Topological non-Abelian gauge structures in Cayley-Schreier lattices

Abstract

Crystalline constructions known as Cayley-Schreier lattices have been suggested as a platform for realizing arbitrary gauge fields in synthetic crystals with real hopping amplitudes. Here, we reveal that Cayley-Schreier lattices can naturally give rise to implementable lattice systems that incorporate non-Abelian gauge structures transforming into a space-group symmetry. We show that their symmetry sectors can be interpreted as blocks of pseudospin models–some of which correspond to true spinors–that can realize a wealth of different topological invariants in a single setup. We underpin these general results with concrete models and illustrate how they can be implemented in technologically available experimental platforms. Our work sets the stage for a systematic investigation of topological insulators and metals with non-Abelian gauge structures.

Cayley-Schreier lattices have been shown to enable the inclusion of non-commutative translations into Abelian systems. Here, authors theoretically demonstrate that these crystalline systems can also bring to the emergence of non-Abelian gauge structures, providing a perspective on realistic experimental implementations.

Introduction

Although a fairly uniform understanding of conventional topological band theory^1–6 has emerged over the past decade, recent results have opened further active research directions within this field. These, in particular, include multi-gap topological phases^7–13 in which invariants emerge from simultaneous involvement of multiple bands, typified by systems with $\mathcal{I}=\mathcal{PT}$ symmetry (product of parity and time reversal) with ${\mathcal{I}}^2=1$, and systems with projectively represented space-group symmetry^14–19. The latter direction is well exemplified by spinless systems whose spatial-temporal symmetry obeys ${\mathcal{I}}^2=-1$^14,17 due to modifying the symmetry operators with gauge transformations. This line of investigations is particularly compelling, as it combines the concepts of gauge theory²⁰—a topic currently heavily explored in quantum simulators²¹—with symmetry representations of band theory. In particular, when the symmetry group is represented projectively, meaning that the representation of translations, reflections, or other symmetries admits an additional phase factor, these phases can add up to a finite value along a closed contour, or, in other words, to a gauge flux^14–19. This shows that altered symmetry can underlie topological band classification, which has been experimentally verified^22–28 in various realizations. Nonetheless, this progress has, by and large, been limited to the Z₂ gauge group, thus posing the question of whether larger gauge groups—possibly non-Abelian ones—may enable previously inaccessible phenomena.

In a separate context, an alternative route of incorporating non-commutativity of translations and effective synthetic gauge structures has been explored through constructions called Cayley-Schreier lattices (CSLs)^29,30. Such lattices, in essence, proceed by generalizing the “on-site” symmetry group, producing a Hamiltonian whose translations allow for more generic structures. However, here also the focus has been on Abelian Z_n cases. Given the freedom in defining the on-site symmetry group, this provokes the central question of whether non-Abelian structures can similarly be introduced while enabling new physical manifestations.

Building on these developments, we here affirmatively answer the central question presented above and formulate a broader class of non-Abelian CSLs. As a main result, we find that CSLs provide a platform in which non-Abelian gauge structures naturally emerge from space group symmetries. As a next step, we show that the dynamics on a CSL separates into decoupled blocks, each corresponding to a symmetry sector of the gauge group. The different symmetry sectors can be interpreted as pseudospin-N models—some of which correspond to true (2N  +  1)-component spinors—where N is specified by the dimension of the block. This moreover allows for a route towards topology. Indeed, we motivate that for the two-component spinor (that is, spin-$\frac{1}{2}$ fermions exemplified by electrons) all known types of band topology can be induced within this single framework. We finally illustrate this general theory with concrete examples in one and two spatial dimensions. These models can directly serve as input for meta-material realizations. To that end, we also provide specific pathways for experimental validation in electric-circuit networks^31–39 with concrete proposals.

Results

Cayley-Schreier lattices

We now review the concept of a Cayley-Schreier lattice (CSL). The key insight here is that the conventional sites are replaced with “pillars” of internal degrees of freedom that transform under a finite group G. While our results can be directly generalized, we focus in particular on the case where this symmetry group entails the quaternion group

$$Q_8=\left\{\pm 1,\pm \mathrm{i},\pm \mathrm{j},\pm \mathrm{k}\right\},$$ 1

where i, j, and k = i ⋅ j are pairwise anticommuting units squaring to i² = j² = k² = − 1. We also adopt the group Z₂ = { ± 1} to clarify the building blocks of CSLs.

In the CSL construction, illustrated in Fig. 1a, each pillar consists of ∣G∣ orbitals (depicted as black dots), with each orbital labeled by a distinct group element g  ∈  G. Here ∣G∣ is the number of elements in the group G. For G = Z₂, as an elementary example, each pillar has two orbitals, whereas for G = Q₈, each pillar hosts eight orbitals corresponding to the eight elements in Eq. (1). A hopping Hamiltonian is then formed by connecting the orbitals of different pillars, and thereby generating the bonds of the lattice with a given space group ($SG$).

Fig. 1. Synthetic gauge fields in Cayley-Schreier lattices.

a Pair of connected pillars of a Cayley-Schreier lattice (CSL) with gauge group G = Z₂. Each pillar represents a lattice site (indices n and ℓ) and contains two orbitals (one for each group element  ± 1 ∈ Z₂). The bonds between the constituent orbitals are `horizontal' (top) or `swapped' (bottom), depending on the value of the connection a_n,ℓ = ± 1 ∈ Z₂. b Illustration of a CSL with G = Z₂ over two-dimensional lattice. One of the two choices of connection from panel a is made for the bonds between any pair of connected pillars. c Schematics of a CSL over a one-dimensional chain with gauge group G = Q₈. There are ∣Q₈∣ = 8 choices of connection possible on each bond, of which four (namely a_n+1,n ∈ { + 1, + i, + j, − j}) are illustrated. The components highlighted with orange indicate that the orbital g_n = + j at site n is coupled by connection a_n+1,n = + i to the orbital g_n+1 = a_n+1,n ⋅ g_n = ( + i) ⋅ ( + j) = + k at site n + 1. d Alternating arrangement of trivial and non-trivial connections in a one-dimensional CSL with gauge group Z₂ (top). Translation by one site flips the connection pattern (middle). Applying gauge transformation χ_i = − 1 to every second site (red rectangles) restores the original connection (bottom). The coloring of orbitals and hoppings within the two selected sites serve as a guide to follow the application of the translation and of the subsequent gauge transformation. In panels b–d, dashed lines represent couplings of the same strength t.

