Euler Band Topology

• Euler band topology is defined for real Bloch bands under space-time inversion symmetries, using the Euler class as an integer invariant to characterize fragile and non-Abelian topological features.
• It employs mathematical formulations based on Euler curvature and quantum geometric tensors to capture nodal points, skyrmion textures, and band-node braiding in two- and three-band systems.
• Experimental realizations using trapped ions, acoustic metamaterials, and electric circuits have validated its role in producing exotic edge states and bulk responses.

Euler band topology denotes a class of topological structures of real Bloch bands or real Bloch subspaces in systems with space-time inversion symmetries such as $PT$, $C_2T$, $C_{2z}T$, or $I_{ST}$, where the Bloch Hamiltonian can be chosen real and the usual Berry-curvature-based Chern number vanishes. Its central invariant is the Euler class of an isolated pair of real bands, an integer-valued multigap invariant that underlies fragile topology, stable band nodes, non-Abelian braiding, and unconventional boundary phenomena; related literature also employs an Euler characteristic number built from the quantum metric of a Bloch-state manifold, which is conceptually distinct from the Euler class of real-band subspaces (Ahn et al., 2018, Zhao et al., 2022, Ma et al., 2012).

1. Terminological scope and conceptual distinctions

The modern literature uses the term “Euler” in two closely related but non-identical senses. In the real-band topology literature, the Euler class $e_2$, $E$, $\xi$, or $\chi$ characterizes a pair of real bands, or equivalently an orientable real rank-2 bundle over a two-dimensional Brillouin zone. In this setting, the invariant is intrinsically interband: unlike the Chern number, which is defined for an individual complex band, the Euler invariant is defined for a set of real bands, and it becomes natural precisely when $PT$, $C_2T$, or $C_2T$0 imposes a real gauge and forces the Berry curvature and Chern number to vanish (Ezawa, 2021, Kwon et al., 2023).

A second usage, originating from quantum geometry, defines an Euler characteristic number for a Bloch-state manifold through the Gauss-Bonnet theorem applied to the Riemannian metric furnished by the real part of the quantum geometric tensor. This invariant is attached to the geometry of a band manifold rather than to the orientability of a real two-band subspace. In that line of work, non-integer values in trivial phases are not interpreted as fractional topology; rather, they indicate that the quantum metric becomes degenerate and the Gauss-Bonnet formula ceases to apply, so the “non-integer Euler number” is ill-defined (Ma et al., 2012, Ma, 2020).

Quantity	Typical setting	Defining structure
Euler class $C_2T$1, $C_2T$2, $C_2T$3	Two real bands or a real two-band subspace in 2D	Euler curvature / real Berry connection
Euler number in three-band models	2D three-band real Hamiltonians	Often reducible to a Pontryagin or skyrmion number
Euler characteristic number $C_2T$4	Bloch-state manifold with quantum metric	Gauss-Bonnet curvature integral

This distinction matters because the dominant usage in contemporary “Euler band topology” concerns real multigap topology, fragile topology, and non-Abelian nodal structure, whereas the quantum-metric Euler characteristic emphasizes band geometry and Riemannian curvature. The two programs are related by quantum geometry, but they are not interchangeable.

2. Mathematical structure of the Euler invariant

For two real bands $C_2T$5 over a two-dimensional Brillouin zone, the Euler class can be written as an integral of the Euler curvature, or equivalently of the curl of the real Berry connection. A standard form is

$C_2T$6

with

$C_2T$7

while a patchwise formulation on a region $C_2T$8 is

$C_2T$9

where $C_{2z}T$0 and $C_{2z}T$1 (Breach et al., 2024, Jankowski et al., 2023).

In three-band two-dimensional models, the Euler number can reduce to a Pontryagin number or skyrmion number. For a real vector $C_{2z}T$2,

$C_{2z}T$3

and in explicit three-band constructions the invariant is carried by skyrmion textures in momentum space (Ezawa, 2021, Zhao et al., 2022). In the trapped-ion realization of a mapped Euler Hamiltonian,

$C_{2z}T$4

the nontrivial phase at $C_{2z}T$5 exhibits a full skyrmion wrapping and yields $C_{2z}T$6, while the trivial phase at $C_{2z}T$7 yields $C_{2z}T$8 (Zhao et al., 2022). By contrast, the acoustic kagome realization reports an odd Euler class $C_{2z}T$9 associated with a meron pattern, i.e. a half-skyrmion characterization (Jiang et al., 2022). This suggests that the detailed normalization and bundle realization are model-dependent.

