Schreiber Bilkent2025

A talk that I have given:

---

• Urs Schreiber$\;\;\;$ (on work with H. Sati):

  
  

  Fragile Topological Phases and Topological Order
  of 2D Crystalline Chern Insulators

  
  

  Zoom talk at Bilkent Topology Seminar,

  15 December 2025

Abstract: We apply methods of equivariant homotopy theory, that may not previously have found due attention in condensed matter physics, to classify first the fragile topological phases of 2D crystalline Chern insulator materials, and second the potential topological order of their fractional cousins. We highlight that the phases are given by the equivariant 2-Cohomotopy of the Brillouin torus of crystal momenta (with respect to wallpaper point group actions) — which, despite the attention devoted to crystalline Chern insulators, seems not to have been considered before. Arguing then that any topological order must be reflected in the adiabatic monodromy of gapped quantum ground states over the covariantized moduli space of these band topologies, we compute the latter in various examples where this group is non-abelian, showing that potential anyons must be localized in momentum space. We close with an outlook on the relevance for the search for topological quantum computing hardware.

Based on:

• Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems

  Applied Physics Letters (2025, in press)

• Fragile Topological Phases and Topological Order of 2D Crystalline Chern Insulators

  (in preparation)

Related talks:

• at QMATH16

• at ISQS29

• Rethinking FQH Anyons

• Rethinking Topological Quantum (lightning talk)

• Homotopy Theory for Topological Quantum Computing (aimed at algebraic topologists)

Script:

Idea of Topological Quantum Gates

The quantum adiabatic theorem entails:

Given a quantum system (e.g., a quantum material)

depending on external parameters $p$ in a parameter space $P$

with a gapped Hilbert space $\mathscr{H}_p$ of quantum ground states

then sufficiently gentle (“adiabatic”) tuning $\gamma$ of parameters

brings about unitary transformation $U_\gamma$ of the quantum state

$\begin{array}{ccc} \mathscr{H}_p &\overset{U_\gamma}{\longrightarrow}& \mathscr{H}_{p'} \\ p &\overset{\gamma}{\rightsquigarrow}& p' \end{array}$

This is called “topological” (really: “homotopical”) if

$U_\gamma$ depends only on the homotopy class of $\gamma$ (rel. endpoints),

whence the $\mathscr{H}_p$ form a local system of Hilbert spaces over $P$,

which over each connected component $[\phi] \in \pi_0(P)$

is a unitary representation of the fundamental group

$U_{(-)} \,\colon\, \pi_1 \big(P, [\phi]\big) \longrightarrow \mathrm{U}(\mathscr{H})$

Regarded as quantum gates, these $U_\gamma$ are “topologically protected”

in that they are undisturbed by noise in control parameter $\gamma$.

There are arguments that some such topological protection

is necessary for quantum computers to reach interesting scale.

Big question: How to find/build quantum systems with such

nontrivial (nonabelian) topological monodromy $U_{(-)}$

(jargon: “topological order”) ??

Classical proposal by Kitaev 2003: If one could arrange that

$P \sim Conf(\mathbb{R}^2)$

is a configuration space of points in the plane, such as of

some kind of soliton/defect positions in a $\sim$2D material

then

$Br(n) \simeq \pi_1(P, n) \longrightarrow U(\mathscr{H})$

is a braid representation of a braid group on $n$ strands

if non-trivial, the solitons/defects are called “anyons”

It remains underappreciated that: $P$ could be different.

Let’s have a closer look which $P$ actually arise in practice.

Idea of Topological Phases of Matter

The Bloch theorem entails that:

The Hamiltonians $H$ of electrons in a crystal are direct integrals

$H \;=\; \underset {{[k] \in \widehat{T}{}^d}} {\displaystyle{\int}} H_k$

over the Brillouin torus $\widehat{T}{}^d$ of crystal momenta

of Bloch Hamiltonians, given by continuous maps

$H_{(-)} \;\colon\; \widehat{T}{}^d \longrightarrow Herm(\mathscr{H}_{Bl}) \mathrlap{\,.}$

Here $H_{[k]}$ encodes the energy states available to electrons

with a well-defined (plane wave) momentum $[k]$.

The graph of eigenvalues of $H_{(-)}$ are the energy bands.

If all the $H_{[k]}$ are gapped, say at $E = 0$,

then the valence bundle is the negative eigenspace bundle

$\mathcal{V} \,\coloneqq\, Eig_{\lt 0}\big( H_{(-)} \big) \longrightarrow \widehat{T}{}^d$

The relevant equivalence class of the valence bundle

depends on how fine/coarse the resolution is. Given a choice

then the class $[\mathcal{V}]$ is the observed topological phase of matter.