Most importantly, these connections are chosen so that they respect the group multiplication in the following sense: we assign a group element a_n,ℓ  ∈  G to the bond from site ℓ to site n and ensure that orbital g_ℓ at ℓ is connected to orbital a_n,ℓ ⋅ g_ℓ at n (the dot stands for the binary composition in G). For the Z₂ example, this yields two types of bonds: ‘horizontal’ for a_n,ℓ = + 1 and ‘swapped’ for a_n,ℓ = − 1, both featured as dashed lines in Fig. 1a and b. Several allowed connections a_n,ℓ between the pillars for the case G = Q₈ are illustrated in Fig. 1c. This construction ensures that between any two connected pillars n and ℓ, there are ∣G∣ hopping terms; namely, one for each g_ℓ  ∈  G. This pairing is one-to-one because group multiplication is a bijection. Crucially, we assign the same hopping amplitude t_n,ℓ to all the hopping terms between sites n and ℓ. Different pillar-to-pillar bonds may carry different connections and amplitudes, and this is what allows non-trivial flux patterns.

The single-particle Hilbert space is $\mathcal{H}{=⨁}_n{\mathcal{H}}_n$ with ${\mathcal{H}}_n=span\left\{\mid n,g⟩:g\in G\right\}$, where $\mid n,g⟩$ is the orbital labeled by group element g at site n. Each pillar contributes ∣G∣ basis states, and N pillars generate a Hilbert space of $\dim \mathcal{H}=N\times \mid G\mid$. The hopping between pillars is then described by the CSL Hamiltonian

$$H={\sum }_{n,\ell \in sites}{t}_{n\ell }{\sum }_{g,h\in G}{c}_{n,g}^{†}{\left[\rho \left(a_{n,\ell }\right)\right]}_{g,h}{c}_{\ell ,h}.$$ 2

Here, ρ(a_n,ℓ) is the regular representation⁴⁰ of G, which describes the ∣G∣ × ∣G∣ permutation matrix describing left multiplication by a_n,ℓ. Specifically, we have [ρ(a)]_g,h = 1 if g = a ⋅ h and zero otherwise, so the sum over g and h picks out exactly the hoppings from $\mid \ell ,h⟩$ to $\mid n,a_{n,\ell }\cdot h⟩$. For instance, [ρ(+i)]_+k,+j = 1 because  + k = ( + i) ⋅ ( + j). [See Supplementary Note 1 of the Supplementary Information for explicit constructions of the regular representations of Z₂ and Q₈.] The resulting internal structure of pillars of group-labeled orbitals connected by regular representations is simple enough to enable universal feasibility in meta-material implementations.

We point out that CSLs as considered in this work extend the earlier constructions of the same name^29,30 in two important aspects. First, we do not require the entirety of the orbitals (that is, all physical sites times the pillar content) to transform as elements of a single group, but merely that the construction respects a specified $SG$ in a sense that will be specified shortly. Second—and crucially for the realization of topological models—we allow the hopping amplitudes t_n,ℓ to depend on the sites ℓ and n in a manner that preserves $SG$.

Synthetic gauge structure

CSLs naturally give rise to synthetic gauge structures with the chosen group G, henceforth called the gauge group. Specifically, we can permute the orbitals within a pillar as $g_n\mapsto {\chi }_n\cdot g_n={\tilde{g}}_n$ where χ_n ∈ G is the gauge transformation at site n (see also Supplementary Note 3). In the elementary case of gauge group Z₂, the non-trivial gauge transformation χ_n = − 1 exchanges the two orbitals at site n, which results in a change of the connection on bonds adjacent to n (see lower two rows in Fig. 1d). For a general choice of G and χ, the connection transforms as

$${\tilde{a}}_{n,\ell }={\chi }_n\cdot a_{n,\ell }\cdot {\chi }_{\ell }^{-1}.$$ 3

This gauge freedom allows us to reorder orbitals within a pillar and to apparently reshuffle the couplings between orbitals of adjacent pillars while still describing the same physical system. Nevertheless, certain characteristics remain invariant in this process. Specifically, the ‘integral’ of connection on a closed loop γ (assumed to be based at site n and passing through sites ℓ₁, …, ℓ_M), defined as

$$W_{\gamma }=a_{n,{\ell }_M}\cdot a_{{\ell }_M,{\ell }_{M-1}}\cdot \dots \cdot a_{{\ell }_2,{\ell }_1}\cdot a_{{\ell }_1,n},$$ 4

transforms under a general choice of gauge transformation to ${\tilde{W}}_{\gamma }={\chi }_n\cdot W_{\gamma }\cdot {\chi }_n^{-1}$. It follows that the integral, called Wilson loop or flux on γ, is well-defined as a conjugacy class in the gauge group G. (Note that we here slightly deviate from the standard lattice gauge theory nomenclature⁴¹ where these terms are instead reserved for the trace of W_γ.) For Abelian groups like Z₂, each element of G constitutes its own conjugacy class. For Q₈, one finds five conjugacy classes, namely {1}, { ± i}, { ± j}, { ± k} and { − 1}, which were previously related to non-trivial braiding of vortex defects in biaxial nematic liquids^42–44 and of band nodes in metals^7–12,45. This noncommutativity also plays a pivotal role when introducing concrete models with quaternion fluxes in the sections below.

Symmetry representations

The inherent presence of a synthetic gauge structure implies that representations of symmetries in CSLs can be modified by gauge transformations. With this, we mean that simply translating the pillar locations with symmetry $s\in SG$ from position vectors r_n to s(r_n) may not preserve the arrangement of the connections; however, the connections could be restored if composing the geometric transformation with a gauge transformation χ. To illustrate the concept of gauge-modified symmetry, consider the one-dimensional chain with G = Z₂ and alternating connections a_n+1,n = ± 1 shown in Fig. 1d (top row). Translation by one pillar (middle row) results in an exchange of the horizontal vs. swapped bonds; however, applying a gauge transformation χ_n = − 1 at every second site (red rectangles) results in the restoration of the original connection (bottom row). The presence of gauge-modified symmetries typically implies that $SG$ is represented projectively. Projectively represented $SGs$ in lattices with synthetic Z₂ gauge have been explored over the past five years in one^17,18, two^{14–17,23–26}, and three^17,19,27,28 spatial dimensions, resulting in a non-trivial extension of band theory.

