nLab unstable topological phase of matter

Idea

A crystalline quantum material in a gapped ground state is said to be in a topological phase if the Bloch Hamiltonian map $H_{(-)} \colon \widehat{T}{}^d \longrightarrow \mathscr{A}$ (from the Brillouin torus of crystal momenta to the space $\mathscr{A}$ of accessible individual Hamiltonians $H_{[\vec k]}$) has a non-trivial homotopy class.

Whether and how this is the case depends crucially on the choice of classifying space $\mathscr{A}$, which in turn reflects the resolution or accuracy with which the material is measurable in experiment: The larger $\mathscr{A}$, the more freedom is admitted to the system to transition between topological phases which may be disconnected for smaller choices of $\mathscr{A}$.

In the extreme case one allows the system to undergo deformations during which the electrons may occupy arbitrarily high energy bands. This coarsest resolution of the situation gives the most stable classification. This is the situation most commonly considered, notably in the K-theory classification of topological phases of matter.

All other classifications are unstable. In the literature such unstable topological phases have also been called fragile (PWV 2018) or delicate (NNBA 2021):

• fragile band topology: unstable when accessing further conduction bands,

• delicate band topology: unstable when accessing further conduction bands or valence bands (generally: both types).

In mathematical terminology, unstable topological phased are classified (SS 2025) by the (equivariant) nonabelian cohomology

$$H_G\big(\widehat{T}{}^d; \mathscr{A} \big) \coloneqq \pi_0 Map\big(\widehat{T}{}^d, \mathscr{A} \big)^G$$

(of the Brillouin torus with respect to the crystal’s point group $G$) with classifying spaces $\mathscr{A}$ that are indeed unstable in the sense of stable homotopy theory: they are not infinite loop spaces (not stages of spectra of spaces).

Details

The classifying space

The assumption that the system and all its accessible deformations are

1. gapped

2. with $v \in \mathbb{N}$ valence bands filled

3. and $c \in \mathbb{N}$ conduction bands available

means that the space of available fiber Bloch Hamiltonians is (taking the energy gap to be around zero, without restriction of generality)

(1)$$AccBlchHams \coloneqq \Big\{ H \in \mathcal{B}\big(\mathbb{C}^{v+c}\big) \,\Big\vert\, H^\dagger = H ,\, Eig_{\lt 0}(H) \simeq \mathbb{C}^{v} ,\, Eig_{\gt 0}(H) \simeq \mathbb{C}^c \Big\} \,,$$

where

• $\mathcal{B}\big(\mathbb{C}^{v+c}\big) \simeq Mat_{v+c \times v + c}(\mathbb{C})$ denotes the space of bounded linear operators, hence of matrices,

• “$Eig_{(-)}(H) \subset \mathbb{C}^{v+c}$” denotes the eigenspace of the Hamiltonian operator for the given range of eigenvalues.

Since the homotopy classes of maps to this space depend only on its homotopy type, we observe that (1) is evidently homotopy equivalent to the space of normalized Bloch Hamiltonians whose eigenvalues have unit absolute value:

(2)$$NrmAccBlchHams \;\coloneqq\; \Big\{ H \in \mathcal{B}\big(\mathbb{C}^{v+c}\big) \,\Big\vert\, H^\dagger = H ,\, Eig_{-1}(H) \simeq \mathbb{C}^{v} ,\, Eig_{+1}(H) \simeq \mathbb{C}^c \Big\} \,,$$

via

(3)$$\begin{array}{ccc} AccBlchHams &\xrightarrow{\phantom{-} \sim \phantom{-}}& NrmAccBlchHams \\ H &\mapsto& N_H \coloneqq \sqrt{H^2}^{-1} \circ H \mathrlap{\,,} \end{array}$$

where

• $\sqrt{H^2}$ denotes the unique positive definition square root of $H$,

• $(-)^{-1}$ denotes the inverse operator.

