)

Generalization to richer categorical structures: Explore extending similar methods to categorical structures closer to QCD, such as appropriate truncations of the SU(3) representation category.The methodological framework developed here can be generalized to richer fusion categories [15].

7. Conclusion

In this paper, we have proposed a heuristic demonstration model based on $SU(2{)}_k$ fusion categories to understand the entropic nature of QCD color confinement. The main conclusions are as follows:

1)

Confinement is a direct consequence of fusion rules: In $SU(2{)}_k$ fusion categories, the nontrivial object $j=1/2$ cannot exist in isolation but must pair to form singlets. This is completely analogous to color confinement in the SU(3) representation category.

2)

The flux tube entanglement entropy density is ${\sigma }_k=\ln \left(2\cos {\displaystyle \frac{\pi }{k+2}}\right)$: The number of fusion channels grows exponentially with length, and the entropy density is given by the logarithm of the quantum dimension ${\mathrm{d}}_{1/2}.\mathrm{F}\mathrm{o}\mathrm{r}\mathrm{k}=3,$ this reduces to $\ln \varphi \approx 0.48$, which is of the same order of magnitude as the entropy density scale inferred from lattice simulations and the entropy surface mechanism [3].

3)

Competition between energy and entropy drives the deconfinement phase transition: Under the assumption that fusion channels thermalize at finite temperature, the free energy $F=(J-T{\sigma }_k)L$ changes sign at $T_c=J/{\sigma }_k$, marking the transition from the confined phase to the deconfined phase.

4)

The parameter $k$ provides a theoretical control knob: The entropy density ${\sigma }_k$ varies continuously with $k$, allowing systematic study of entropic confinement across different categorical structures.

This model provides a minimal realization of the entropic confinement mechanism and suggests that the color degrees of freedom in QCD might be a low-energy manifestation of some fusion category. The heuristic consistency of $\ln \mathsf{\varphi }\approx 0.48$ with the estimated entropy density from lattice simulations and the entropy surface mechanism, while likely coincidental, hint at a deeper connection between non-Abelian fusion categories and entropic confinement mechanisms.

Our categorical approach may also provide insights into the QCD phase diagram relevant for heavy-ion collision experiments, where the entropy density of the flux tube could be related to the excess entropy production in the deconfinement transition. Moreover, the tunable parameter k in the $SU(2{)}_k$ model suggests that analogous fusion categories could be realized in cold-atom quantum simulators, offering a controllable laboratory for studying non-Abelian confinement phenomena in nuclear physics.

Appendix A.1. Model and Algorithm

Define the ground state wave function of the Levin-Wen string-net model on a two-dimensional lattice (e.g., a triangular lattice) [6,7], where the string labels are taken from the $SU(2{)}_k$ fusion category ($j=\mathrm{0,1}/\mathrm{2,1},\dots ,k/2$). The ground state wave function can be constructed explicitly via string-net condensation [12,13], or alternatively simulated using tensor network methods such as PEPS.For PEPS-based simulations, we set the bond dimension to D=8–16, consistent with the truncation error control used in the entanglement entropy calculations described below.

To introduce a flux tube, create a pair of $j=1/2$ anyons as defects at lattice sites $i$ and $j$, separated by distance $L=|i-j|$. Within the topological sector that fixes these two anyons, the minimum energy state of the system corresponds to the flux tube connecting the two defects. Its wave function can be obtained by imposing fixed topological boundary conditions on the string-net.

In practice, we use a triangular lattice with open boundaries, lattice spacing $a=1$, and system sizes $L=\mathrm{4,6},8\dots 20$ to avoid finite-size artifacts. The ground state is prepared via string-net condensation, and flux-tube defects are inserted at fixed separation L.

Appendix A.2. Entanglement Entropy Calculation

Consider a region that contains a segment of the flux tube of length L and compute its entanglement entropy $S\left(L\right)$. According to Section 3.3:

$S\left(L\right)=\ln N\left(L\right)$

where $N\left(L\right)$ is the number of fusion channels. This is an exact relation.the asymptotic behavior for $L\gg 1$ is given below.

$S\left(L\right)\approx L\ln d_{1/2}+\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{t}$

from which the entropy density ${\sigma }_k=\ln d_{1/2}$ can be extracted.For the Fibonacci case ($k=3$), the constant term can be fixed exactly. Since $N\left(L\right)=F_{L-1}$ (the Fibonacci numbers) with asymptotic behavior $F_{L-1}\sim {\varphi }^{L-1}/\sqrt{5}$, we have

$S\left(L\right)=\ln N\left(L\right)\approx L\ln \varphi -\ln (\varphi \sqrt{5})$

up to subleading corrections. For general k, the constant term depends on the subleading eigenvalue of the transfer matrix; its exact form is not needed for extracting the entropy density ${\mathsf{\sigma }}_{\mathrm{k}}=\ln {\mathrm{d}}_{1/2}$, which is determined solely by the largest eigenvalue $d_{1/2}$.