We find that going beyond G = Z₂ introduces the additional option of automorphism-modified symmetries. In this context, automorphism φ ∈ Aut(G) of G is an isomorphism of the group such that φ(g₁ ⋅ g₂) = φ(g₁) ⋅ φ(g₂). A non-trivial example for Q₈ is the cyclic permutation of the imaginary units,

$${\phi }_{cyclic}:+\mathrm{i}\mapsto +\mathrm{j}\mapsto +\mathrm{k}\mapsto +\mathrm{i}.$$ 5

[The complete characterization of Aut(Q₈) and a graphical comparison of gauge transformations to gauge automorphisms appear in Supplementary Note 2 and 3.] The key observation is that a global permutation of orbitals within all pillars with automorphism φ ensures that the transformed CSL maintains a synthetic gauge structure with connection valued in G. Therefore, one can attempt to compose the real-space translations $s\in SG$ of the pillars with such a global permutation within the pillars to restore the original connection. The possibility of automorphism-modified space group symmetries constitutes an additional avenue in band theory, and it naturally arises in the two models introduced in the next pages.

Peter-Weyl decomposition

Owing to the interpretation of the orbitals within each pillar as elements of the gauge group, the dynamics of a CSL decouples into several blocks. To see this, let ψ_n,g be the amplitude of a wave function at site n in orbital g. Gauge transformation by χ_n at site n acts on the amplitudes by the regular representation⁴⁰ρ(χ_n). A standard result known as the Peter-Weyl theorem^30,46 ensures that the regular representation is reducible, with every irreducible representation (‘irrep’) K of the group appearing a number of times equal to its dimension d_K. Specifically, the decomposition

$$U^{†}\rho \left({\chi }_n\right)U={⨁}_K{\left[D^K\left({\chi }_n\right)\right]}^{\oplus d_K}$$ 6

is achieved with the unitary U in Eq. (11) of the Methods section, where D^K(χ) is the matrix representation of χ in irrep K. We show in the Methods section that the connection a_n,ℓ, by virtue of its compatibility with the group structure, couples the same irrep D^K at various pillars. It therefore follows that the unitary U results in block decomposition of the Hamiltonian of a CSL as well as of the Wilson operators in Eq. (4).

We investigate the implications of Eq. (6) for gauge group Q₈. The quaternion group has five irreps: the trivial one-dimensional irrep (A₀; maps all elements to  + 1), three non-trivial one-dimensional irreps (A_x,y,z; each maps half of the elements to  + 1 and the other half to  − 1 = e^iπ), and one two-dimensional irrep (E). Higher-dimensional irreps are unique only up to unitary equivalence; here we specify the matrices D^E(g) as

$$\begin{array}{cc}{D}^E\left(\pm 1\right)=\pm \mathbb{1},\hfill & D^E\left(\pm \mathrm{i}\right)=\mp i{\sigma }_x,\hfill \\ D^E\left(\pm \mathrm{j}\right)=\mp i{\sigma }_y,\hfill & D^E\left(\pm \mathrm{k}\right)=\mp i{\sigma }_z,\hfill \end{array}$$ 7

where σ_x,y,z are the Pauli matrices, and $\mathbb{1}$ is the 2 × 2 identity matrix. The Hamiltonian therefore decouples into six sectors as $U^{†}HU{=⨁}_K{\left[H^K\right]}^{\oplus d_K}$ with physical interpretation as follows. Each of the sectors $H^{A_{0,x,y,z}}$ supports a single (spinless) degree of freedom per pillar. Due to the reduction of the Wilson operators, sectors $H^{A_{x,y,z}}$ perceive an effective U(1) flux quantized to ϕ ∈ {0, π} on every closed loop (thus replicating lattice structures with synthetic Z₂ gauge^{14–19,22–28}), while fluxes are absent in sector $H^{A_0}$.

In addition, the two-dimensional irrep E results in two identical sectors, H^E,1 = H^E,2 ≡ H^E, each supporting a two-level degree of freedom per pillar. We reveal by virtue of concrete models that this sector models spin-$\frac{1}{2}$ electrons with spinful time-reversal symmetry, and elaborate further in the Methods section. The connection a_n,ℓ in these sectors takes the values listed in Eq. (7), reminiscent of Rashba-type antisymmetric spin-orbit coupling for spinful electrons moving in a locally inversion-breaking environment^47–49. In this work, we adopt an alternative interpretation as spin-$\frac{1}{2}$ particle moving in SU(2) gauge field whose flux on any closed loop is quantized to the right-hand sides of Eq. (7).

We thus find that a CSL with gauge group Q₈ naturally encompasses models of spinless particles moving on a lattice with quantized flux Z₂ < U(1) (where ‘ < ’ denotes a subgroup) as well as spinful particles with quantized flux Q₈ < SU(2). By default, the energetics of these sectors overlap. We present in Supplementary Note 4 a possible approach to splitting the energetics of the sectors by decorating each site with an on-site ‘pillar Hamiltonian’. As an experimentally practical alternative, we also discuss within the context of envisioned experimental simulations how a desired sector could be extracted even in the absence of the energy splitting through means of signal interference. In the remainder of our work we therefore focus solely on the sector H^E as it accommodates a faithful representation of the quaternion group, realizing a synthetic gauge structure with a non-Abelian gauge group.

Triangular ladders with quaternion fluxes

To illustrate the concept of CSL with a non-Abelian synthetic gauge structure, we introduce two one-dimensional models with translation symmetry, both based on decorating a triangular ladder as shown in Fig. 2a and b. The unit cell of each model (beige region) contains two sites (white dots), each representing an eight-component pillar as in Fig. 1c. The dashed (solid) bonds indicate hoppings of strength t₁ (t₂), and the connection is indicated with color (legend below Fig. 2b). The Bloch Hamiltonians of the two models are given by Eqs. (19) and (20) of the Methods section.

Fig. 2. Triangular ladders with quaternion fluxes.

a, b Translationally symmetric triangular ladders, both exhibiting flux W_△ = { ± i} on the lower and W_▿ = { ± j} on the upper set of triangles, with unit cells higlighted in beige. Hopping amplitude is t₁ on the dashed (t₂ on the solid) bonds. The coloring of the bonds indicates the connection a_n,ℓ according to the legend below panel b. The two models are distinguished by the flux W_γ on path γ (orange). c,d Energy spectra ε(k) of the model in a (black) and in b (green) with parameters t₁ = 1 and t₂ = 2 for the H^E sector, in which the non-Abelian quaternion fluxes are faithfully represented. Both models have a gapped band structure in the bulk (panel c) while finite chains support a pair of topological zero-energy modes on each boundary (panel d).