In fact, (3) is a deformation retraction: its right inverse is given by the inclusion

$$\begin{array}{ccc} NrmAccBlchHams &\xhookrightarrow{\phantom{--}}& AccBlchHams \\ N &\mapsto& N \mathrlap{\,,} \end{array}$$

and a homotopy

$$(H \mapsto N_H) \Rightarrow (H \mapsto H)$$

is given by

(4)$$\begin{array}{ccc} AccBlchHams \times [0,1] &\longrightarrow& AccBlchHams \\ (H, t) &\mapsto& \big( (1-t)\sqrt{H^2}^{-1} + t \mathrm{I}_{c+v} \big) \circ H \mathrlap{\,.} \end{array}$$

But this space of normalized Bloch Hamiltonians (2) is furthermore homotopy equivalent (homeomorphic, even) to the space of hermitian rank=$c$ projection operators

(5)$$Proj_v^{c+v} \;\coloneqq\; \Big\{ P \in \mathcal{B}\big(\mathbb{C}^{v+c}\big) \,\Big\vert\, P^\dagger = P ,\, P \circ P = P ,\, rnk(P) = c \Big\} \mathrlap{\,,}$$

via

$$\begin{array}{ccc} NrmAccBlchHams &\xrightarrow{\phantom{-} \sim \phantom{-}}& Proj_{v}^{v+c} \\ N &\mapsto& \tfrac{1}{2}\big(1_{c+v} + N\big) \mathrlap{} \end{array}$$

and furthermore to the “Grassmannian space”

$$Gr_v^{v+c} \;\coloneqq\; \Big\{ V \subset \mathbb{C}^{v + c} \,\Big\vert\, dim(V) = v \Big\} \;\simeq\; \frac{ \mathrm{U}(v+c) }{ \mathrm{U}(v) \times \mathrm{U}(c) } \mathrlap{\,,}$$

by passage to kernels $ker(-)$:

$$\begin{array}{ccc} Proj_{v}^{v+c} &\xrightarrow{\phantom{-} \sim \phantom{-}}& Gr_v^{v+c} \\ P &\mapsto& ker(P) \mathrlap{\,.} \end{array}$$

The symmetry group action

Furthermore assuming that the system and all its available deformations respects (“is protected” by) a subgroup $G$ of the point group of the crystalline material, acting on crystal momenta as

$$\begin{array}{ccc} G \times \widehat{T}{}^d &\longrightarrow& \widehat{T}{}^d \\ \big(g, [\vec k]\big) &\mapsto& g \cdot [\vec k] \end{array} \mathrlap{\,,}$$

means to specify a unitary representation group representation

(6)$$\rho \;\colon\; G \longrightarrow \mathrm{U}\big(\mathbb{C}^{c+v}\big)$$

such that the Bloch Hamiltonian satisfies

$$H_{ g \cdot [k] } \;=\; U_g \circ H_{[\vec k]} \circ U_g^{-1} \,,$$

hence that as a map

$$H_{(-)} \,\colon\, \widehat{T}{}^d \longrightarrow AccBlchHams$$

it is equivariant, for $AccBlchHams$ regarded as a G-space via the conjugation action of (6).

Again, the equivariant homotopy class of such a map depends only on the equivariant homotopy type of $AccBlchHams$ — but that is the same as that of $NrmAccBlchHams$, because the above normalizing homotopy (4) is itself equivariant.

