We report on a class of gapped projected entangled pair states (PEPS) with non-trivial Euler topology motivated by recent progress in band geometry. In the non-interacting limit, these systems have optimal conditions relating to saturation of quantum geometrical bounds, allowing for parent Hamiltonians whose lowest bands are completely flat and which have the PEPS as unique ground states. Protected by crystalline symmetries, these states evade restrictions on capturing tenfold-way topological features with gapped PEPS. These PEPS thus form the first tensor network representative of a non-interacting, gapped two-dimensional topological phase, similar to the Kitaev chain in one dimension. Using unitary circuits, we then formulate interacting variants of these PEPS and corresponding gapped parent Hamiltonians. We reveal characteristic entanglement features shared between the free-fermionic and interacting states with Euler topology. Our results hence provide a rich platform of PEPS models that have, unexpectedly, a finite topological invariant, forming the basis for new spin liquids, quantum Hall physics, and quantum information pursuits.

The formation of two-band nodal points in gapless topological phases, referred to as conventional Weyl nodes, relies solely on translational symmetry. However, when coupled with other spatial and spatiotemporal symmetries, unconventional Weyl nodes with pronounced chirality, high degeneracy, and complementary quaternion charges can manifest. In this paper, we identify ReO₃ as an ideal unconventional Weyl semimetal in which rotation and screw symmetries as well as their combination with time-reversal symmetry play a crucial role. To show this, we first revisit in detail the algebraic determination of the chirality of doubly and triply degenerate Weyl nodes from the spinful irreducible representations of the occupied and unoccupied bands, and then combine it with the complementary 𝐶₂⁢𝑇-symmetry-protected patch Euler class and non-Abelian frame charges that indicate the pinning of the Weyl nodes on 𝐶₂⁢𝑇-invariant planes. Notably, we find one of the Weyl nodes in ReO₃ as the first example of a node with multiple nontrivial quaternion charges. Supporting our findings with first-principles calculations, we furthermore reveal very clear Fermi arc signatures of the high-chirality Weyl nodes at the Fermi level for different surface orientations. We finally investigate the effect of strain upon which the robustness of Weyl nodes clearly demonstrates their Chern (i.e., chirality conservation) and quaternionic (i.e., symmetry-plane pinning) topological nature.

We analyze quantum-geometric bounds on optical weights in topological phases with pairs of bands hosting nontrivial Euler class, a multigap invariant characterizing non-Abelian band topology. We show how the bounds constrain the combined optical weights of the Euler bands at different dopings and further restrict the size of the adjacent band gaps. In this process, we also consider the associated interband contributions to dc conductivities in the flat-band limit. We physically validate these results by recasting the bound in terms of transition rates associated with the optical absorption of light, and demonstrate how the Euler connections and curvatures can be determined through the use of momentum and frequency-resolved optical measurements, allowing for a direct measurement of this multiband invariant. Additionally, we prove that the bound holds beyond the degenerate limit of Euler bands, resulting in nodal topology captured by the patch Euler class. In this context, we deduce optical manifestations of Euler topology within k·p models, which include quantized optical conductivity, and third-order jerk photoconductivities. We showcase our findings with numerical validation in lattice-regularized models that benchmark effective theories for real materials and are realizable in metamaterials and optical lattices.

Many previous studies of Andreev reflection have demonstrated that unusual effects can occur in media which have a nontrivial bulk topology. Following this line of investigation, we study Andreev reflection by analysing a simple model of a bulk node with a generic winding number n > 0, where the even cases directly relate to topological Euler materials. We find that the magnitudes of the resultant reflection coefficients depend strongly on whether the winding is even or odd. Moreover this parity dependence is reflected in the differential conductance curves, which are highly suppressed for n even but not n odd. This gives a possible route through which the recently discovered Euler topology could be probed experimentally.

We study the effect of disorder in systems having a nontrivial Euler class. As these recently proposed multigap topological phases come about by braiding non-Abelian charged band nodes residing between different bands to induce stable pairs within isolated band subspaces, novel properties may be expected. Namely, a modified stability and critical phases under the unbraiding to metals can arise when the disorder preserves the underlying C2T or PT symmetry on average. Employing elaborate numerical computations, we verify the robustness of associated topology by evaluating the changes in the average densities of states and conductivities for different types of disorders. Upon performing a scaling analysis around the corresponding quantum critical points, we retrieve a universality for the localization length exponent of ν=1.4±0.1 for Euler-protected phases, relating to two-dimensional percolation models. We generically find that quenched disorder drives Euler semimetals into critical metallic phases. Finally, we show that magnetic disorder can also induce topological transitions to quantum anomalous Hall plaquettes with local Chern numbers determined by the initial value of the Euler invariant.