For fragile topological phases one considers deformations

that explore only the given space of gapped Bloch Hamiltonians

$\mathcal{B}\big(\mathbb{C}^{v+c}\big)_{gap} \,\coloneqq\, \Bigg\{ H \in Herm(\mathbb{C}^{v+c}) \,\Bigg\vert\, \substack{ Eig_{\lt 0}(H) \simeq \mathbb{C}^v \\ Eig_{\gt 0}(H) \simeq \mathbb{C}^c } \, \Bigg\} \sim Gr_{v}^{v+c} \,,$

which is homotopy equivalent to the Grassmannian

$Gr_{v}^{v+c} \,\coloneqq\, \left\{ \substack{ v\;\text{dimensional linear} \\ \text{subspaces of}\;\mathbb{C}^{v+c} } \right\} \mathrlap{\,,}$

whence the parameter space of fragile topological couplings is

the mapping space $P \,\equiv\, Map\big( \widehat{T}{}^d ,\, Gr_{v}^{v+c} \big) \mathrlap{\,.}$

Whereas the coarser stable topological phases classify the

deformation classes where the system is allowed to explore

1. any of the higher conduction bands, where

   $\bigcup_{c \in \mathbb{N}} Gr_v^{v + c} \,\simeq\, B \mathrm{U}(v)$

2. moreover any number of valence bands, where

   $\bigcup_{v \in \mathbb{N}} \bigcup_{c \in \mathbb{N}} Gr_v^{v + c} \,\simeq\, \bigcup_{v \in \mathbb{N}} B \mathrm{U}(v) \,\equiv\, B \mathrm{U}$

For $\left\{\begin{aligned}c & \to \infty \\v & = 1\end{aligned}\right.$

these phases are in ordinary cohomology:

$H^2\big( \widehat{T}{}^d ;\, \mathbb{Z} \big) \,\simeq\, \pi_0\, Map\big( \widehat{T}{}^d ,\, Gr_{1}^\infty \big) \mathrlap{\,,}$

while for $\left\{\begin{aligned}c & \to \infty \\ v & \to \infty\end{aligned}\right.$ these phases are

in (reduced) topological K-theory

$KU^0\big( \widehat{T}{}^d \big) \,\simeq\, \pi_0\, Map\big( \widehat{T}{}^d ,\, B \mathrm{U} {\color{gray} \times \mathbb{Z}} \big) \mathrlap{\,,}$

but general fragile phases are

in extraordinary nonabelian cohomology (FSS23)

classified by any (pointed connected) topological space $\mathcal{A}$:

$H^1\big( X ;\, \Omega \mathcal{A} \big) \,\coloneqq\, \pi_0\, Map\big( X ,\, \mathcal{A} \big)$

here specifically:

$H^1\big( \widehat{T}{}^d ;\, \Omega Gr_{v}^c \big) \,\simeq\, \pi_0\, Map\big( \widehat{T}{}^d ,\, Gr_{v}^\infty \big) \mathrlap{\,.}$

For example:

In the case of “2 accessible bands”, $\mathscr{H}_{Bl} \simeq \mathbb{C}^2$,

where the classifying space $Gr_1^{2} \simeq \mathbb{C}P^2 \simeq S^2$ is the 2-sphere,

the fragile topological phases are classified in 2-Cohomotopy:

$\pi^2(X) \,\equiv\, \pi_0\, Map\big(X, S^2\big) \mathrlap{\,.}$

For 2D Chern materials this coincides already with the stable phases

$\begin{array}{ccc} \pi^2\big( \widehat{T}{}^2 \big) & \xrightarrow{\phantom{-}\sim\phantom{-}} & H^2\big( \widehat{T}{}^2 ,\, \mathbb{Z} \big) & \simeq \mathbb{Z} \\ = && = \\ \pi_0 \, Map\big( \widehat{T}{}^2 ,\, \mathbb{C}P^1 \big) &\xrightarrow[\iota_\ast]{\phantom{-}\sim\phantom{-}}& \pi_0 \, Map\big( \widehat{T}{}^2 ,\, \mathbb{C}P^\infty \big) \\ S^2 \simeq Gr_1^2 &\xhookrightarrow{\phantom{-}\iota\phantom{-}}& Gr_1^{\infty} \simeq B \mathrm{U}(1) \end{array}$

This integer class is the Chern class of the valence line bundle

whence one speaks of topological Chern insulators.