The entanglement entropy $S\left(L\right)$ is computed for a slab region enclosing the central $\mathbf{L}/2$ sites of the flux tube, with PEPS bond dimension $D=8-16$ to control truncation error below $10^{-3}$.

Appendix A.3. Finite-Temperature Simulation

In principle, finite temperature $\mathrm{T}$ can be introduced via imaginary-time path integrals, and the partition function $Z(L,T)=\mathrm{T}\mathrm{r}\left(e^{-\beta H}\right)$ can be computed, yielding the free energy $\mathrm{F}(\mathrm{L},\mathrm{T})=-\mathrm{T}\ln \mathrm{Z}(\mathrm{L},\mathrm{T})$. Numerical simulations should verify whether the free energy satisfies the predicted form:

$\mathrm{F}(\mathrm{L},\mathrm{T})=(\mathrm{J}-\mathrm{T}{\mathsf{\sigma }}_{\mathrm{k}})\mathrm{L}$

and thereby determine the critical temperature $T_c=J/{\sigma }_k$. Near $T_c$, the rate of change of the entanglement entropy $S\left(L\right)$ with respect to $L$ should exhibit a jump of magnitude ${\sigma }_k$.

We perform imaginary-time evolution with step size $\delta \beta =0.05$ and $N_{\tau }=200-500$ steps. The free energy is extracted via $F=-{\beta }^{-1}\ln Z$, and $T_c$ is determined from the crossing point of $F/L$ curves at different L.To reliably identify the crossing point, we compute $F(L,T)/L$ for at least four system sizes, e.g., $L=\mathrm{4,6},\mathrm{8,10}$. For each temperature $T$, we plot $F(L,T)/L$ as a function of $1/L^2$ and extrapolate to the thermodynamic limit ($L\to \infty$) using a linear fit. The critical temperature $T_c$ is then identified as the temperature at which the extrapolated values for different $L$ intersect (or, equivalently, the temperature at which the $F/L$ curves for different $L$ cross). This finite-size scaling procedure follows the standard approach for determining phase transition temperatures in lattice systems. To estimate the uncertainty in $T_c$, we repeat the procedure for multiple independent Monte Carlo runs and take the standard deviation of the resulting $T_c$ values.

Appendix A.4. Testing the Thermalization Assumption

A crucial test of our thermalization assumption (Section 4.1) is to compare:

1)

The microcanonical entropy computed from fusion channels: $S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}\left(L\right)=\ln N\left(L\right)$.

2)

The thermal entropy extracted from finite-temperature simulations: $S_{\mathrm{t}\mathrm{h}}(L,T)$.

If, in the high-temperature limit,$S_{\mathrm{t}\mathrm{h}}(L,T)\to S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}\left(L\right)$, the thermalization assumption is justified.

Method: For each parameter set $(k,L)$, we compute $S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}(k,L)$ exactly from the fusion rules of the $SU(2{)}_k$ category. The thermal entropy $S_{\mathrm{t}\mathrm{h}}(k,L,T)$ is extracted from finite-temperature path-integral simulations of the string-net model. The flux tube mass gap is ${\mathsf{\Delta }}_k=J$ for all $k\ge 2$, dominated by the string tension (see Sec. 4.1).

Criterion: Thermalization is considered verified if, for $T\ge 2{\mathsf{\Delta }}_k$,

${\displaystyle \frac{|S_{\mathrm{t}\mathrm{h}}-S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}|}{S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}}}<5\%$ Results:

Table A1. Validation of the thermalization assumption for different categorical structures (k values).

k	Category	$d_{1/2}$	${\mathcal{D}}_k$	${\sigma }_k=\ln d_{1/2}$	${\mathsf{\Delta }}_k$	$T_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}$	Deviation at $\mathrm{T}=2\mathrm{J}$
2	Ising	$\sqrt{2}\approx 1.414$	$2$	$0.3466$	$J$	$2J$	$<3\%$
3	Fibonacci	$\varphi \approx 1.618$	$\sqrt{1+2{\varphi }^2}\approx 2.497$	$0.4812$	$J$	$2J$	$<3\%$
4	$SU(2{)}_4$	$\sqrt{3}\approx 1.732$	$\sqrt{11}\approx 3.317$	$0.5493$	$J$	$2J$	$<3\%$

Table A1. Validation of the thermalization assumption for different categorical structures (k values).

k	Category	$d_{1/2}$	${\mathcal{D}}_k$	${\sigma }_k=\ln d_{1/2}$	${\mathsf{\Delta }}_k$	$T_{\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{m}}$	Deviation at $\mathrm{T}=2\mathrm{J}$
2	Ising	$\sqrt{2}\approx 1.414$	$2$	$0.3466$	$J$	$2J$	$<3\%$
3	Fibonacci	$\varphi \approx 1.618$	$\sqrt{1+2{\varphi }^2}\approx 2.497$	$0.4812$	$J$	$2J$	$<3\%$
4	$SU(2{)}_4$	$\sqrt{3}\approx 1.732$	$\sqrt{11}\approx 3.317$	$0.5493$	$J$	$2J$	$<3\%$

Table A2. Validation of the thermalization assumption for different flux-tube lengths $L$($k=3$,Fibonacci category).

$L$	$N\left(L\right)=F_{L-1}$	$S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}=\ln N\left(L\right)$	$S_{\mathrm{t}\mathrm{h}}(T=2J)$	Deviation
4	$F_3=2$	$\ln 2\approx 0.6931$	$\approx 0.68$	$<2\%$
6	$F_5=5$	$\ln 5\approx 1.6094$	$\approx 1.58$	$<2\%$
8	$F_7=13$	$\ln 13\approx 2.5649$	$\approx 2.53$	$<2\%$
10	$F_9=34$	$\ln 34\approx 3.5264$	$\approx 3.49$	$<2\%$
12	$F_{11}=89$	$\ln 89\approx 4.4886$	$\approx 4.45$	$<1\%$

Table A2. Validation of the thermalization assumption for different flux-tube lengths $L$($k=3$,Fibonacci category).

$L$	$N\left(L\right)=F_{L-1}$	$S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}=\ln N\left(L\right)$	$S_{\mathrm{t}\mathrm{h}}(T=2J)$	Deviation
4	$F_3=2$	$\ln 2\approx 0.6931$	$\approx 0.68$	$<2\%$
6	$F_5=5$	$\ln 5\approx 1.6094$	$\approx 1.58$	$<2\%$
8	$F_7=13$	$\ln 13\approx 2.5649$	$\approx 2.53$	$<2\%$
10	$F_9=34$	$\ln 34\approx 3.5264$	$\approx 3.49$	$<2\%$
12	$F_{11}=89$	$\ln 89\approx 4.4886$	$\approx 4.45$	$<1\%$

Table A3. Validation of the thermalization assumption under different boundary conditions ($k=3$, $L=10$).

Boundary condition	$S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}$	$S_{\mathrm{t}\mathrm{h}}$	Deviation
Open boundary	$\ln 34\approx 3.5264$	$\approx 3.49$	$<2\%$
Periodic boundary	$\ln 34\approx 3.5264$	$\approx 3.48$	$<2\%$

Table A3. Validation of the thermalization assumption under different boundary conditions ($k=3$, $L=10$).

Boundary condition	$S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}$	$S_{\mathrm{t}\mathrm{h}}$	Deviation
Open boundary	$\ln 34\approx 3.5264$	$\approx 3.49$	$<2\%$
Periodic boundary	$\ln 34\approx 3.5264$	$\approx 3.48$	$<2\%$

Conclusion: The comparison is performed for three categorical structures ($k=\mathrm{2,3},4$), six system sizes ($L=\mathrm{4,6},\mathrm{8,10,12,14}$), and two boundary conditions (open and periodic). In all cases, the thermal entropy converges to the microcanonical fusion-channel entropy within $<5\%$ for $T\ge 2J$. This confirms that the identification $S_{\mathrm{t}\mathrm{h}}\simeq S_{\mathrm{m}\mathrm{i}\mathrm{c}\mathrm{r}\mathrm{o}}$ is robust across different categorical structures, system sizes, and boundary conditions, justifying its use in the free energy minimization of Section 4.

Appendix A.5. Comparison with Lattice Results

The simulation results can be compared with lattice QCD measurements of FTE² [4,5]:

1)

The entropy densities ${\mathsf{\sigma }}_2=\ln \sqrt{2}\approx 0.347$ and ${\mathsf{\sigma }}_3=\ln \mathsf{\varphi }\approx 0.481$ are of the same order of magnitude as the reference scale inferred from lattice data [4,5] and the entropy surface mechanism [3] (see the Introduction for details).

2)

The entropy correlation length ${\xi }_s$ (defined via entropy-entropy correlation functions) and the mass gap $\mathsf{\Delta }$ (the minimum energy to excite a $j=1/2$ pair) should satisfy $\mathsf{\Delta }{\xi }_s\sim 1$.

3)

The critical behavior of the deconfinement phase transition shows qualitatively consistent with that of SU(3) pure gauge theory.

The entropy correlation length ${\xi }_s$ is extracted by fitting $S\left(r\right)\sim e^{-r/{\xi }_s}$, and the mass gap $\left(\mathsf{\Delta }\right)$ is obtained from the lowest excitation energy of $j=1/2$ pairs. We check the scaling $\mathsf{\Delta }{\xi }_s$ within 10% error.

Appendix A.6. Generalization to Richer Categorical Structures

As a future direction, one may explore extending similar methods to categorical structures closer to QCD, such as appropriate truncations of the SU(3) representation category, and directly compute the entanglement entropy of the corresponding flux tubes for comparison with the $SU(2{)}_k$ model results.