We discuss a class of three-band non-Abelian topological insulators in three dimensions that carry a single bulk Hopf index protected by spatiotemporal (PT) inversion symmetry. These phases may also host subdimensional topological invariants given by the Euler characteristic class, resulting in real Hopf-Euler insulators. Such systems naturally realize helical nodal structures in the three-dimensional Brillouin zone, providing a physical manifestation of the linking number described by the Hopf invariant. We show that, by opening a gap between the valence bands of these systems, one finds a fully-gapped "flag"phase, which displays a three-band multigap Pontryagin invariant. Unlike the previously reported PT-symmetric four-band real Hopf insulator, which hosts a Z⊕Z invariant, these phases are not unitarily equivalent to two copies of a complex two-band Hopf insulator. We show that such uncharted phases can be obtained through dimensional extension of two-dimensional Euler insulators, and that they support (i) an optical bulk integrated circular shift effect quantized by the Hopf invariant, (ii) quantum-geometric breathing in the real-space Wannier functions, and (iii) surface Euler topology on boundaries. Consequently, our findings pave the way for novel experimental realizations of real-space quantum geometry, as these systems may be directly simulated by utilizing synthetic dimensions in metamaterials or ultracold atoms.

Topological band theory has conventionally been concerned with the topology of bands around a single gap. Only recently non-Abelian topologies that thrive on involving multiple gaps were studied, unveiling a new horizon in topological physics beyond the conventional paradigm. Here, we report on the first experimental realization of a topological Euler insulator phase with unique meronic characterization in an acoustic metamaterial. We demonstrate that this topological phase has several nontrivial features: First, the system cannot be described by conventional topological band theory, but has a nontrivial Euler class that captures the unconventional geometry of the Bloch bands in the Brillouin zone. Second, we uncover in theory and probe in experiments a meronic configuration of the bulk Bloch states for the first time. Third, using a detailed symmetry analysis, we show that the topological Euler insulator evolves from a non-Abelian topological semimetal phase via. the annihilation of Dirac points in pairs in one of the band gaps. With these nontrivial properties, we establish concretely an unconventional bulk-edge correspondence which is confirmed by directly measuring the edge states via. pump-probe techniques. Our work thus unveils a nontrivial topological Euler insulator phase with a unique meronic pattern and paves the way as a platform for non-Abelian topological phenomena.

Two-dimensional Euler insulators are novel kinds of systems that host multigap topological phases, quantified by a quantized first Euler number in their bulk. Recently, these phases have been experimentally realized in suitable two-dimensional synthetic matter setups. Here, we introduce the second Euler invariant, a familiar invariant in both differential topology (Chern-Gauss-Bonnet theorem) and in four-dimensional Euclidean gravity, whose existence still needs to be explored in condensed matter systems. Specifically, we first define two specific models in four dimensions that support a nonzero second Euler number in the bulk together with peculiar gapless boundary states. Second, we discuss its robustness in general spacetime-inversion invariant phases and its role in the classification of topological degenerate real bands through real Grassmannians. Our results naturally generalize the second Chern and spin Chern numbers to the case of four-dimensional phases that are characterized by real Hamiltonians and open doors for implementing such higher-dimensional phases in artificial engineered systems, ranging from ultracold atoms to photonics and electric circuits.

We report on a certain class of three-dimensional topological insulators and semimetals protected by spinless 𝒫⁢𝒯 symmetry, hosting an integer-valued bulk invariant. We show using homotopy arguments that these phases host multigap topology, providing a realization of a single ℤ invariant in three spatial dimensions that is distinct from the Hopf index. We identify this invariant with the Pontryagin index, which describes Belavin-Polyakov-Schwartz-Tyupkin (BPST) instantons in particle physics contexts and corresponds to a three-sphere winding number. We study naturally arising multigap linked nodal rings, topologically characterized by split-biquaternion charges, which can be removed by non-Abelian braiding of nodal rings, even without closing a gap. We additionally recast the describing winding number in terms of gauge-invariant combinations of non-Abelian Berry connection elements, indicating relations to Pontryagin characteristic class in four dimensions. These topological configurations are furthermore related to fully nondegenerate multigap phases that are characterized by a pair of winding numbers relating to two isoclinic rotations in the case of four bands and can be generalized to an arbitrary number of bands. From a physical perspective, we also analyze the edge states corresponding to this Pontryagin index as well as their dissolution subject to the gap-closing disorder. Finally, we elaborate on the realization of these novel non-Abelian phases, their edge states, and linked nodal structures in acoustic metamaterials and trapped-ion experiments.

We study the relationship between momentum-space spin textures projected onto the occupied bands and Wilson loop winding, proving a map between band topology and spin topology in certain restricted symmetry settings relevant to fragile topology. Our results suggest that, in specific scenarios, the spin gap may act as a smoking gun bulk indicator for fragile topology.