Both models realize the same flux arrangement, with consecutive triangles characterized by alternating Wilson loops W_Δ = { ± i} and W_▿ = { ± j}. However, the models are not related by a gauge transformation, which is revealed by comparing the Wilson loop along closed path γ (orange) that encircles two corner-sharing triangles with the same flux W_Δ. In Fig. 2a, the Wilson loop along γ is W_γ = − 1, while W_γ = + 1 in the case of Fig. 2b. This distinction is clearly manifested by the different bulk energy spectra of the ${\mathcal{H}}^E\left(k\right)$ sector, with $\mathcal{H}$ the Bloch Hamiltonian and k the one-dimensional momentum, plotted in Fig. 2c. That two fluxes { ± i} can combine into a net flux  + 1 or  − 1 is enabled by non-commutativity of the quaternion group^{7–12,42–44}, and it illustrates that the connection (rather than the flux arrangement) is necessary to uniquely fix the gauge structure⁴¹.

We next use the triangular ladders to illustrate how the synthetic gauge forces $SG$ symmetries to be modified by gauge and automorphisms. We explicitly consider the model of Fig. 2a (discussion of the model in Fig. 2b is included in the Methods section and in Supplementary Note 6). Adopting first t₁ = t₂, the model exhibits mirror symmetry

$$M_x=\left[\begin{array}{cc}\hfill -i{\sigma }_z\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{0}\hfill & \hfill -i{\sigma }_z\hfill \end{array}\right]\left[\begin{array}{cc}\hfill \mathbb{1}\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{0}\hfill & \hfill e^{ik}\mathbb{1}\hfill \end{array}\right],$$ 8

that acts on the Bloch Hamiltonian as $M_x\left(k\right){\mathcal{H}}^E\left(k\right)M_x^{†}\left(k\right)={\mathcal{H}}^E\left(-k\right)$. The expression in Eq. (8) is interpreted as follows. The right matrix corresponds to reflecting the site locations (with site 1 kept in place and site 2 moved by one unit cell to the left). Looking at Fig. 2a, we find that the reflection reverts the sign of the connections  ± i and  ± j on the horizontal bonds. This sign flip is corrected by virtue of Eq. (3) with gauge transformation χ₁ = χ₂ = + k, implemented with the left matrix in Eq. (8). The utilization of a gauge transformation in Eq. (8) means that $M_x$ is gauge-modified.

On the other hand, the model exhibits a glide reflection, which acts as $G_y\left(k\right){\mathcal{H}}^E\left(k\right)G_y^{†}\left(k\right)={\mathcal{H}}^E\left(k\right)$, represented by

$$G_y=\left[\begin{array}{cc}\hfill \left({\sigma }_y-{\sigma }_x\right)/\sqrt{2}\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{0}\hfill & \hfill \left({\sigma }_y-{\sigma }_x\right)/\sqrt{2}\hfill \end{array}\right]\left[\begin{array}{cc}\hfill \mathbb{0}\hfill & \hfill e^{ik}\mathbb{1}\hfill \\ \hfill \mathbb{1}\hfill & \hfill \mathbb{0}\hfill \end{array}\right].$$ 9

The right matrix in Eq. (9) again encodes the real-space action, with site 2 shifted to the location of site 1 in the next unit cell. However, in contrast to Eq. (8), the left matrix in Eq. (9) does not correspond to a gauge transformation with elements in Q₈. Specifically, observe that the glide reflection replaces bonds with connections i ↔ − j, which cannot be corrected with a gauge transformation; however, the original connection is restored with the automorphism that maps ( + i, + j, + k) ↦ ( − j, − i, − k). Adopting the representation of quaternions in Eq. (7), such automorphism is achieved through conjugation with the term ‘$\left({\sigma }_y-{\sigma }_x\right)/\sqrt{2}$’ that appears on the diagonal of the left matrix in Eq. (8). Therefore, the glide reflection provides an example of an automorphism-modified symmetry.

We next generalize to t₂ ≠ t₁. This breaks the $M_x$ and $G_y$ symmetries, but retains time-reversal, chiral, and π-rotation symmetries represented as $\mathcal{T}={\sigma }_0\otimes i{\sigma }_y\mathcal{K},\mathcal{S}={\sigma }_z\otimes {\sigma }_z$, and $C_{2z}={\sigma }_x\otimes i\left({\sigma }_x+{\sigma }_y\right)/\sqrt{2}$, respectively, where we used $\mathcal{K}$ for complex conjugation. It is noteworthy that the time-reversal operator acquires the familiar spinful form, despite the spinlessness of the orbitals constituting the CSL. We discuss the origin of ${\mathcal{T}}^2=-1$ in the Methods section, with further clarifications included in Supplementary Note 5. Observe further that the π-rotation $C_{2z}$ is automorphism-modified, with the second factor in the tensor product encoding the automorphism ( + i, + j, + k) ↦ ( + j, + i, − k).

We finally reveal that the triangular ladder with quaternion fluxes is topological. Namely, the presence of spinful time-reversal (${\mathcal{T}}^2=-1$) that anticommutes with spinful rotation ($C_{2z}^2=-1$) enables Z₂-valued partial polarization of Kramers-degenerate bands (see refs. ^50,51., and Supplementary Note 7). We find that the partial polarization at half-filling is non-trivial for ∣t₂∣ > ∣t₁∣, implying the appearance of a topological Kramers doublet inside the bulk energy gap at each edge, visible in the open-boundary spectra in Fig. 2d. The Kramers doublets are pinned to ε = 0 by the chiral symmetry.

Honeycomb model with quaternion fluxes

CSLs extend naturally to higher spatial dimensions. To demonstrate this, we construct a model on the honeycomb lattice with a non-Abelian synthetic gauge field as shown in Fig. 3. The notation and the labeling follow that of Fig. 2, and we additionally include the sublattice mass term  ± m. In the H^E sector, with Bloch Hamiltonian given in Eq. (24), the non-Abelian gauge connection generates terms reminiscent of spin-orbit coupling, resulting in a Hamiltonian akin to the Kane-Mele model⁴⁷. We probe the model’s band topology by evaluating the Kane-Mele invariant using the Wilson-loop technique [see Code availability]^52,53. Setting t₁ = 1, we find that the band topology is non-trivial when $\mid t_2\mid >\sqrt{6}\mid m\mid$. In addition, we verify the existence of helical edge states in the topological phase, shown in Fig. 3b. Owing to the presence of non-Abelian gauge structure, the model exhibits automorphism-modified symmetries. For example, the sixfold symmetry $C_{6z}$ around the center of a hexagon (whose representation is included in the Methods section) involves the cyclic automorphism in Eq. (5).

