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arXiv:2603.16799

The weakly interacting tenfold way

Lucas C. P. A. M. Müssnich, Renato Vasconcellos Vieira

math-phmath.ATmath.KTquant-phSubmitted Mar 17, 2026• Announced Mar 18, 2026• Revised Mar 31, 2026 (v2)
v1 · Mar 17, 2026v2 · Mar 31, 2026
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Abstract

The tenfold way is a classification scheme for the building blocks of free fermion systems. More precisely, it classifies the isomorphism classes of spaces of equivariant free Hamiltonians in irreducible fermion systems with symmetries. This classification scheme naturally leads to the K-theoretical classification of topological phases of matter, known as the periodic table of topological insulators and superconductors. Topological K-theory is represented by spectra $KU$ and $KO$, and in this article we present realizations of these spectra in terms of time evolution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. We introduce a geometric definition of the space of weakly interacting time evolution operators, as the complement of the cut locus of the subspace of free operators. Our main result is that spectra $KU^{wi}$ and $KO^{wi}$ of weakly interacting time evolution operators deformation retract to $KU$ and $KO$. We thus have a stable homotopy theoretical proof that the tenfold way is stable to weak interactions.

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29 pages, 2 Tables

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The tenfold way is a classification scheme for the building blocks of free fermion systems. More precisely, it classifies the isomorphism classes of spaces of equivariant free Hamiltonians in irreducible fermion systems with symmetries. This classification scheme naturally leads to the K-theoretical classification of topological phases of matter, known as the periodic table of topological insulators and superconductors. Topological K-theory is represented by spectra KUKUKU and KOKOKO, and in this article we present realizations of these spectra in terms of time evolution operators of irreducible free fermion systems with symmetries, with explicit formulas for the structural suspension maps. We introduce a geometric definition of the space of weakly interacting time evolution operators, as the complement of the cut locus of the subspace of free operators. Our main result is that spectra KUwiKU^{wi}KUwi and KOwiKO^{wi}KOwi of weakly interacting time evolution operators deformation retract to KUKUKU and KOKOKO. We thus have a stable homotopy theoretical proof that the tenfold way is stable to weak interactions.
The celebrated tenfold way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of nonspatial symmetries.It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly interacting electrons' noncrystalline topological phases for all symmetry classes.Over recent years, a plethora of topological phenomena with diverse surface states has been discovered in crystalline materials.In this paper, we obtain an exhaustive classification of topologically distinct ground states as well as topological phases with anomalous surface states of crystalline topological insulators and superconductors for key space-groups, layer-groups, and rod-groups.This is done in a unified manner for the full tenfold way of Altland-Zirnbauer nonspatial symmetry classes.We establish a comprehensive paradigm that harnesses the modern mathematical framework of equivariant spectra; it allows us to obtain results applicable to generic topological classification problems.In particular, this paradigm provides efficient computational tools that enable an inherently unified treatment of the full tenfold way.
We extend the periodic table of phases of free fermions in the tenfold way symmetry classes to a classification of free fermionic phases protected by an arbitrary on-site unitary symmetry $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{G}$ in an arbitrary dimension. The classification is described as a function of the real representation theory of $\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{G}$ and the data of the original periodic table. We also systematically study in low dimensions the relationship between the free invariants and the invariants of short-range entangled interacting phases of fermions. Namely we determine whether a given symmetry protected phase of free fermions is destabilized by sufficiently strong interactions or it remains stable even in the presence of interactions. We also determine which interacting fermionic phases cannot be realized by free fermions. Examples of both destabilized free phases and intrinsically interacting phases are common in all dimensions.
The celebrated tenfold-way of Altland-Zirnbauer symmetry classes discern any quantum system by its pattern of non-spatial symmetries. It lays at the core of the periodic table of topological insulators and superconductors which provided a complete classification of weakly-interacting electrons' non-crystalline topological phases for all symmetry classes. Over recent years, a plethora of topological phenomena with diverse surface states has been discovered in crystalline materials. In this paper, we obtain an exhaustive classification of topologically distinct groundstates as well as topological phases with anomalous surface states of crystalline topological insulators and superconductors for key space-groups, layer-groups, and rod-groups. This is done in a unified manner for the full tenfold-way of Altland-Zirnbauer non-spatial symmetry classes. We establish a comprehensive paradigm that harnesses the modern mathematical framework of equivariant spectra; it allows us to obtain results applicable to generic topological classification problems. In particular, this paradigm provides efficient computational tools that enable an inherently unified treatment of the full tenfold-way.
We provide a systematic treatment of the tenfold way of classifying fermionic systems that naturally allows for the study of those with arbitrary N-body interactions. We identify four types of symmetries such systems can possess, which consist of one ordinary type (usual unitary symmetries), and three non-ordinary symmetries (such as time reversal, charge conjugation and sublattice). Focusing on systems that possess no non-trivial ordinary symmetries, we demonstrate that the non-ordinary symmetries are strongly constrained. This approach not only leads very naturally to the tenfold classes, but also obtains the canonical representations of these symmetries in each of the ten classes. We also provide a group cohomological perspective of our results in terms of projective representations. We then use the canonical representations of the symmetries to obtain the structure of Hamiltonians with arbitrary N-body interactions in each of the ten classes. We show that the space of N-body Hamiltonians has an affine subspace (of a vector space) structure in classes which have either or both charge conjugation and sublattice symmetries. Our results can help address many open questions including the topological classification of interacting fermionic systems. (C) 2017 Elsevier Inc. All rights reserved.
Building on the ten-way symmetry classification of disordered fermions, the authors have recently given a homotopy-theoretic proof of Kitaev's ‘periodic table’ for topological insulators and superconductors. The present paper offers an introduction to the physical setting and the mathematical model used. Basic to the proof is the so-called diagonal map, a natural transformation akin to the Bott map of algebraic topology, which increases by one unit both the momentum-space dimension and the symmetry index of translation-invariant ground states of gapped free-fermion systems. This mapping is illustrated here with a few examples of interest. (Based on a talk delivered by the senior author at the Nobel Symposium on ‘New Forms of Matter: Topological Insulators and Superconductors’; Stockholm, 13–15 June, 2014.)
Abstract Guided by the many-particle quantum theory of interacting systems, we develop a uniform classification scheme for topological phases of disordered gapped free fermions, encompassing all symmetry classes of the Tenfold Way. We apply this scheme to give a mathematically rigorous proof of bulk-boundary correspondence. To that end, we construct real C ∗^*∗ ∗ -algebras harbouring the bulk and boundary data of disordered free-fermion ground states. These we connect by a natural bulk-to-boundary short exact sequence, realising the bulk system as a quotient of the half-space theory modulo boundary contributions. To every ground state, we attach two classes in different pictures of real operator KKK K -theory (or KRKRKR KR -theory): a bulk class, using Van Daele’s picture, along with a boundary class, using Kasparov’s Fredholm picture. We then show that the connecting map for the bulk-to-boundary sequence maps these KRKRKR KR -theory classes to each other. This implies bulk-boundary correspondence, in the presence of disorder, for both the “strong” and the “weak” invariants.