nLab bulk-boundary correspondence

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In solid state physics, specifically in the discussion of topological phases of matter, the bulk-boundary correspondence (or bulk-edge correspondence, due to its historical roots in 2D systems such as exhibiting the quantum Hall effect) states, broadly, that for topological systems on a domain with boundary the topological bulk observables correspond to certain boundary observables (edge modes).

The general physics intuition is that the nontrivial topological twist in the bulk must somehow “unwind” at the boundary in order to interpolate to the trivial topological situation beyond the boundary. That “unwinding” is exhibited by “edge mode” dynamics on the boundary, which hence corresponds to the bulk topological phase.

A general mathematical formalization of the correspondence goes back to Kellendonk, Richter & Schulz-Baldes 2002, in the context of the K-theory classification of topological phases of matter: Here one considers a short exact sequence of C * C^\ast -algebras (typically a Toeplitz extension, cf. Arici & Mesland 2020)

(1)0→A bdr⟶A full⟶A blk→0, 0 \to A_{bdr} \longrightarrow A_{full} \longrightarrow A_{blk} \to 0 \mathrlap{\,,}

whose entries are meant to reflect (cf. Prodan & Schulz-Baldes 2016 §3), respectively the (noncommutative) geometry of the bulk (A blkA_{blk}) and the boundary (A bdrA_{bdr}) inside the full bulk-boundary system (A fullA_{full}). Then in the induced long exact sequence in operator K-theory (discussed by Pimsner & Voiculescu 1980 when (1) is a Toeptlitz extension)

⋯→K n(A bdr)⟶K n(A full)⟶K n(A blk)⟶∂K n−1(A bdr)⟶K n−1(A full)⟶K n−1(A blk) \cdots \to K_{n}(A_{bdr}) \longrightarrow K_{n}(A_{full}) \longrightarrow K_{n}(A_{blk}) \overset{ \partial }{\longrightarrow} K_{n-1}(A_{bdr}) \longrightarrow K_{n-1}(A_{full}) \longrightarrow K_{n-1}(A_{blk})

the connecting homomorphism

K 0(A blk)⟶∂K −1(A bdr) K_{0}(A_{blk}) \overset{ \partial }{\longrightarrow} K_{-1}(A_{bdr})

maps bulk observables to boundary observables. With due care, the image of this connecting homomorphism under (schematically) certain trace operations 𝒯\mathcal{T} becomes an equality between bulk and boundary “invariants”:

𝒯(∂):𝒯(K 0(A blk))=𝒯(K −1(A bdr)). \mathcal{T}(\partial) \,\colon\, \mathcal{T}\big( K_{0}(A_{blk}) \big) = \mathcal{T}\big( K_{-1}(A_{bdr}) \big) \mathrlap{\,.}

(cf. P&S-B ‘16 Thm. 5.5.1)

References

General

Monographs:

More on the C * C^\ast -algebras involved:

Original discussion for integer quantum Hall systems:

using:

  • M. Pimsner and D. Voiculescu: Exact sequences for K-groups of certain cross-products of C *C^\ast-algebras, J. Op. Theory 4 (1980) 93–118 [pdf, pdf]

Further discussion for quantum Hall systems:

Monograph account:

Further examples:

Via Kasparov K-theory:

See also:

  • Zixian Zhou, Liang-Liang Wan: Proof of bulk-edge correspondence for band topology by Toeplitz algebra, Journal of Physics A: Mathematical and Theoretical 57 46 (2024) 465203 [doi:10.1088/1751-8121/ad8aab, arXiv:2410.19539]

Under T-Duality

On T-duality in the K-theory classification of topological phases of matter (related to the Fourier transform between crystals and their Brillouin torus) as expressing the bulk-boundary correspondence:

Review:

Last revised on April 14, 2026 at 14:12:14. See the history of this page for a list of all contributions to it.

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