From Multipartite Entanglement to TQFT

Abstract

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.

Figures (17)

• Figure 1: Here we have graphically represented the equation $|\psi\rangle=|\psi_1\rangle\otimes |\psi_2\rangle$ where all the states are $q=4$ partite states.
• Figure 2: Graphical presentation of $\psi$ and $\bar{\psi}$ coefficients.
• Figure 3: A sample $3$-partite multi-invariant with $n=3$.
• Figure 4: (a) The graphical representation of $\alpha\in \mathrm{Hom}_\mathcal{C}(o_1,o_2)$. (b) An equivalent diagram in the case when $o_1=\ell_1\otimes \ell_2\otimes \ell_3$ and $o_2=\ell_1'\otimes \ell_2'\otimes \ell_3'\otimes \ell_4'$. (c) The graphical representation of the evaluation morphism $o^*\otimes o\rightarrow\mathbf{1}$. (d) The graphical representation of the coevaluation morphism $\mathbf{1} \rightarrow o\otimes o^*$.
• Figure 5: (a) The dual graph $\mathcal{G}$ for a tetrahedron show in red. The edges are decorated by simple objects $\ell_i\in \mathrm{Irr}(\mathcal{C})$ and vertices by the morphisms from $\mathrm{Hom}_{\mathcal{C}}(\ell_i\otimes \ell_j,\ell_k)$ or $\mathrm{Hom}_{\mathcal{C}}(\ell_k,\ell_i\otimes \ell_j)$. (b) The graph $\mathcal{G}$ embedded in a plane. Read from bottom to top, it provides a composition of the morphisms of the form shown in Figure \ref{['fig:hom']}. The result is a morphism $\mathbf{1}\rightarrow \mathbf{1}$, which is necessarily a multiple of the identity. In the case when $\mathcal{C}$ is (the semisimplification of) the category of finite-dimensional representations of the quantum $\mathfrak{sl}_2$ at a root of unity, the labels $\ell_i$ can be identified with half-integer spins and the choice of morphisms at the trivalent junctions is unique (up to a multiple). With the standard choice (given by quantum Clebsch-Gordon coefficients), the result of the composition is then given by a properly normalized quantum 6j-symbol turaev1992state.