Hopf Number: Topological Linking Invariant

• Hopf number is an integer-valued topological invariant defined by the linking of preimages in continuous maps from S³ to S², clearly classifying distinct homotopy classes.
• It is computed via Chern–Simons 3-form integrals and explicit constructions in momentum space, playing a critical role in characterizing hopfions, Hopf insulators, and semimetals.
• Its experimental relevance is highlighted through quantum quenches and solitonic measurements in liquid crystals and magnetic systems, confirming its theoretical predictions.

The Hopf number, or Hopf invariant, is an integer-valued topological invariant associated with continuous maps from the three-dimensional sphere to the two-dimensional sphere. It is a central quantity in algebraic topology and finds direct realization in the band theory of topological phases, field configurations of solitons, and the experimental observation of knot-like structures in diverse physical systems, including quantum matter and liquid crystals. The Hopf number measures the linking of preimage fibers: distinct loops in the domain whose images coincide with different points on the target sphere, with the integer value quantifying their mutual linking. This invariant underpins the stability, classification, and physical manifestation of hopfions, Hopf insulators, and Hopf semimetals.

1. Topological Definition and Linking Interpretation

Given a continuous map $\varphi: S^3\to S^2$, the Hopf number $H(\varphi)$ is defined as the linking number between the preimages in $S^3$ of two regular values $q_1, q_2\in S^2$ not in the image of the singular locus of $\varphi$. Each preimage set $L_i=\varphi^{-1}(q_i)$ is typically a link, possibly with multiple components. The Hopf number is given by

$H(\varphi)=\mathrm{Lk}(L_1,L_2),$

the total linking number, which is independent of the choice of $q_1$ and $q_2$ (Nozaki et al., 20 Jul 2025). In homotopy-theoretic terms, $H$ classifies maps up to homotopy: $\pi_3(S^2) \cong \mathbb Z$ and the Hopf number assigns to each equivalence class an integer, quantifying how many times the domain wraps and links around the target (Wang et al., 2014).

2. Explicit Constructions and Integral Formulae

In the context of topological phases, the Hopf number is computed for maps emerging from two-band models with Pauli-matrix-valued Bloch Hamiltonians: $$H(\mathbf{k})=d_i(\mathbf{k})\,\sigma^i,\quad |d(\mathbf{k})|\neq 0.$$ Defining the normalized vector $n(\mathbf{k})=d(\mathbf{k})/|d(\mathbf{k})|\in S^2$, the map $T^3\to S^2$ (with $T^3$ the Brillouin zone) is established. This construction is routinely recast as a composition with spinors $u(\mathbf{k})$ normalized to $z(\mathbf{k})\in S^3$, yielding

$$T^3 \xrightarrow{u} S^3 \xrightarrow{z\mapsto d(z)} S^2,$$

with $d_i(z) = z^* \sigma^i z$ (Wang et al., 2014, Ezawa, 2017).

The Hopf number is extracted via a Chern–Simons 3-form over momentum space: $$H = \frac{1}{32\pi^2}\int_{T^3} d^3k\;\epsilon^{\mu\nu\rho}A_\mu(k)F_{\nu\rho}(k)$$ with $A_\mu(k)$ the Berry connection and $F_{\nu\rho}(k)$ its curvature, or, equivalently,

$$H = -\frac{1}{4\pi^2}\int d^3k\;\epsilon^{\mu\nu\rho}\,A_\mu\,\partial_\nu A_\rho$$

(Wang et al., 2014, Ezawa, 2017, Yi et al., 2019). In the most general topological setting, the Hopf invariant may also be expressed as a four-dimensional winding number or in terms of singularity theory using Stein factorizations for generalized fold maps (Nozaki et al., 20 Jul 2025).

3. Generalized Hopf Maps, High Hopf Number, and Fiber Structure

Explicit families of maps $\varphi_n:S^3\to S^2$ of arbitrary integer Hopf number $n$ are engineered by decomposing $S^3$ into toroidal pieces and successively twisting their fibers, introducing $n$-fold saddle singularities. The Stein factorization of such a map produces a 2D polyhedron composed of $n+1$ disks glued along their boundaries, encoding the global structure of the preimages (Nozaki et al., 20 Jul 2025).

The map’s linking properties are as follows:

• For generic points in the upper hemisphere of $S^2$, preimages under $\varphi_n$ form $(2,2n)$-torus links in $S^3$, with linking number $n$.
• Six canonical arrangements of two-fiber preimages emerge, corresponding to the distributions of points on $S^2$: both on the equator, both in northern or southern hemisphere, and mixed cases. Each diagram encodes a spatial graph whose total linking number is $n$ (Nozaki et al., 20 Jul 2025).