Quantum geometry enters Euler band topology through a distinct but powerful inequality. For two real bands in an $I_{ST}$0-symmetric system,

$I_{ST}$1

so the quantum volume provides a lower bound on the Euler invariant. Saturation of this bound defines an “ideal condition” for Euler bands,

$I_{ST}$2

which was proposed as a geometric criterion for stabilizing interaction-driven fractional topological phases in Euler bands (Kwon et al., 2023).

3. Stable nodes, fragile topology, and non-Abelian braiding

A defining consequence of a nonzero Euler class is the existence of stable nodal points between the relevant bands. In a real two-band subspace with Euler class $I_{ST}$3, the system must support $I_{ST}$4 stable band crossings, and in two-band systems with $I_{ST}$5 symmetry the total winding number of Dirac points is

$I_{ST}$6

This directly underlies the statement that the conventional Nielsen-Ninomiya theorem fails in two-dimensional systems with nonzero Euler class: the Dirac points need not come in pairs of opposite winding (Ahn et al., 2018, Guan et al., 2021).

This nodal topology is fragile rather than stable. Adding a trivial band destroys the integer Euler classification and reduces it to the second Stiefel-Whitney invariant,

$I_{ST}$7

Thus an even Euler class becomes $I_{ST}$8, an odd Euler class becomes $I_{ST}$9, and the full integer obstruction is lost under band addition (Ahn et al., 2018, Zhao et al., 2022). In twisted bilayer graphene at the magic angle, the Wannier obstruction and fragile topology of the nearly flat bands were explicitly tied to this Euler-class structure (Ahn et al., 2018).

The instability of Euler topology under band addition is tied to its non-Abelian nodal charges. In real multiband systems, band nodes are characterized not by simple Abelian signs but by frame-rotation charges that obey quaternionic algebra. Dirac strings act as branch cuts across which the sign of a node’s charge flips, and pair annihilation becomes path-dependent: braiding nodes in adjacent gaps can change whether a pair of nodes can annihilate (Breach et al., 2024). In three- and four-band minimal models, phase transitions between inequivalent Euler phases are mediated by linked nodal rings or braiding trajectories in adjacent gaps, and the linking numbers match the relevant Euler-class changes (Bouhon et al., 2022).

The patch Euler class is the natural language for this local obstruction. On a region containing several nodes, it detects whether the enclosed nodes carry a net same-charge obstruction and therefore cannot be pairwise annihilated without closing an adjacent gap or changing the braiding history (Breach et al., 2024).

4. Bulk response, boundary correspondence, and dynamical manifestations

The simplest explicit “topological Euler insulator” was formulated as a two-dimensional three-band tight-binding model in which the momentum-dependent vector $e_2$0 forms a skyrmion texture for $e_2$1. In that regime the Euler number is nontrivial, and topological edge states appear, establishing a bulk-edge correspondence for the Euler invariant (Ezawa, 2021). The same work mapped the Hamiltonian to an electric-circuit Laplacian,

$e_2$2

with edge states signaled by strong impedance resonances at

$e_2$3

Euler topology also leaves a characteristic imprint in magnetic response. In two-dimensional $e_2$4-symmetric Euler insulators subject to a magnetic field, the Hofstadter butterfly is robustly gapless at half filling in the flat-band limit whenever the Euler class is nontrivial. For dispersive bands, the gapping of the Landau levels is controlled by a hidden symmetry in balanced Euler phases, while imbalanced phases remain robustly gapless without fine-tuning. The same analysis established that the Chern number of the main magnetic subgap obeys a lower bound proportional to the Euler class, $e_2$5, thereby linking fragile real-band topology to quantum Hall response (Guan et al., 2021).

Out-of-equilibrium dynamics furnish a second family of signatures. Quenching into an Euler Hamiltonian generates stable monopole-antimonopole pairs and linked momentum-time trajectories under the first Hopf map, making the Euler class observable through dynamical tomography in optical lattices (Ünal et al., 2020). In trapped-ion simulations, quench dynamics were used to probe skyrmion-antiskyrmion pairs and Hopf links in momentum-time space, extending static Euler topology into a dynamical setting (Zhao et al., 2022).

Optical response encodes further constraints. Quantum-geometric bounds imply that the combined optical weights of Euler bands are lower-bounded by the magnitude of the Euler class, and the same framework restricts the sizes of the adjacent band gaps. In nodal Euler phases, effective $e_2$6 models yield quantized optical conductivity and universal ratios of third-order jerk photoconductivities, with the patch Euler class controlling the low-frequency response (Jankowski et al., 2023).

