Universal versus geometry- and anyon-dependent contributions to multipartition entanglement at trijunctions of two-dimensional topological orders

When the ground state of a (2+1)-dimensional topologically ordered phase is partitioned into three or more spatial regions meeting at junction points, the constant, sub-area-law pieces of multipartition correlation measures - entanglement entropy, logarithmic negativity, reflected entropy, and the Markov gap - are expected to encode topological data: anyon quantum dimensions and fusion rules, and the chiral central charge of ungappable edge modes. What is not known in general is which parts of these constants are universal topological invariants and which are geometry-dependent corner/junction contributions or artifacts of anyonic excitations pinned to the junction by topological charge conservation. The problem is to establish this decomposition for general topological liquids - including Abelian fractional quantum Hall states, non-Abelian orders, and trijunctions of distinct topological orders - and to determine whether the same inventory emerges from the edge theory (vertex states) and from complementary bulk approaches (TQFT, anyon-diagrammatic, or surgery). Substantial partial results exist (corner contributions computed, bulk diagrammatic negativity for doubled orders, chirality probes from the bulk wavefunction), but a complete characterization remains open.

Background

Topologically ordered phases of matter, such as fractional quantum Hall states, gapped quantum spin liquids, and chiral p-wave superconductors (two-dimensional superconductors whose vortices trap Majorana zero modes), are characterized not by local order parameters but by patterns of long-range entanglement and by anyons - quasiparticles whose exchange statistics and fusion properties are captured by a modular tensor category. The classic entanglement diagnostic is the topological entanglement entropy: the entanglement entropy of a spatial region obeys an area law $S_A=\alpha L-\gamma$, whose constant subleading piece $\gamma=\ln\mathcal{D}$ (with $\mathcal{D}$ the total quantum dimension of the anyon theory) is universal (Kitaev and Preskill, 2006; Levin and Wen, 2006). This single number is far from a complete characterization, motivating the question of whether multipartite entanglement measures reveal finer topological data.

Three correlation measures are central here. The logarithmic negativity $\mathcal{E}_{A:B}$ quantifies entanglement in a mixed state: it is computed from the partial transpose of the reduced density matrix $\rho_{AB}$ (transposition acting only on subsystem $B$), whose negative eigenvalues signal entanglement. The reflected entropy $S_R(A:B)$ is defined via the canonical purification of $\rho_{AB}$ - a purification in which each of $A$ and $B$ is doubled - as the entanglement entropy between the two copies of $A$ and the two copies of $B$. The Markov gap $h_{A:B}=S_R(A:B)-I(A:B)$, with $I(A:B)$ the mutual information, measures tripartite correlations beyond classical GHZ-type (Greenberger-Horne-Zeilinger) three-party correlations. For two adjacent regions in the ground state of a (2+1)-dimensional topological liquid, Siva, Zou, Soejima, Mong, and Zaletel (2022) conjectured, and verified in examples, that the Markov gap - after non-universal short-range correlations near the junction are removed by minimizing over local unitaries - is quantized at $h^{\mathrm{IR}}_{A:B}=(c/3)\ln 2$, where $c$ is the minimal total central charge of the ungappable edge degrees of freedom; the central charge measures the anomaly of chiral edge modes that cannot be gapped without closing the bulk gap. They proved that $h=0$ for string-net models (exactly solvable lattice realizations of doubled topological orders, whose edges are gappable), so a non-vanishing Markov gap diagnoses intrinsic edge chirality.