Experimentally, this fiber structure is manifest as linked disclination loops in chiral liquid crystals or magnetic materials, with high-index hopfion structures matching the topological predictions for preimage linkages.

4. Hopf Number in Band Topology: Hopf Insulators and Semimetals

The Hopf number serves as the primary topological invariant classifying the so-called Hopf insulators. In three-dimensional two-band Hamiltonians with vanishing Chern numbers along all planar cuts, the integer valued Hopf invariant $H$ precludes any adiabatic deformation to a trivial insulator without a gap-closing, ensuring robustness of topological phases without symmetry protection (Wang et al., 2014).

Explicit lattice models—e.g., those assigning spinor components raised to integer powers $p,q$—yield Hopf number $H=\pm pq$ (Ezawa, 2017, Wang et al., 2014). Semimetal realizations permit generalization to torus links and knots:

• $(p,q)$ integer leads to Hopf links, Solomon’s knots, double links/knots, with linking number $pq$.
• Half-integer $p$ or $q$ supports torus knot Fermi surfaces (e.g., trefoil).
• Rational $p$ or $q$ yields open-string Fermi surfaces (nodal lines that do not close).

The Fermi surface thereby becomes the preimage of specific points on $S^2$, with the linking structure directly encoding the Hopf number (Ezawa, 2017). This mapping from topological invariants to physical structures extends to signatures in ARPES, Landau level phenomena, and optical responses.

5. Dynamical and Experimental Measurement of the Hopf Number

The Hopf number finds direct physical realization and measurement in quantum quenches and dynamical evolutions. In 2D quantum anomalous Hall systems subject to a quench, the combination of quasimomentum $(q_x, q_y)$ and evolution time $t$ parameterizes a $T^3\to S^2$ Hopf map for the evolving Bloch vector. The dynamical Hopf invariant $I_H$ is rigorously equal to the Chern number $C$ of the post-quench Hamiltonian: $$I_H = \frac{1}{4\pi}\int_{T^2}dq_x dq_y\, \hat{\bm h}_f \cdot (\partial_{q_x}\hat{\bm h}_f \times \partial_{q_y}\hat{\bm h}_f) = C$$ (Yi et al., 2019).

Experimental procedure:

• Time-dependent spin-polarization $P_z(q_x, q_y, t)$ is extracted via time-of-flight imaging.
• Trajectories in quasimomentum-time space at $P_z=+1$ and $P_z=-1$ are traced, corresponding to Hopf fibers for the North and South Poles.
• The number of mutual linkings directly yields the Hopf number (absolute Chern number), visualizing the abstract invariant in concrete, measureable structures.

Nested Hopf tori at non-extremal latitudes on the sphere further evidence the full fibration, observed as mutually nested toroidal surfaces in the parameter space (Yi et al., 2019).

6. Dimensional Reduction and $\mathbb Z_2$ Classification in 2D Systems

Dimensional reduction from 3D Hopf insulators to 2D insulators with vanishing Chern number allows for the definition of a derived $\mathbb Z_2$ topological index. Viewing $k_z$ as a parameter interpolating between $k_z=0$ and $k_z=\pi$ slices, the Hopf number $H$ defined over the periodic $k_z$ loop gives rise to

$\nu = (-1)^H \in \{+1, -1\}$

as the $\mathbb Z_2$ index. For odd $H$, a 2D model is "dimensionally reduced Hopf-nontrivial" and can present protected edge states even though the conventional Chern number vanishes. This extension considerably broadens the scope of topological classification beyond symmetry-protected topological insulators (Wang et al., 2014).

7. Physical and Geometric Impact Across Systems

The Hopf invariant is not only a descriptor of abstract topological maps but provides a quantitative link between mathematical topology and observed structure in materials:

• In magnetic and liquid crystal hopfions, measured linking of director field preimages matches theoretical preimage link patterns for $\varphi_n$ with $H=n$.
• Composite solitonic objects with net $Q=0$ can be constructed and classified via hybridization of maps with $H=+1$ and $H=-1$, in line with experimental findings (Nozaki et al., 20 Jul 2025).
• Modifications of the singular locus or perturbations in field configurations lead to richer sets of linking geometries, all constrained by the global topological classification.

This unification of linkage, Chern–Simons-type invariants, and physical degeneracy structures, underpins much of the ongoing research in three-dimensional topological matter, field theory solitons, and the topology of differentiable maps (Wang et al., 2014, Ezawa, 2017, Nozaki et al., 20 Jul 2025, Yi et al., 2019).