Fragile Crystalline Topological Phases

More precisely, in general a crystalline symmetry $G \curvearrowright \widehat{T}{^2}$

is respected by the Bloch Hamiltonian and its deformations:

$\underset{[k] \in \widehat{T}{}^d}{\forall} \;\;\;\;\;\; H_{g \cdot [k]} \,=\, U_g \circ H_{[k]} \circ U_g^{-1}$

for unitary operators $U_{(-)}$, in which case one speaks of

“symmetry protected topological phases”.

Observation: This means that

the classifying maps are equivariant maps

$P \;\equiv\; Map\big(\widehat{T}{}^d, Gr_{v}^{c+c}\big)^G$

and the classification is in equivariant nonabelian cohomology

$H^1_G\big( \widehat{T}{}^d ;\, \Omega Gr_{v}^{v+x} \big) \,\equiv\, \pi_0\, Map\big(\widehat{T}{}^d, Gr_{v}^{c+c}\big)^G \mathrlap{\,.}$

In particular:

Fragile crystalline 2-band phases are classified by equivariant 2-Cohomotopy.

Evident as this is, it has not been classified or even stated before.

While filling such gaps we also also point out that the topological

parameter space really ought to be the homotopy quotient

$P \;=\; Map\big( \widehat{T}{}^2 ,\, \mathcal{A} \big)^G { \color{purple} \sslash Diff\big(\widehat{T}{}^2\big)^G }$

by the equivariant diffeomorphism group (acting by precomposition)

as befits the modular functor of a topological field theory.

Identifying Anyons in FQAH Materials

Hence if there is topological order/anyons

in crystalline fractional quantum Hall systems it ought to be

reflected in nontrivial parameter fundamental groups $\pi_1(P,[\phi])$,

in a given topological phase $[\phi] \in \pi_0(P)$

$\pi_1\big( P,\, [\phi] \big) \;\equiv\; \pi_1\Big( Map\big( \widehat{T}{}^2 ,\, Gr_{1}^2 \big)^G \sslash Diff\big( \widehat{T}{}^2 \big)^G ,\, [\phi] \Big)$

Two examples (1.) solitonic and (2.) defect anyons:

Proposition 1 (potential solitonic FQAH anyons):

When all symmetry is broken ($G = 1$ the trivial group):

$\pi_1\Big( Map\big( \widehat{T}{}^2 ,\, S^2 \big) \sslash Diff\big( \widehat{T}{}^2 \big) \Big) \,\simeq\, \widehat{\mathbb{Z}^2} \rtimes Mod \mathrlap{\,,}$

where $\widehat{\mathbb{Z}^2}$ is the integer Heisenberg group at level=2

$\widehat{\mathbb{Z}^2} \,=_{{}_{Set}}\, (\mathbb{Z}_a \times \mathbb{Z}_b) \times \mathbb{Z}$

with generators

$\left. \begin{aligned} W_a & \coloneqq \big((1,0),0\big) \\ W_b & \coloneqq \big((0,1),0\big) \\ \zeta & \coloneqq \big((0,0),1\big) \end{aligned} \right\} \,\in\, (\mathbb{Z}_a \times \mathbb{Z}_b) \times \mathbb{Z}$

and the only nontrivial group commutator being

$[W_a, W_b] \,=\, \zeta^2 \,.$

This $\widehat{\mathbb{Z}^2}$ is precisely the relation characterizing FQH anyons on the torus!

And $\widehat{\mathbb{Z}^2} \rtimes Mod$ imposes their expected modular data.

(SS25)

Proposition 2 (potential defect FQAH anyons):

For p3 symmetry

$\pi_1\Big( Map\big( \widehat{T}{}^2 ,\, S^2 \big)^{\mathbb{Z}_{/3}} \sslash Diff\big( \widehat{T}{}^2 \big)^{\mathbb{Z}_{/3}} \Big) \,\supset\, \mathbb{Z}^3 \rtimes Sym(3) \mathrlap{\,,}$

where the symmetric group $Sym(3)$ acts by permutation of the high symmetry points.

This implies that the high symmetry points themselves

may constitute para- defect anyons.

Computing Equivariant 2-Cohomotopy

of $G$-tori is among the most basic questions

not just for the physics of crystalline fragile topological phases

but also in pure Cohomotopy theory as such

but seems not to have found any attention before.

One approach:

For point group-symmetries $G$ of wallpaper groups.

Determine minimal G-CW complex-structure on $G$-torus

(worked out here.)

Observe that filtering the equivariant mapping space into $S^2$

by the skeleta of this G-CW complex gives homotopy fiber

realizations from which the homotopy groups may be deduced.

The example of p3 symmetry:

Classification so far: