Topological Entanglement Entropy

• Topological entanglement entropy is a universal invariant in quantum many-body systems that captures long-range entanglement beyond conventional symmetry breaking.
• Recent methodologies employ mapping class group invariance, the replica trick, and surgical techniques to reduce complex knot bipartitions to tractable algebraic computations using the modular S-matrix.
• A robust lower bound of -2 ln D is established, with state-dependent corrections highlighting the interplay between universal quantum dimensions and ground state structure.

Topological entanglement entropy (TEE) characterizes long-range entanglement in gapped quantum many-body systems and serves as a universal, subleading term distinguishing topological order. In two-dimensional topological phases, the ground state entanglement entropy of a region with a smooth boundary of length $L$ assumes the form $S = \alpha L - \gamma + \cdots$ where the nonuniversal boundary term $\alpha L$ encodes short-range physics and the constant $\gamma$ is the TEE—a robust invariant signifying quantum order beyond symmetry breaking. Recent advances generalize TEE extraction to arbitrary subdivisions, link it to mapping class group invariance, and reveal deeper algebraic structures via knot bipartitions and Verlinde-like formulas. Its precise definition, computation, and universal lower bounds are now established in a variety of topological quantum field theory (TQFT) settings.

1. Mapping Class Group Invariance and Knot Bipartitions

In (2+1)d TQFTs on the torus, the entanglement and Rényi entropies for a given ground state depend on both the manifold bipartition and the choice of basis. However, when the bipartition is transformed alongside the manifold via a mapping class group action (specifically, $SL(2,\mathbb{Z})$ on the torus), the spectrum of the reduced density matrix and the resulting entropies are invariant. This property is pivotal for interfaces defined by torus knots—nontrivial cycles labeled $K(k_a, k_b)$ with coprime winding numbers. By applying an $SL(2,\mathbb{Z})$ transformation (a sequence of modular $S$ and $T$ moves), any bipartition along a torus knot (or set of knots) can be mapped to a simpler scenario: for instance, a pair of meridian (or longitude) cuts. The unitary transformation on the Hilbert space induced by mapping class group elements provides a correspondence between complicated “entangled” bipartitions and a canonical reference frame, substantially simplifying the analysis.

2. Replica Trick and Surgery Approach to TEE

The replica trick computes the $n$th Rényi entropy as $$S_n = \frac{1}{1-n} \ln \operatorname{Tr}(\rho_A^n)$$, from which the von Neumann entanglement entropy follows as $n\to 1$. In TQFT, this amounts to evaluating partition functions over replica manifolds assembled by gluing $n$ copies of the original space along the specified bipartition. For nontrivial knot bipartitions, the surgery method decomposes such manifolds into standard pieces (e.g., $S^3$, $S^2\times S^1$) whose partition functions are known exactly in the TQFT, and their combination is dictated by modular data—specifically the modular $S$-matrix and total quantum dimension $D = \sqrt{\sum_a d_a^2}$, where the $d_a$ are quantum dimensions of anyon types. This process ensures that, despite topological complexity, TEE computation is tractable and reducible to algebraic operations involving the modular group.

3. Verlinde-like Formulas and the Quantum Dimension

TEE for multi-interface or knot bipartitions is encoded by algebraic structures related to the modular $S$-matrix. In the untwisted basis (the canonical meridian cut), the ground state can be written as $|0\rangle = \sum_a S_{0a} |a\rangle$, with $S_{0a} = d_a / D$ up to normalization, so that the weights in $\rho_A$ are determined by the quantum dimension distribution. For a system divided by two non-intersecting S$^1$ knots (e.g., two cycles wrapped along different torus directions), the Rényi entropy is

$$S_n = \frac{1}{1-n} \ln \left[ \sum_{a_1,\ldots,a_n} \left( \prod_{i=1}^n\frac{d_{a_i}}{D} \right) W(a_1,\ldots,a_n) \right]$$

where the weight $W$ depends on any fusion constraints. The entire sum often collapses, via fusion algebra trace identities, to a structure directly analogous to the Verlinde formula (i.e., expressing fusion multiplicities in terms of modular $S$-matrix elements):

$$N_{ab}^c = \sum_{d} \frac{S_{ad} S_{bd} S_{cd}^*}{S_{0d}}$$

In the most general torus-knot bipartition, an $SL(2,\mathbb{Z})$ transformation relates the calculation to an effective ground state $|\psi\rangle=\sum_a \mathcal{O}_{0a}|a\rangle$ for some modular operator $\mathcal{O}$. The TEE in these cases splits into universal and state-dependent contributions.

4. Universal Lower Bound for TEE and State Dependence

A principal result is that, for bipartitions with two non-intersecting torus-knot interfaces, the topological entanglement entropy exhibits a universal lower bound, regardless of interface winding:

$S_{TEE} \geq -2\ln D$

For a generic ground state $|\psi\rangle = \sum_a \psi_a |a\rangle$ (e.g., arising from basis change under modular transformations), the TEE takes the form

$S_{TEE} = -2\ln D + S_{gs}$

where

$$S_{gs} = \sum_a 2|\psi_a|^2\left( \ln d_a - \ln |\psi_a| \right)$$

Notably, $S_{gs} \geq 0$ and is bounded above by $2\ln D$, so the universal term $-2\ln D$ always constitutes the entanglement minimum; the remaining corrections depend solely on the effective ground state in the new “outside” basis after untwisting the interfaces. In minimal cases (trivial sector), the lower bound is saturated.

5. Summary of Methodology and Implications

The invariant property of TEE with respect to mapping class group transformations allows the problem of general knot bipartitions to be reduced to the simplest possible geometry. The replica trick, combined with surgery, ensures entanglement measures only depend on modular data, making TEE both computable and interpretable in terms of fusion category theory. The Verlinde-like structures make explicit the relationship between global entanglement structure and ground state topological invariants (quantum dimensions, modular $S$-matrix). The presence of a state-dependent correction in $S_{TEE}$ demonstrates that while the area-law contributions can always be canceled algebraically, the residual topological piece is both universal and physically meaningful—providing a robust lower bound tied to quantum dimension, and a window into how nontrivial ground state structure (as induced by knot interfaces or modular transformations) can further enrich long-range entanglement content.

Key LaTeX expressions:

• $$S_{n} = \frac{1}{1-n} \ln \operatorname{Tr}(\rho_A^n)$$
• $S_{TEE} = -2\ln D + S_{gs}$ with $$S_{gs} = \sum_a 2|\psi_a|^2\left( \ln d_a - \ln |\psi_a| \right )$$
• $S_{TEE} \geq -2\ln D$

These results solidify the interpretation of topological entanglement entropy as a universal invariant under mapping class group action, algebraically linked to the modular data of the TQFT, and bounded below by the fundamental quantum dimension—regardless of the topological complexity of the bipartition (Lo et al., 2023).