Liu, Sohal, Kudler-Flam, and Ryu (2022) computed these measures for tripartitions in which the three regions are mutually adjacent and meet at junction points. Using the cut-and-glue construction - in which the entanglement cut is physically opened and re-healed by interactions, so that the entanglement reduces to that of counter-propagating edge modes of a (1+1)-dimensional conformal field theory (CFT) - they showed that a network of gapped interfaces meeting at a junction is described by a vertex state, a generalization of conformal boundary states (Ishibashi states) imported from the interaction vertices of string field theory. For chiral p-wave superconductors and Chern insulators (free-fermion bands with quantized Hall conductance), they verified the $(c/3)\ln 2$ Markov-gap law in the sector where all edge fermions obey Neveu-Schwarz (antiperiodic) boundary conditions, and found that in the R-R-R sector - periodic (Ramond) boundary conditions on all three edges - topological charge conservation forces an Ising $\sigma$ anyon (the non-Abelian anyon of quantum dimension $\sqrt{2}$) to sit at the junction, producing extra contributions to the entanglement measures that depend on how the four-fold degenerate junction zero-mode subspace is occupied; only one occupation choice was computed. Their lattice numerics found the Markov gap taking a value of order $2.27$ times $(c/3)\ln 2$ rather than being quantized, which they attributed to non-universal contributions from the sharp corners of the trijunction. Relatedly, Berthiere, Chen, Liu, and Chen (2021) computed the topological reflected entropy in Chern-Simons theory using boundary-state and surgery (cut-and-reglue) methods, but for tripartitions without trijunctions.

The unresolved target is a complete inventory: which parts of the constant terms of $S_A$, $\mathcal{E}_{A:B}$, $S_R$, and $h_{A:B}$ at junctions are genuine universal topological data (anyon data and chiral central charge), which are geometry-dependent corner/junction contributions, and which reflect anyonic excitations pinned to the junction - and whether this inventory is the same when computed from the edge theory and from the bulk.

Problem Statement

Determine the decomposition of the sub-area-law (constant) terms of the multipartition correlation measures - entanglement entropy $S_A$, logarithmic negativity $\mathcal{E}_{A:B}$, reflected entropy $S_R(A:B)$, and the Markov gap $h_{A:B}=S_R(A:B)-I(A:B)$ (with $I(A:B)$ the quantum mutual information) - for ground states of (2+1)-dimensional topologically ordered phases partitioned into three or more spatial regions that meet at junction points, into (i) universal topological contributions (anyon quantum dimensions and fusion data such as the F-symbols, the associativity matrices relating the different total-fusion channels of three anyons, and the chiral central charge of ungappable edge modes - edge modes that cannot be gapped without closing the bulk gap - generalizing $h=(c/3)\ln 2$), (ii) geometry-dependent corner/junction contributions controlled by junction angles and the number of junctions, and (iii) contributions from anyonic excitations pinned to the junction by topological charge conservation (such as an Ising $\sigma$ anyon, the non-Abelian anyon of quantum dimension $\sqrt{2}$, in the R-R-R sector, i.e. the sector with periodic (Ramond) boundary conditions on all three edge modes), including their dependence on the anyon type, on the occupation of the degenerate junction zero-mode subspace, and on the short-distance regularization. Both edge-theory descriptions (vertex states or conformal interfaces) and complementary bulk descriptions (anyon diagrammatics, string-net lattice models, or TQFT surgery) are relevant to this characterization, and it is open whether they yield the same inventory; the characterization should extend beyond chiral p-wave superconductors and Chern insulators to general topological liquids - Abelian fractional quantum Hall states and non-Abelian orders - including trijunctions at which regions with distinct topological orders meet, where gapped interfaces and parafermion-like zero modes can occur.

These parts are not interchangeable. The base-case decomposition for trijunctions of chiral p-wave superconductors and Chern insulators, the extension of that decomposition to the general class of topological liquids named above - Abelian fractional quantum Hall states, non-Abelian orders, and trijunctions of distinct orders - and the comparison of the edge-theory and bulk inventories are three separate claims: establishing the base case settles the decomposition only for the base class, resolving the edge-versus-bulk question settles only whether the two descriptions agree, and the complete inventory is obtained only when the general decomposition and the edge-versus-bulk comparison are both settled. The region count is itself part of the requirement. The measures are taken for multipartitions into three or more spatial regions meeting at junction points, so the required inventory covers multipartitions with more than three regions (and their junctions) as well as the three-region trijunction; the three-region trijunction is the geometry computed explicitly in the cited edge-theory and lattice work, but a decomposition restricted to it does not by itself complete the requirement.

