nLab short-range entanglement

Idea

In solid state physics, where the nature of quantum materials depends crucially on the entanglement of their ground states, it has been proposed (Kitaev 11, 13) that “short-range entanglement” (SRE) in a quantum material means or implies that the “ground state is non-degenerate”, hence that the Hilbert space of quantum states with the same lowest energy has dimension $1$. See also Chen, Gu & Wen 10, Sec. II.

If such an SRE material is in a topological phase of matter, with the ground state separated by an energy gap from the excited states, then this means (Freed 14, cf. G & JF 19, ftn 2) that the (extended) topological field theory which describes the topological phase is an invertible topological field theory, hence one also speaks of an invertible topological phase.

In particular, short-range entanglement thus implies the absence of “topological order” (at least by two of three common definitions of the latter, see there).

In contrast, long-range entanglement in the ground state is supposedly the hallmark of topological order (Chen, Gu & Wen 10, Sec. V)

Related concepts

• entanglement entropy

• long-range entanglement

References

Short-range entanglement

The notion was introduced (only) in a series of talks:

• Alexei Kitaev, Toward Topological Classification of Phases with Short-range Entanglement, talk at KITP (2011) [video]

• Alexei Kitaev, On the Classification of Short-Range Entangled States, talk at Simons Center (2013) [video]

• Alexei Kitaev: Homotopy-theoretic approach to SPT phases in action: $\mathbb{Z}_{16}$ classification of three-dimensional superconductors, Talk at: Symmetry and Topology in Quantum Matter Workshop, Institute for Pure and Applied Mathematics, University of California (2015) [web]

• Alexei Kitaev: Topological quantum phases, Talk at IAS (2019) [video]

Formalization in terms of invertible topological field theories:

• Daniel S. Freed, Short-range entanglement and invertible field theories $[$arXiv:1406.7278 $]$

See also:

• Davide Gaiotto, Theo Johnson-Freyd, around footnote 2 of: Symmetry Protected Topological phases and Generalized Cohomology, J. High Energ. Phys. 2019 7 (2019) ]arXiv:1712.07950, doi:10.1007/JHEP05%282019%29007]

• Yosuke Kubota: Stable homotopy theory of invertible gapped quantum spin systems I: Kitaev’s $\Omega$-spectrum [arXiv:2503.12618]

• Yosuke Kubota: Stable homotopy theory of invertible gapped quantum spin systems, talk at CQTS@NYUAD (Nov 2025) [slides:pdf]

Long-range entanglement

The opposite notion of long-range entanglement:

• Xie Chen, Zheng-Cheng Gu, Xiao-Gang Wen, Section V of: Local unitary transformation, long-range quantum entanglement, wave function renormalization, and topological order Phys. Rev. B 82 155138 (2010) $[$arXiv:1004.3835$]$

Further discussion:

• Tsung-Cheng Lu, Sagar Vijay, Characterizing Long-Range Entanglement in a Mixed State Through an Emergent Order on the Entangling Surface $[$arXiv:2201.07792$]$