Fig. 3. Honeycomb model with quaternion fluxes.

a Model on the honeycomb lattice with nearest-neighbor (amplitude t₁, dashed lines) and next-nearest-neighbor hoppings (amplitude t₂, solid lines; only a few are explicitly shown). Unit cell is displayed in beige. Orange arrows b_1,2,3 indicate nearest-neighbor displacement vectors from sublattice `1' to sublattice `2'. The model exhibits quaternion-valued connection according to the color legend. b Spectrum ε(k_∥) of a zigzag-terminated ribbon for parameters t₁ = t₂ = m = 1 with helical edge modes inside the bulk energy gap.

Blueprint for electric-circuit realization

Experimental realization of CSLs with arbitrary gauge fields requires systems that can faithfully capture their inner structure. Electric-circuit networks—assemblies of resistors, inductors, and capacitors—provide an ideal platform, even enabling lattice geometries impossible in natural materials³⁶. The circuit Laplacian maps directly to tight-binding Hamiltonians^32,33, transforming standard electrical measurements into direct probes of band structures^34,35, density of states³¹, eigenstate profiles³⁶, and topological boundary modes³³. A particular advantage lies in controlling the boundary condition: direct edge wiring implements periodic boundaries whereas grounding of the edge sites creates open boundaries, with the specific choice potentially controllable within a single platform through a switch³⁸. Alternatively, variable impedance elements based on operational amplifiers³⁷ or analog multipliers³⁹ enable continuous tuning of the boundary condition.

Realization of CSLs in electric circuits is straightforward and proceeds according to the schematic in Fig. 4a. Each orbital within a pillar becomes a node of the network, with nodes in adjacent pillars n and ℓ connected via capacitors C_n,ℓ (proportional to the hopping amplitudes t_n,ℓ) and with connectivity that mimics the connection a_n,ℓ. In addition, each node in pillar n couples to ground through inductor L_n that sets the on-site mass (determining the frequency at which the bands form); choosing uniform L_n ≡ L₀ corresponds to the absence of sublattice potentials.

Fig. 4. Circuit implementation of a Cayley-Schreier lattice with quaternion fluxes.

a Every orbital of each pillar is replaced by a circuit node, and every hopping amplitude t_n,ℓ between pillars n and ℓ is replaced with a capacitor C_n,ℓ while maintaining the connectivity dictated by the connection. Compare also the connection a_ℓ,n = + i here to a_n+1,n = + i in Fig. 1b. In addition, all nodes are coupled to ground with an inductor L₀. b States transforming in the spinful irrep E can be selectively excited with the knowledge of the Peter-Weyl decomposition in Eq. (6) [also Eq. (13) in Methods]. To excite the state $\mid \uparrow ⟩$ (state $\mid \downarrow ⟩$), one needs to inject AC current to the nodes marked with black arrows (with orange arrow) with relative phase shifts indicated to the left.

While bands from the different irrep sectors generically overlap, electric circuits enable selective excitation and probing of individual sectors through engineered current injection patterns. Specifically, the Peter-Weyl decomposition [Eq. (6), with the unitary U for the quaternion group listed in Eq. (13)] specifies how every irrep corresponds to a specific linear combination of the orbitals within the pillars, thus dictating the relative amplitudes and phases of the injection currents necessary to excite states transforming in that irrep. For example, the trivial irrep A₀ (first column of the matrix U) receives equal contributions from all group elements, necessitating in-phase injection currents of the same amplitude at all nodes. Conversely, irrep A_y (third column of U) requires same-amplitude currents but with a relative π phase shift between odd and even layers. To excite the two states in irrep E—corresponding in the spin-$\frac{1}{2}$ language to the states $\mid \uparrow ⟩$ and $\mid \downarrow ⟩$—we read columns 5 and 6 (or equivalently columns 7 and 8) of the U in Eq. (13). For each of the two states, only half of the orbitals in the pillar receive an injection current, with their relative phases shown in Fig. 4b (orange for state $\mid \uparrow ⟩$, black for state $\mid \downarrow ⟩$).

With the relative phase shifts and amplitudes of the injection currents of the desired irrep determined, probing of the spectrum corresponds to electrical measurements at variable drive frequency ω. Specifically, the ground impedance Z(ω) = V(ω)/I(ω) exhibits sharp resonances at eigenmode frequencies, providing direct spectral mapping. In the triangular ladder CSL realization (Fig. 2a), topological edge modes manifest as isolated peaks within the bulk gap. The Kramers degeneracy of the edge mode implies that the impedance is the same for injection currents corresponding to both states $\mid \uparrow ⟩$ and $\mid \downarrow ⟩$. Lastly, to measure the wave function of the edge modes, one simply tunes the drive to the mode’s resonance frequency and measures the spatial profile of the induced voltage.

Discussion

We showed that Cayley-Schreier lattices allow one to construct crystalline structures with space group symmetry while naturally incorporating non-Abelian fluxes. In addition, these models can be decomposed into sectors that correspond to pseudospin-N particles coupled to quantized U(2N  +  1) fluxes, with N set by the dimension of the sector. Some of these pseudospins correspond to true spinors; in particular, the quaternion group Q₈ is the smallest gauge group that enables the simulation of spin-$\frac{1}{2}$ fermions. Each of these (pseudo-)spin models can, in principle, realize any kind of band topology within a single setup. We have demonstrated this by formulating representative models in one and two spatial dimensions that simultaneously exhibit both spinless and spinful excitations. Due to the simple internal structure of CSLs, the construction of these models in fact only requires elementary hopping terms, making them suitable for meta-material realizations as we have also detailed.

The above results set the stage for multifold follow-up directions, both theoretically and experimentally. In this regard, an immediate pursuit is to realize the presented models, as well as generalizations thereof, in electric-circuit setups or other meta-material platforms along the specific details outlined. From a theoretical angle, our work invites a broad range of generalizations. To appreciate the prospects of CSLs, note that the inclusion of Z₂ fluxes in crystalline structures—corresponding to the simplest choice of gauge theory—has led to a substantial body of work in topological band theory^{14–19,22–28}, which includes the discovery of Möbius topological insulators²⁴, Klein Brillouin zones^15,25, shifting of high-symmetry points¹⁶, and the formation of platycosms in the momentum space¹⁹, and whose topological implications continue to be investigated at present. While all these phenomena can be reproduced by CSLs by setting the gauge group to G  =  Z₂, the presented platform has the notable flexibility of implementing arbitrary finite gauge groups G—including those with non-trivial automorphisms φ ∈ Aut(G) and even non-Abelian ones—which we anticipate to vitally broaden the scope of topological band theory. In particular, the presented models are directly generalizable to gauge structures supporting higher-spin particles, which lack simple equivalents in electronic and photonic band structures. Besides topological insulators, we anticipate such high-spin particles to enable exotic types of band degeneracies and semimetals. The presented models also call for a systematic characterization of non-Abelian gauge structures even if narrowing attention to the quaternion group: to enumerate symmetry-compatible synthetic gauge structures for all wallpaper groups and space groups on the one hand, and to classify topological bands in every such gauge arrangement on the other.