References

1. Yuhan Liu, Ramanjit Sohal, Jonah Kudler-Flam, Shinsei Ryu, "Multipartitioning topological phases by vertex states and quantum entanglement", Phys. Rev. B 105, 115107 (2022). DOI: 10.1103/PhysRevB.105.115107; arXiv:2110.11980; https://arxiv.org/abs/2110.11980
2. Karthik Siva, Yijian Zou, Tomohiro Soejima, Roger S. K. Mong, Michael P. Zaletel, "Universal tripartite entanglement signature of ungappable edge states", Phys. Rev. B 106, L041107 (2022). DOI: 10.1103/PhysRevB.106.L041107; arXiv:2110.11965; https://arxiv.org/abs/2110.11965
3. Alexei Kitaev, John Preskill, "Topological Entanglement Entropy", Phys. Rev. Lett. 96, 110404 (2006). DOI: 10.1103/PhysRevLett.96.110404; https://doi.org/10.1103/PhysRevLett.96.110404
4. Michael Levin, Xiao-Gang Wen, "Detecting Topological Order in a Ground State Wave Function", Phys. Rev. Lett. 96, 110405 (2006). DOI: 10.1103/PhysRevLett.96.110405; https://doi.org/10.1103/PhysRevLett.96.110405
5. Clément Berthiere, Hongjie Chen, Yuefeng Liu, Bin Chen, "Topological reflected entropy in Chern-Simons theories", Phys. Rev. B 103, 035149 (2021). DOI: 10.1103/PhysRevB.103.035149; arXiv:2008.07950; https://arxiv.org/abs/2008.07950
6. Yuhan Liu, Yuya Kusuki, Jonah Kudler-Flam, Ramanjit Sohal, Shinsei Ryu, "Multipartite entanglement in two-dimensional chiral topological liquids", Phys. Rev. B 109, 085108 (2024). DOI: 10.1103/PhysRevB.109.085108; arXiv:2301.07130; https://arxiv.org/abs/2301.07130
7. Ramanjit Sohal, Shinsei Ryu, "Entanglement in tripartitions of topological orders: A diagrammatic approach", Phys. Rev. B 108, 045104 (2023). DOI: 10.1103/PhysRevB.108.045104; arXiv:2301.07763; https://arxiv.org/abs/2301.07763
8. Chao Yin, Shang Liu, "Mixed-state entanglement measures in topological order", Phys. Rev. B 108, 035152 (2023). DOI: 10.1103/PhysRevB.108.035152; arXiv:2301.08207; https://arxiv.org/abs/2301.08207
9. Shang Liu, "Anyon quantum dimensions from an arbitrary ground state wave function", Nat. Commun. 15, 5134 (2024). DOI: 10.1038/s41467-024-47856-7; arXiv:2304.13235; https://arxiv.org/abs/2304.13235
10. Nico Kirchner, Wonjune Choi, Frank Pollmann, "Characterizing the entanglement of anyonic systems using the anyonic partial transpose", Phys. Rev. B 110, 085143 (2024). DOI: 10.1103/PhysRevB.110.085143; arXiv:2403.12121; https://arxiv.org/abs/2403.12121
11. Bowei Liu, Junjia Zhang, Shuhei Ohyama, Yuya Kusuki, Shinsei Ryu, "Multiwavefunction overlap and multientropy for topological ground states in (2+1) dimensions", Phys. Rev. B 112, 125160 (2025). DOI: 10.1103/tjcf-yryh; arXiv:2410.08284; https://arxiv.org/abs/2410.08284
12. Yarden Sheffer, Ady Stern, Erez Berg, "Extracting Topological Spins from Bulk Multipartite Entanglement", Phys. Rev. Lett. 135, 086601 (2025). DOI: 10.1103/h53w-64dy; arXiv:2502.12259; https://arxiv.org/abs/2502.12259
13. Yarden Sheffer, Ruihua Fan, Ady Stern, Erez Berg, Shinsei Ryu, "Probing chiral topological states with permutation defects", Phys. Rev. B 113, 195123 (2026). DOI: 10.1103/3bz4-fbxr; arXiv:2512.04649; https://arxiv.org/abs/2512.04649
14. Christian Northe, Paolo Rossi, "Entanglement Through Topological Defects: Reconciling Theory with Numerics" (2025). arXiv:2512.07974; https://arxiv.org/abs/2512.07974
15. Christian Northe, Riccardo Poletti, Paolo Rossi, "Entanglement in Presence of Topological Interfaces and Dualities" (2026). arXiv:2607.22451; https://arxiv.org/abs/2607.22451