Interaction-robust classification of gapped fermionic phases beyond one dimension (the higher-dimensional Fidkowski–Kitaev program)

For noninteracting fermions, topological insulators and superconductors are classified in each spatial dimension and symmetry class by Kitaev's periodic table. Fidkowski and Kitaev showed that in one dimension many-body interactions collapse the integer classification of the time-reversal-invariant Majorana chain (Altland–Zirnbauer class BDI) from $\mathbb{Z}$ to $\mathbb{Z}_8$, gave in the same work a classification of all one-dimensional gapped fermionic phases with unitary and anti-unitary symmetries, and closed the paper by posing the open direction of generalizing the interaction-robust classification — and, ideally, the matrix-product-state and graded-central-extension machinery behind it — to two and three spatial dimensions. The one-dimensional table and the classification of invertible (short-range-entangled) phases in higher dimensions are now settled at the level of widely accepted physics results, given by spin/Pin cobordism and supercohomology, but the completeness of these classifications for microscopic interacting lattice systems remains a conjecture, gapped phases with intrinsic topological order are classified only partially, and constructive lattice-level formulations of the classifying invariants are still being developed. The surviving open problem is a complete, rigorous, interaction-robust classification of gapped fermionic phases with symmetry in two and three dimensions.

Background

Free fermions with a gap, in band insulators and in superconductors described at the Bogoliubov–de Gennes (BdG, the mean-field quasiparticle) level, fall into ten Altland–Zirnbauer (AZ) symmetry classes — the possible combinations of time-reversal symmetry ($T^2=\pm1$), particle-hole symmetry ($C^2=\pm1$), and chiral symmetry, with or without charge conservation. Kitaev's periodic table classifies their topological phases in each spatial dimension: for example, a time-reversal-invariant Majorana chain (class BDI, $T^2=+1$) carries an integer band index $k$, and $k$ unpaired Majorana zero modes (self-conjugate fermionic edge modes) appear at each end of an open chain.

Interactions change this picture. Fidkowski and Kitaev showed that for the one-dimensional BDI chain the band classification $\mathbb{Z}$ breaks down to $\mathbb{Z}_8$: local interaction terms adiabatically connect phases whose indices differ by a multiple of 8, while the eight remaining classes are distinct. Their proof introduced techniques that became standard: matrix product states (MPS, the variational ansatz that describes ground states of one-dimensional gapped systems, whose bond degrees of freedom carry projective — i.e., up-to-phase — representations of the symmetry group), the entanglement spectrum of a half-chain (the eigenvalue spectrum of the reduced density matrix of a half-chain, which itself realizes an AZ symmetry class), and graded central extensions (group extensions of the symmetry group $G$ by $U(1)$ that track fermion parity, governed by twisted group cohomology — cohomology with anti-unitary group elements acting on coefficients by complex conjugation — and Wall's invariants, which classify extensions of $\mathbb{Z}_2$-graded algebras). This paralleled the group-cohomology classification of bosonic symmetry-protected topological (SPT) phases — gapped, short-range-entangled (SRE) phases that become trivial only if the symmetry is broken or the gap closed.

In their 2011 paper the authors state that the one-dimensional classification framework applies to all gapped fermionic phases with anti-unitary symmetries, and close with the explicit open direction: generalizing these methods to higher dimensions. In two and three dimensions the question is strictly larger than in one dimension, because gapped phases there split into invertible phases (the direct analogues of 1D SRE phases) and phases with intrinsic topological order — long-range-entangled phases with anyonic (fractionalized, statistically non-trivial) excitations. Candidate higher-dimensional frameworks that grew out of this program are group supercohomology (a fermionic extension of group cohomology) and spin/Pin cobordism, which proposes that invertible fermionic phases are classified by bordism groups of manifolds equipped with spin or Pin structure — equivalently, by invertible topological quantum field theories. Establishing the completeness of such a classification for microscopic interacting systems, and extending it beyond the invertible sector, is what remains open.

Problem Statement

The problem, as derived from Fidkowski and Kitaev's 2011 one-dimensional classification of interacting fermionic topological phases, is the higher-dimensional completion of that program. The authors' explicitly stated open direction, in the closing sentence of the paper, is to generalize the methods developed there — matrix-product-state and projective edge representations and the graded-central-extension algebra — to higher spatial dimensions. The one-dimensional classification, including the Altland–Zirnbauer classes beyond the BDI example, is completed within that same paper and confirmed by later work, so it enters below only as a boundary condition that any answer must reproduce; the open target is two and three spatial dimensions.

The invertible (short-range-entangled) part of the higher-dimensional classification is now settled at the level of accepted physics results: invertible phases in 2D/3D are classified by spin/Pin cobordism and supercohomology-type theories. What remains open, and constitutes the problem to be solved, is three linked targets that together complete the classification program in two and three dimensions: (1) a rigorous microscopic foundation — prove that the cobordism/topological-field-theory classification is complete for interacting lattice systems, i.e., that no gapped interacting fermionic phase with the given symmetry escapes it (this is currently a conjecture resting on effective-field-theory assumptions, and the analogous completeness is open even for two-dimensional bosonic SPT phases); (2) extension of the classification to gapped fermionic phases beyond the invertible sector, i.e., phases with intrinsic topological order enriched by the symmetry, in two and three spatial dimensions; and (3) constructive, lattice-level realizations of the classifying invariants and explicit models for every predicted phase, in the spirit of the original matrix-product-state construction. These three targets share one scientific objective — a complete, interaction-robust, and rigorous classification of gapped fermionic phases with symmetry in two and three dimensions — with the free-fermion periodic table and the one-dimensional interacting table as boundary conditions any answer must reproduce. All three targets are jointly required and none is independently sufficient: a complete answer must address (1), (2), and (3) together.

References

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