The classification statement

In consequence, we find that the unstable $G$-symmetry protected crystalline topological phases seen when $v$ valence bands are filled and $c$ conduction bands are accessible are given by the equivariant homotopy classes of maps from the Brillouin torus to $Proj_{v}^{c+v} = Gr_{v}^{c+v}$, hence by the equivariant nonabelian cohomology of the Brillouin torus with coefficients in $Proj_v^{v+c}$

(7)$$(v,c)Phases^G(d) \;\simeq\; H_G\big( \widehat{T}{}^d;\, Proj_{v}^{c+v} \big) \;\coloneqq\; \pi_0\, Map\big( \widehat{T}{}^d ,\, Proj_{v}^{c+v} \big) \,.$$

For example: For $v = 1$ and $c = 1$ (the usual case of Chern insulators, when $d = 2$), the classifying space (5) is the (Riemann) 2-sphere

$$Proj_{1}^{2} \;\simeq\; \mathbb{C}P^1 \;\simeq\; S^2 \mathrlap{\,,}$$

and the nonabelian cohomology theory in (7) is equivariant 2-Cohomotopy.

Stabilization

The stable situation is obtained by allowing access to any number of conduction bands and valence bands.

First, allowing access to arbitrary conduction bands means to consider the union (colimit) of the Grassmannians with respect to their canonical inclusions $Gr_v^{v + c} \hookrightarrow Gr_v^{v + c + 1}$:

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

This yields the classifying space for complex vector bundles of rank=$v$: the valence bundle.

The corresponding classifying nonabelian cohomology theory (7) is also known as unstable K-theory.

Then also allowing an arbitrary number of valence bands means to further pass to the union (colimit)

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

which is known as the classifying space of the stable unitary group. This is the classifying space for reduced complex K-theory (of virtual vector bundles with vanishing virtual rank).

References

Unstable classification of topological phases

Arguments for unstable topological phases of matter, saying that that some effects in topological phases of matter are “unstable” (“fragile” or “delicate”) in that the relevant deformation class of their valence bundles over the Brillouin torus is not their class in topological K-theory (as assumed by the K-theory classification of topological phases of matter) but an un-stable homotopy class (what may be called a class in generalized nonabelian cohomology) such as of maps to a Grassmannian space (or more general flag variety) classifying (systems of) sub-bundles of a trivial vector bundle of fixed finite rank:

Influential precursor discussion:

• Andreas P. Schnyder, Shinsei Ryu, Akira Furusaki, Andreas W. W. Ludwig: Classification of topological insulators and superconductors in three spatial dimensions, Phys. Rev. B 78 195125 (2008) [doi:10.1103/PhysRevB.78.195125, arXiv:0803.2786]

More explicit highlighting of the role of the unstable case and coinage of the term “fragile topologucal phase”:

• Hoi Chun Po, Haruki Watanabe, Ashvin Vishwanath: Fragile Topology and Wannier Obstructions, Phys. Rev. Lett. 121 (2018) 126402 [doi:10.1103/PhysRevLett.121.126402]

• Hoi Chun Po, Liujun Zou, T. Senthil, Ashvin Vishwanath: Faithful tight-binding models and fragile topology of magic-angle bilayer graphene, Phys. Rev. B 99 (2019) 195455 [doi:10.1103/PhysRevB.99.195455, arXiv:1808.02482]

• Adrien Bouhon, Tomáš Bzdušek, Robert-Jan Slager: Geometric approach to fragile topology beyond symmetry indicators, Phys. Rev. B 102 (2020) 115135 [doi:10.1103/PhysRevB.102.115135, arXiv:2005.02044]

Coinage of the term “delicate topological phase”:

• Aleksandra Nelson, Titus Neupert, Tomáš Bzdušek, Aris Alexandradinata: Multicellularity of delicate topological insulators, Phys. Rev. Lett. 126 (2021) 216404 [doi:10.1103/PhysRevLett.126.216404, arXiv:2009.01863]

• Aleksandra Nelson, Titus Neupert, Aris Alexandradinata, Tomáš Bzdušek: Delicate topology protected by rotation symmetry: Crystalline Hopf insulators and beyond, Phys. Rev. B 106 (2022) 075124 [doi:10.1103/PhysRevB.106.075124, arXiv:2111.09365]

• Aleksandra Nelson: Delicate topological insulators: a new world of phases between trivial and fragile, PhD thesis, Zürich (2022) $[$webpage, pdf]

