Topological and Long-Range Entanglement Interfaces

Topological entanglement subtraction can isolate a long-range constant after local boundary contributions cancel. It is one continuum information signature among several: it does not by itself classify a phase, distinguish intrinsic from invertible order, or remove boundary and interface contributions.

Required background. Use Mutual Information and Regulator-Independent Correlations. Helpful background. Universal Terms and Entangling-Surface Geometry supplies the subtraction and scheme criteria.

Evidence cutoff. Research-sensitive comparisons on this page include primary literature available through 10 August 2026.

Long-range subtraction

For a gapped $2+1$-dimensional phase and a smooth simply connected region much larger than the correlation length,

$$S(A)=\alpha\,\lvert\partial A\rvert-\gamma+O(e^{-L/\xi}).$$

The perimeter coefficient is nonuniversal. In an intrinsic topological phase, $\gamma=\log\mathcal D$ under the standard assumptions, where $\mathcal D$ is the total quantum dimension. Kitaev–Preskill and Levin–Wen combinations cancel local boundary terms by combining several regions; see Kitaev and Preskill 2006, pp. 110404-1–110404-4 and Levin and Wen 2006, pp. 110405-1–110405-4.

The structure map locates this subtraction on the universal-geometry branch while keeping it connected to multipartite combinations.

Topological entropy is a carefully balanced multipartite subtraction of local boundary terms. Its interpretation assumes a gapped bulk, regions large compared with the correlation length, and controlled boundaries and interfaces. Schematic.

Intrinsic, invertible, and interface contributions

Intrinsic topological order has nontrivial superselection sectors and long-range entanglement. An invertible phase may have protected response or a chiral edge without nontrivial bulk anyons and can have $\gamma=0$. A trivial gapped phase also has $\gamma=0$. Thus the constant alone cannot distinguish invertible from trivial phases.

Interfaces introduce further structure. Boundary superselection sectors, edge zero modes, anyon condensation, and interface entropy can contribute constants or logarithms. The subtraction regions must remain far from physical boundaries unless those terms are the object of study. Long-range correlations from gaplessness can also mimic slow finite-size drift and invalidate the gapped expansion.

Compare three controlled families—an intrinsic topological fixed point, an invertible phase, and a trivial gapped phase—using identical region geometries and several sizes $L/\xi$. Combine the entropy result with response, quasiparticle, and boundary evidence in any phase claim. The present page owns the information-theoretic subtraction, not the material classification.

Failure controls

The validity figure shows why an apparently stable constant is not automatically topological.

Local perimeter and corner terms cancel only for the intended geometry. Boundary modes, interfaces, small $L/\xi$, symmetry breaking, and unmatched regulators can leave a residual constant unrelated to intrinsic topological order. Schematic.

Report every region, sign in the subtraction, correlation-length range, boundary distance, regulator, extrapolation uncertainty, and competing non-topological explanations.

References

• Kitaev, Alexei, and John Preskill. “Topological Entanglement Entropy.” Physical Review Letters 96 (2006): 110404. DOI. Open preprint.
• Levin, Michael, and Xiao-Gang Wen. “Detecting Topological Order in a Ground State Wave Function.” Physical Review Letters 96 (2006): 110405. DOI. Open preprint.