Topological insulators and superconductors: ten-fold way and dimensional hierarchy

TL;DR

This work provides a unified framework for classifying topological insulators and superconductors across all spatial dimensions by constructing Dirac-Hamiltonian representatives for the tenfold symmetry classes and organizing their topological content via complex and real invariants. It establishes a dimensional hierarchy (dimensional reduction) that relates higher-dimensional parent phases to lower-dimensional descendants, clarifying how $ ext{Z}$ and $ ext{Z}_2$ invariants descend and how Bott periodicity governs the real case. The paper links bulk invariants (Chern numbers, winding numbers) to boundary phenomena through Chern-Simons and $ heta$ terms in spacetime, and it discusses the roles of inversion and other discrete symmetries in modifying the classification. The framework yields comprehensive predictions for strong/weak topological phases, zero modes on defects, and topological field theories describing linear responses, with implications for both condensed matter and related high-energy contexts.

Abstract

It has recently been shown that in every spatial dimension there exist precisely five distinct classes of topological insulators or superconductors. Within a given class, the different topological sectors can be distinguished, depending on the case, by a Z or a Z_2 topological invariant. This is an exhaustive classification. Here we construct representatives of topological insulators and superconductors for all five classes and in arbitrary spatial dimension d, in terms of Dirac Hamiltonians. Using these representatives we demonstrate how topological insulators (superconductors) in different dimensions and different classes can be related via dimensional reduction by compactifying one or more spatial dimensions (in Kaluza-Klein-like fashion). For Z-topological insulators (superconductors) this proceeds by descending by one dimension at a time into a different class. The Z_2-topological insulators (superconductors), on the other hand, are shown to be lower-dimensional descendants of parent Z-topological insulators in the same class, from which they inherit their topological properties. The 8-fold periodicity in dimension d that exists for topological insulators (superconductors) with Hamiltonians satisfying at least one reality condition (arising from time-reversal or charge-conjugation/particle-hole symmetries) is a reflection of the 8-fold periodicity of the spinor representations of the orthogonal groups SO(N) (a form of Bott periodicity). We derive a relation between the topological invariant that characterizes topological insulators/superconductors with chiral symmetry and the Chern-Simons invariant: it relates the invariant to the electric polarization (d=1), or to the magnetoelectric polarizability (d=3). Finally, we discuss topological field theories describing the space time theory of linear responses, and study how the presence of inversion symmetry modifies the classification.

Figures (5)

• Figure 1: Topological distinction among quantum ground states. (a) $\mathbb{Z}$ classification and (b) $\mathbb{Z}_2$ classification.
• Figure 2: (a) The $d=2n+2$ dimensional Brillouin zone, $\mathrm{BZ}^{d=2n+2}\sim S^{2n+2}= S^{2n+1}\wedge S^1$, and its $d=2n+1$ dimensional descendant $\mathrm{BZ}^{d=2n+1}\sim S^{2n+1}$ located at the equator of $S^{2n+2}$. (b) Splitting the $d=2n+2$ dimensional Brillouin zone in half, or embedding $d=2n+1$ dimensional Brillouin zone into a higher-dimensional one.
• Figure 3: (a) Winding number $\nu_3$ for Hamiltonian (\ref{['matrixAIII']}) as a function of $m_5$. (b) Two-dimensional energy spectrum of the surface states of model (\ref{['matrixAIII']}) with mass $m_5=+0.5$. There are two inequivalent surface modes in agreement with the winding number $\nu_3 (m_5 =+0.5) = -2$.
• Figure 4: (a) One-parameter interpolation of two topological insulators represented by the projectors $q_1$ and $q_2$. In (b), the one-way interpolation (a) is extended by making use of the discrete symmetry. (c) Two different interpolations, each colored red and blue, respectively. In (d) the original interpolations are rearranged into two different interpolations.
• Figure 5: The energy spectrum with $t=\Delta=1$ and $\mu=-1$ [panel (a)] and $\mu=+1$ [panel (b)].