From Multipartite Entanglement to TQFT

At long distances, a gapped phase of matter is described by a topological quantum field theory (TQFT). We conjecture a tight and concrete relationship between the genuine $(d+1)$-partite entanglement – labelled by a $d$-dimensional manifold $M$ – in the ground state of a $(d-1)+1$-dimensional gapped theory and the partition function of the low energy TQFT on $M$. In particular, the conjecture implies that for $d=3$, the ground state wavefunction can determine the modular tensor category description of the low energy TQFT. We verify our conjecture for general (2+1)-dimensional Levin-Wen string-net models.

💡 Research Summary

The paper “From Multipartite Entanglement to TQFT” establishes a concrete and quantitative bridge between genuine multipartite entanglement in the ground state of a gapped quantum many‑body system and the partition function of the low‑energy topological quantum field theory (TQFT) that describes its long‑distance physics. The authors focus on a (d‑1)+1 dimensional gapped theory whose ground state |ψ⟩ₕ is regarded as a (d+1)‑partite quantum state by partitioning the spatial manifold into d+1 regions that correspond to the faces of a d‑simplex. They introduce the notion of “multi‑invariants” – polynomial functions of |ψ⟩ that are invariant under local unitary transformations and multiplicative under tensor products. Each multi‑invariant is specified by a tuple of permutations (σ₁,…,σ_q) acting on n replicated copies of the state and its complex conjugate; graphically it is represented by a q‑regular, edge‑coloured bipartite graph.

From these multi‑invariants the authors construct “signals” – additive combinations of the logarithms of multi‑invariants that vanish on any state that factorises into a (q‑1)‑partite piece and a single‑partite piece. Such a signal therefore detects genuine q‑partite entanglement (GME). The key insight is that when the permutation data are chosen to encode a bipartite triangulation Δ of a d‑dimensional manifold M, the resulting multi‑invariant Z(M_Δ;|ψ⟩ₕ) can be interpreted as a GME signal associated with the manifold. The triangulation gives rise to a graph‑encoded manifold (GEM), a (d+1)‑regular edge‑coloured graph that mirrors the combinatorial structure of M.

The central conjecture, equation (1.1) in the paper, states that in the large‑region limit (L → ∞) the GME signal extracted from the ground state satisfies
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