Finally, our work also raises additional structural questions. Indeed, the CSL construction invites refined mathematical methods to capture non-Abelian synthetic gauge structures and CSLs, because conventional methods such as group cohomology evaluations^54–56 are specific to Abelian gauge groups. Additionally, such analyses do not include symmetries generated by automorphisms of the gauge group, suggesting that a more general framework is at play. Not in the least place, taking the non-Abelian CSL identified so far and considering quantum and correlated phases as well as disorder in this geometry, in which case all sectors interplay, sets yet another intriguing direction to consider.

Methods

Block-diagonalization of CSL Hamiltonians

The Peter-Weyl decomposition of CSL Hamiltonians is achieved via a unitary rotation in the orbital basis. The original basis, where each basis vector corresponds to a group element g ∈ G, is transformed to a basis labeled by the irreducible representations of the gauge group. Following ref. ⁴⁶, the transformation is defined as

$${\left[{\tilde{\psi }}^{†}\right]}_{\mu }={\tilde{c}}_{\mu }^{†}={\sum }_g{U}_{g,\mu }{c}_g^{†}={\sum }_g{U}_{g,\mu }{\left[{\psi }^{†}\right]}_g,$$ 10

where the ${\left[\tilde{\psi }\right]}_{\mu }={\tilde{c}}_{\mu }$ are components of the annihilation operator vector in the irrep basis and U_g,μ are the components of the unitary matrix of the basis transformation. We introduced μ as a composite index representing the different (K, λ, ν) tuples, where K denotes an irreducible representation of G, λ enumerates the occurrence of K in the decomposition of the regular representation, and ν ∈ 1, …, d_K labels the basis states within the irrep of dimension d_K. In this way, Eq. (10) is compactly written as ${\tilde{\psi }}^{†}={\psi }^{†}U$. Generally, the matrix U is given by ref. ⁴⁶

$$U_{g,\left(K,\lambda ,\nu \right)}=\sqrt{\frac{d_K}{\mid G\mid }}\left[D_{\nu \lambda }^K\left(g\right)\right]*,$$ 11

where $D_{\nu \lambda }^K\left(g\right)$ are the matrix elements of the irrep, and the star denotes complex conjugation.

CSL Hamiltonians are kept invariant under such unitary transformations, since each quadratic term is invariant,

$${\psi }_n^{†}\rho \left(a_{n,\ell }\right){\psi }_{\ell }={\psi }_n^{†}U{U}^{†}\rho \left(a_{n,\ell }\right)U{U}^{†}{\psi }_{\ell }={\tilde{\psi }}_{\ell }^{†}\tilde{\rho }\left(a_{n,\ell }\right){\tilde{\psi }}_{\ell },$$ 12

where ρ(a_n,ℓ) is the regular representation of the connection that encodes hopping amplitudes from site ℓ to site n. Importantly, rotation to the irrep basis $\tilde{\rho }\left({\chi }_{n,\ell }\right)=U^{†}\rho \left({\chi }_{n,\ell }\right)U$ results in a block-diagonal form given by the decomposition in Eq. (6). Since the tight-binding Hamiltonian of CSLs generally takes the form given in Eq. (2), that is, as a sum of regular representations of connections a_n,ℓ over all bonds, the entire CSL Hamiltonian is block-diagonalized by 1 ⊗ U, where the identity acts on the site degree of freedom and U acts on the orbitals within pillars.

For the specific choice G = Q₈, and given the ordering of the group elements as ( + 1, + i, + j, + k, − 1, − i, − j, − k) in accordance with Fig. 1b, the matrix U takes the form

$$U=\frac{1}{\sqrt{8}}\left[\begin{array}{cccccccc}\hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill \sqrt{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{2}\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill \sqrt{2}i\hfill & \hfill \sqrt{2}i\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill \sqrt{2}\hfill & \hfill -\sqrt{2}\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill \sqrt{2}i\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{2}i\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill 1\hfill & \hfill -\sqrt{2}\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -\sqrt{2}\hfill \\ \hfill 1\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill -\sqrt{2}i\hfill & \hfill -\sqrt{2}i\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill -\sqrt{2}\hfill & \hfill \sqrt{2}\hfill & \hfill 0\hfill \\ \hfill 1\hfill & \hfill -1\hfill & \hfill -1\hfill & \hfill 1\hfill & \hfill -\sqrt{2}i\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \sqrt{2}i\hfill \end{array}\right].$$ 13

In Eq. (11), the subscripts g and (K, λ, ν) correspond to the rows resp. columns of the displayed unitary.

Spinless vs. spinful time-reversal symmetry

We clarify how spinful time-reversal symmetry (in the sense that ${\mathcal{T}}^2=-1$) emerges in the Hamiltonian block ${\mathcal{H}}^E$ of a CSL with gauge group G = Q₈. This observation may seem unexpected since the time-reversal operator T at the level of the constituent orbitals (black dots in Fig. 1) is manifestly spinless (meaning T² = + 1). Specifically, it acts on a general wave function through complex conjugation,

$$T\left[{\sum }_{n,g}{\psi }_{n,g}\mid n,g⟩\right]={\sum }_{n,g}{\psi }_{n,g}^{*}\mid n,g⟩,$$ 14

where n labels physical sites and g the orbitals within pillars.

To understand how this time-reversal operator acts at the level of individual irreps, we study how it is changed under the unitary transformation in Eq. (6). We find

$$U^{†}TU=U^{†}{U}^{*}\mathcal{K}=\left[\begin{array}{cccccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill \hfill \end{array}\right]\mathcal{K},$$ 15

which is not block-diagonal for the two-dimensional irrep: the off-diagonal form of the bottom-right 4 × 4 elements implies that the two copies of irrep E are swapped under T.