5. Experimental realizations and measurement protocols

The first direct quantum-simulation observation of a topological Euler insulator was achieved with a trapped-ion qutrit simulating a three-band Hamiltonian. Quantum state tomography on a $e_2$7 discretized Brillouin zone reconstructed the highest-energy eigenstate at each momentum and allowed direct evaluation of the Euler class. The same experiment measured Wilson loop flow, entanglement spectra, and Berry phases around the four Dirac points, confirming the predicted topology and the protection of the nodes by the Euler class (Zhao et al., 2022).

An acoustic metamaterial on a kagome lattice provided a complementary observation. There the Euler insulator phase was characterized by a nontrivial Euler class and a meronic bulk Bloch pattern reconstructed from measured response data. Pump-probe measurements directly detected in-gap edge states, and symmetry analysis connected the gapped Euler phase to a non-Abelian topological semimetal through pair annihilation of Dirac points in one of the band gaps (Jiang et al., 2022).

Electric circuits offer a particularly transparent platform because the circuit Laplacian can be engineered to reproduce the target Hamiltonian. In the circuit implementation of a topological Euler insulator, edge-localized states were read out as impedance resonances, providing an experimentally accessible bulk-boundary diagnostic (Ezawa, 2021).

Cold-atom proposals go beyond static spectroscopy. One protocol reconstructs the generalized Bloch vector after a quench and identifies the linked momentum-time trajectories associated with the Euler class; another uses band-node interferometry in ultracold atoms to determine the relative frame charges of nodes and thereby measure the patch Euler class. A complementary consecutive-deflection scheme extracts the sign structure of two frame charges from final band populations after a single wavepacket probes two nodes sequentially (Ünal et al., 2020, Breach et al., 2024).

Optical probes can, in principle, access Euler topology directly through momentum- and frequency-resolved measurements. The optical-bound analysis showed that Euler connections and curvatures can be reconstructed from transition rates, providing a route to direct measurement of this multiband invariant in lattice models, metamaterials, and optical lattices (Jankowski et al., 2023).

6. Three-dimensional, magnetic, superconducting, and classical extensions

Euler band topology now extends well beyond two-dimensional electronic insulators. In three-dimensional $e_2$8-symmetric insulators with two occupied bands, the relevant invariant is

$e_2$9

the difference between Euler classes on the two $E$0-invariant planes. Surface theory and tight-binding calculations show that a phase with $E$1 supports $E$2 chiral hinge modes, a result explicitly demonstrated for $E$3 and $E$4. Because the system has only two occupied bands, these phases are distinct from stacked Chern insulators with the same number of hinge channels (Tanaka et al., 27 Mar 2026).

In spin-orbit-coupled magnetic systems, Euler bands can emerge when a quantum spin Hall insulator with $E$5 symmetry is subjected to an in-plane Zeeman field or in-plane ferromagnetism that preserves $E$6. The resulting magnetic insulator may be a magnetic Euler insulator if the topmost occupied pair carries a nonzero Euler number, or a magnetic Stiefel-Whitney insulator more generally. The topological phase transition is mediated by a stable topological semimetal phase in which Dirac nodes with non-Abelian charges braid across the transition (Lee et al., 2024).

Bogoliubov-de Gennes Hamiltonians with $E$7 symmetry provide a superconducting and superfluid generalization. In symmetry classes DIII and CI, the Euler class becomes an integer-valued invariant of a $E$8 BdG Hamiltonian, and three-dimensional class-DIII phases with odd winding number, including the B phase of superfluid $E$9He, were identified as Euler superconductors or superfluids. In class CI with inversion symmetry, nontrivial Euler class enforces superconducting nodal lines with linking structure (Kobayashi et al., 8 Sep 2025). A related honeycomb-lattice analysis found valley-Euler superconductors for $\xi$0-wave spin-singlet pairing and Euler superconductors for $\xi$1-wave spin-triplet pairing, together with mirror-symmetry-protected helical domain-wall modes and non-Abelian braiding of Dirac nodes in momentum space (Ghadimi et al., 12 May 2026).

Classical platforms exhibit the same multigap logic. Elastic metamaterials have realized coexisting Euler and Stiefel-Whitney phases in a single vector-wave system, where non-Abelian braiding of nodal points and topologically stable Goldstone-mode degeneracy produce simultaneous edge and corner states (Tang et al., 9 Mar 2025). Classical spin liquids provide a further extension: the Euler class enters homotopy-based classifications of dispersive eigenvectors, and band-node braiding alters the topology and stability of pinch points in the spin structure factor (Rüegg et al., 14 May 2025).

Taken together, these developments establish Euler band topology as a unifying framework for real multigap topology. Its defining features are integer-valued topology for real band pairs, enforced stable nodes, path-dependent non-Abelian conversion under braiding, fragility under addition of trivial bands, and a broad range of manifestations in edge states, hinge states, optical response, quench dynamics, and classical-wave analogues.