With focus on Bloch Hamiltonian classifying spaces with non-abelian fundamental groups:

• QuanSheng Wu, Alexey A. Soluyanov, Tomáš Bzdušek: Non-Abelian band topology in noninteracting metals, Science 365 (2019) 1273-1277 [doi:10.1126/science.aau8740, arXiv:1808.07469]

Applications:

• Adrien Bouhon, Robert-Jan Slager, around equation (3) in: Multi-gap topological conversion of Euler class via band-node braiding: minimal models, PT-linked nodal rings, and chiral heirs [arXiv:2203.16741]

• Zory Davoyan, Wojciech J. Jankowski, Adrien Bouhon, Robert-Jan Slager, section II.A in: Three-dimensional $\mathcal{P}\mathcal{T}$-symmetric topological phases with Pontryagin index [doi:10.1103/PhysRevB.109.165125 arXiv:2308.15555]

Expositions with an eye towards non-abelian braiding of band nodes in momentum space:

• Gunnar F. Lange: Multi-gap topology & non-abelian braiding in $\mathbf{k}$-space, talk at University of Oslo (Feb 2023) $[$pdf]

• Adrien Bouhon: Non-Abelian and Euler multi-gap topologies in crystalline materials, talk at: Quantum Information and Quantum Matter, CQTS @ NYU Abu Dhabi (May 2023) $[$pdf]

Further discussion:

• Barry Bradlyn, Zhijun Wang, Jennifer Cano, B. Andrei Bernevig: Disconnected Elementary Band Representations, Fragile Topology, and Wilson Loops as Topological Indices: An Example on the Triangular Lattice, Phys. Rev. B 99 (2019) 045140 [doi:10.1103/PhysRevB.99.045140, arXiv:1807.09729]

• Junyeong Ahn, Sungjoon Park, Bohm-Jung Yang, Failure of Nielsen-Ninomiya theorem and fragile topology in two-dimensional systems with space-time inversion symmetry: application to twisted bilayer graphene at magic angle, Phys. Rev. X 9 (2019) 021013 [doi:10.1103/PhysRevX.9.021013, arXiv:1808.05375]

• Yoonseok Hwang, Junyeong Ahn, Bohm-Jung Yang: Fragile Topology Protected by Inversion Symmetry: Diagnosis, Bulk-Boundary Correspondence, and Wilson Loop, Phys. Rev. B 100 (2019) 205126 [doi:10.1103/PhysRevB.100.205126, arXiv:1905.08128]

• Zhi-Da Song, Luis Elcoro, B. Andrei Bernevig: Real Space Invariants: Twisted Bulk-Boundary Correspondence of Fragile Topology, Science 367 6479 (2020) 794-797 [doi:10.1126/science.aaz7650, arXiv:1910.06869]

• Zhida Song, L. Elcoro, Nicolas Regnault, B. Andrei Bernevig: Fragile Phases As Affine Monoids: Classification and Material Examples, Phys. Rev. X 10 031001 (2020) [doi:10.1103/PhysRevX.10.031001, arXiv:1905.03262]

• Xiaoming Wang, Tao Zhou: Fragile topology in nodal-line semimetal superconductors, New Journal of Physics 24 (2022) [doi:10.1088/1367-2630/ac8306, arXiv:2106.06928]

• Piet W. Brouwer, Vatsal Dwivedi: Homotopic classification of band structures: Stable, fragile, delicate, and stable representation-protected topology, Phys. Rev. B 108 (2023) 155137 [doi:10.1103/PhysRevB.108.155137, arXiv:2306.13713]

• Simon Becker, Zhongkai Tao, Mengxuan Yang: Fragile topology on solid grounds: a mathematical perspective [arXiv:2502.03442]

• Hisham Sati, Urs Schreiber: Identifying Anyonic Topological Order in Fractional Quantum Anomalous Hall Systems [arXiv:2507.00138]