However, the situation can be rectified. Since both copies of E are associated with the same representations D^E(g) and identical Hamiltonians H^E, we can compose U^†TU with a unitary matrix M that mixes the two copies of E while keeping the Hamiltonian blocks invariant. Specifically, choosing

$$M=\left[\begin{array}{cccccccc}\hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill 0\hfill \end{array}\right],$$ 16

we find that the construction of $\mathcal{T}=M{U}^{†}TU$ results in

$$\mathcal{T}=\left[\begin{array}{cccccccc}\hfill +1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill +1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill +1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill +1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill +1\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill +1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill -1\hfill & \hfill 0\hfill \end{array}\right]\mathcal{K}{=⨁}_K{\left[{\mathcal{T}}^K\right]}^{\oplus d_K}.$$ 17

We find the familiar spinful time-reversal operator ${\mathcal{T}}^E=i{\sigma }_y\mathcal{K}$ for the blocks transforming in the two-dimensional irrep.

For a general choice of gauge group G and irrep K, the character of the emergent time-reversal symmetry ${\mathcal{T}}^K$ of the Hamiltonian H^K is dictated by the Frobenius-Schur indicator⁴⁰

$${\iota }_K=\frac{1}{\mid G\mid }{\sum }_{g\in G}tr\left[D^K\left(g^2\right)\right].$$ 18

This indicator always satisfies ι_K ∈ {0, + 1, − 1}; specifically:

1. ι_K = 0 implies that H^K has no time-reversal symmetry,

2. ι_K = + 1 implies spinless time-reversal symmetry, and

3. ι_K = − 1 implies spinful time-reversal symmetry.

We derive this trichotomy of time-reversal operators in Supplementary Note 5. Generally, odd-dimensional irrep D^K cannot have a spinful ${\mathcal{T}}^K$; on the other hand, even-dimensional irrep can have either spinless or spinful ${\mathcal{T}}^K$. We also remark that the quaternion group Q₈, as considered in this manuscript, constitutes the smallest gauge group that results in a CSL exhibiting spinful Hamiltonian blocks.

Details of the triangular ladder model in Fig. 2a

We write the Bloch Hamiltonian of both triangular ladders in the ‘periodic Bloch convention’, which disregards the embedding of the sites within the unit cell⁵⁷. The Bloch Hamiltonian of the model in Fig. 2a reads

$${\mathcal{H}}^E\left(k\right)=\left[\begin{array}{cc}\hfill 2t_1\sin k{\sigma }_x\hfill & \hfill \left(t_1+t_2{e}^{ik}\right)\mathbb{1}\hfill \\ \hfill \left(t_1+t_2{e}^{-ik}\right)\mathbb{1}\hfill & \hfill -2t_1\sin k{\sigma }_y\hfill \end{array}\right].$$ 19

Below, we supplement the discussion of the model’s symmetries beyond the treatment given in the Results section. In particular, we clarify why setting t₁ = t₂ enforces a gap closing at momentum k = π. This band touching, visible in Supplementary Fig. 1a, signals a topological change of the partial polarization.

Observe that setting t₁ = t₂ leads to the appearance of the glide reflection $G_y$ [Eq. (9)], which squares to translation by one unit cell. It follows that all bands carry eigenvalues λ(k) = ± e^ik/2 of the glide symmetry. For convenience, we color the energy bands in Supplementary Fig. 1a, according to the complex phase of their glide eigenvalue. Crucially, the fraction 1/2 in the eigenvalue exponent implies⁵⁸ that energy bands appear in glide doublets, visible in Supplementary Fig. 1a, that are connected together across the Brillouin zone boundary. The glide eigenvalues of the two bands within a glide doublet differ by the  ± sign; specifically, λ(0) = ± 1 are real while λ(π) = ± i are imaginary.

We finally utilize the presence of spinful time-reversal symmetry, which enforces Kramers pairs at time-reversal-invariant momenta (TRIMs) k  ∈  {0, π}. The glide reflection and the time-reversal operators obey $\left\{G_y,\mathcal{T}\right\}=0$ at both TRIMs, which in combination with the antiunitarity of $\mathcal{T}$ implies that the glide eigenvalues of the two states within a Kramers pair are (λ, − λ^*). Observe that the eigenvalues ( + 1, − 1) of a glide doublet at k = 0 are compatible with the eigenvalues of a Kramers pair. In contrast, the glide eigenvalues ( + i, − i) of a glide doublet at k = π are not compatible with a Kramers pair. It follows⁵⁹ that, for t₁ = t₂, the four bands are tied together through a degeneracy of the central two bands. In the presented model, this gap closing occurs in the form of four-fold degeneracy at momentum k = π, pinned by the chiral symmetry $\mathcal{S}$ to energy ε = 0.

Details of the triangular ladder model in Fig. 2b

We next investigate the model in Fig. 2b, whose Bloch Hamiltonian

$${\mathcal{H}}^E\left(k\right)=\left[\begin{array}{cc}\hfill 2t_1\sin k{\sigma }_z\hfill & \hfill i\left(t_1{\sigma }_z+t_2{e}^{ik}{\sigma }_x\right)\hfill \\ \hfill -i\left(t_1{\sigma }_z+t_2{e}^{-ik}{\sigma }_x\right)\hfill & \hfill 2t_1\cos k\mathbb{1}\hfill \end{array}\right]$$ 20

produces the green bands in Fig. 2c. While flux through the triangles implies that this nearest-neighbor model has chiral symmetry, representation of this symmetry requires unit-cell doubling. We thus consider a four-site unit cell consisting of sites 1 and 2 in Fig. 2b together with their translations 1’ and 2’ to the right. The folded Bloch Hamiltonian reads

$${\mathcal{H}}_{2\times }^E\left(k\right)=t_1\left[\begin{array}{cccc}\hfill \mathbb{0}\hfill & \hfill i{\sigma }_z\hfill & \hfill i\left(1-e^{ik}\right){\sigma }_z\hfill & \hfill i\tau e^{ik}{\sigma }_x\hfill \\ \hfill -i{\sigma }_z\hfill & \hfill \mathbb{0}\hfill & \hfill -i\tau {\sigma }_x\hfill & \hfill \left(1+e^{ik}\right)\mathbb{1}\hfill \\ \hfill i\left(e^{-ik}-1\right){\sigma }_z\hfill & \hfill i\tau {\sigma }_x\hfill & \hfill \mathbb{0}\hfill & \hfill i{\sigma }_z\hfill \\ \hfill -i\tau e^{-ik}{\sigma }_x\hfill & \hfill \left(1+e^{-ik}\right)\mathbb{1}\hfill & \hfill -i{\sigma }_z\hfill & \hfill \mathbb{0}\hfill \end{array}\right]$$ 21

where τ = t₂/t₁. Chiral symmetry is represented as $\mathcal{S}={\sigma }_z\otimes {\sigma }_z\otimes {\sigma }_z$, and the model exhibits spinful time-reversal symmetry $\mathcal{T}={\sigma }_0\otimes {\sigma }_0\otimes i{\sigma }_y\mathcal{K}$. We show the band structure of the folded Bloch Hamiltonian for t₁ = 1 and t₂ = 2 in Supplementary Fig. 1b.

Note that the triangular ladder model in Fig. 2b is not symmetric under the operation of a glide plane $G_y$ even if setting t₁ = t₂; this is because two consecutive fluxes { ± i} combine to W_γ = + 1 while two consecutive fluxes { ± j} combine to $W_{G_y\gamma }=-1$. For the same reason, the $C_{2z}$ symmetry is also absent. By contrast, setting t₁ = t₂ enables a gauge-modified mirror symmetry

$$M_x=\left[\begin{array}{cccc}\hfill -i{\sigma }_x\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{0}\hfill & \hfill -i{\sigma }_z\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill -i{\sigma }_x\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill -i{\sigma }_z\hfill \end{array}\right]\left[\begin{array}{cccc}\hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{1}\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{0}\hfill & \hfill \mathbb{1}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{1}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill e^{ik}\mathbb{1}\hfill \end{array}\right],$$ 22

which involves gauge transformation ${\chi }_1={\chi }_{1^{\prime }}=+i$ and ${\chi }_2={\chi }_{2^{\prime }}=+k$ on the four sites within the doubled unit cell.

On the other hand, translation by a single (not doubled) unit cell has to be treated as an additional symmetry of the folded Bloch Hamiltonian¹⁴. This translation, which is a symmetry of the model irrespective of the choice of t₁ and t₂, is represented as

$$L_x=\left[\begin{array}{cccc}\hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill e^{ik}\mathbb{1}\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill e^{ik}\mathbb{1}\hfill \\ \hfill \mathbb{1}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill \\ \hfill \mathbb{0}\hfill & \hfill \mathbb{1}\hfill & \hfill \mathbb{0}\hfill & \hfill \mathbb{0}\hfill \end{array}\right].$$ 23

and it acts on the Hamiltonian as $L_x\left(k\right){\mathcal{H}}_{2\times }^E\left(k\right)L_x^{†}\left(k\right)={\mathcal{H}}_{2\times }^E\left(k\right)$ while squaring to $e^{ik}\mathbb{1}$. We observe by comparing against Eq. (9) that the translation symmetry of ladder b acts on the Hamiltonian (21) in an analogous fashion as the glide symmetry of ladder a acts on the Hamiltonian (19). In particular, energy bands appear in translation doublets which are connected together across the Brillouin zone boundary and that carry translation eigenvalues (e^ik/2, − e^ik/2) ⁵⁸.

We include an extended analysis of the symmetry of the triangular ladder model of Supplementary Fig. 1b in Supplementary Note 6. Therein, we show how the combined effect of symmetries $L_x$ and $\mathcal{T}$ for t₁ ≠ t₂ enforces the butterfly-like spectrum shown in Supplementary Fig. 1b, and how the combined effect of the set $\left\{L_x,\mathcal{T},{\mathcal{M}}_x,\mathcal{S}\right\}$ for t₁ = t₂ enforces a four-fold degeneracy at momentum k = π as shown in Supplementary Fig. 1c.

Details of the honeycomb model

We write the Bloch Hamiltonian of the quaternion-flux honeycomb model in the ‘non-periodic Bloch convention’ which includes the embedding of the sites within the unit cell⁵⁷. We set the nearest-neighbor distance to 1, so that the three displacement vectors from sublattice ‘1’ to sublattice ‘2’ (shown in Fig. 3) are ${\mathbf{b}}_1=\left(\sqrt{3},-1\right)/2,{\mathbf{b}}_2=\left(0,1\right)$, and ${\mathbf{b}}_3=\left(-\sqrt{3},-1\right)/2$, and we introduce their differences (Bravais vectors) a₁ = b₁ − b₂, a₂ = b₂ − b₃, and a₃ = b₃ − b₁. The Hamiltonian can then be expressed as

$$\begin{array}{ccc}\hfill {\mathcal{H}}^E\left(\mathbf{k}\right)& \hfill =\hfill & {\sigma }_z\otimes \left[m{\sigma }_0-2t_2{\sum }_{l=1}^3\sin \left(\mathbf{k}\cdot {\mathbf{a}}_l\right){\sigma }_{\ell }\right]\hfill \\ \hfill & \hfill \hfill & +t_1\left[{\sigma }_1{\sum }_{l=1}^3\cos \left(\mathbf{k}\cdot {\mathbf{b}}_l\right)+{\sigma }_2{\sum }_{l=1}^3\sin \left(\mathbf{k}\cdot {\mathbf{b}}_l\right)\right]\otimes {\sigma }_0\hfill \end{array}$$ 24

where t₁ (t₂) is the nearest-neighbor (next-nearest-neighbor) hopping amplitude and m is the sublattice mass.

The constructed model with non-Abelian fluxes is invariant under the space group of the honeycomb lattice with gauge-modified and automorphism-modified symmetries. For example, rotation by 2π/6 around the center of a hexagon (present when the sublattice mass is set to m = 0) permutes the connection on next-nearest-neighbor bonds as  + i ↦ + k ↦ + j ↦ i. Correcting for this change of the connection demands that the rotation is composed with the inverse automorphism  + i ↦ + j ↦ + k ↦ + i, represented with the matrix $V=\left(\mathbb{1}-i{\sum }_{l=1}^3{\sigma }_l\right)/2$ in the complement SU(2)\Q₈. Therefore, the rotation is automorphism-modified as $C_{6z}={\sigma }_x\otimes V$ with the first factor coming from the exchange of the two sublattices under the symmetry and the second factor encoding the non-trivial automorphism.

Author contributions

L.K.U. and T.B. conceived the project. Z.G. and T.B. analyzed the presented tight-binding models. R.-J.S. advised on the gauge theory aspects. L.K.U. and T.B. devised the electric-circuit blueprint. All authors contributed to developing the theoretical framework, discussed together, and wrote the manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. A peer review file is available.

Data availability

All the data used to arrive at the conclusions presented in this work are publicly available in the following data repository: 10.5281/zenodo.18954513.

Code availability

All the Python code used to generate and analyze the data and to arrive at the conclusions presented in this work is publicly available in the form of annotated notebooks in the following data repository: 10.5281/zenodo.18954513.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-71401-3.