Cayley Lattices & Non-Abelian Gauge Structures

In an ordinary crystal, translating the lattice one step to the right and then one step up gives the same result as translating up first and then right — translations commute. But what if they didn’t? Cayley-Schreier lattices (or CS lattices) are a recently introduced class of periodic structures whose translation operations are drawn from non-Abelian (non-commutative) groups. The lattice is literally the Cayley graph of a chosen group, and the non-commutativity of the generators acts as a built-in, discrete gauge field.The pedagogical picture: a particle hopping around a closed loop on such a lattice does not return to its original internal state — it picks up a “memory” of the path it took, exactly as a charged particle encircling a magnetic flux picks up an Aharonov-Bohm phase. The distinction arises when the chosen group is non-Abelian: the “flux” can then be a matrix rather than a number, so two different loops need not even commute with each other!

This idea bridges the gap between hyperbolic lattices (where translations do not commute) and Euclidean lattices (where they do). Experimentally, hyperbolic lattices are far from trivial to engineer at any desired scale — the exponential growth of sites eats up physical space fast. CS lattices overcome this: they inherit the non-commutative translation structure while living comfortably in flat space. I developed this program in close collaboration with my longtime collaborator Marco Marciani (for Abelian groups); the extension to non-Abelian groups was carried out with T. Bzdušek.

Why is this exciting? Gauge fields are at the heart of modern physics — from electromagnetism to the Standard Model. CS lattices offer a way to engineer non-Abelian gauge structures in simple, flat-space lattice models with purely real hopping amplitudes — no external magnetic fields, no synthetic spin–orbit coupling. My current research focuses on the physical (like transport) and topological properties that emerge in families of Z_n CS lattices, and on their experimentally measurable fingerprints — Bloch oscillation dynamics and impedance signatures — that can serve as direct probes of the underlying non-Abelian structure.

Key publication: Z. Guba, R-J. Slager, L. K. Upreti, T. Bzdušek, “Topological non-Abelian Gauge Structures in Cayley-Schreier Lattices,” Nat. Commun. (2026).

Hyperbolic Lattices

Imagine tiling a floor with identical regular pentagons. In flat (Euclidean) space this is impossible — the angles never fit — but on a surface of constant negative curvature (the hyperbolic plane) it works perfectly. A hyperbolic lattice is such a tiling, and it serves as the discrete analog of a crystal in negatively curved space. The negative curvature radically changes the physics: the circumference of a circle now grows exponentially with its radius (as sinh r, not r), so the number of sites explodes exponentially with distance — a hyperbolic lattice is, in a sense, all boundary! One striking consequence is spectral: modes with higher angular momentum can appear at lower energies than their radially excited counterparts — the ordering is reversed relative to flat space. The pedagogical picture: on a hyperbolic drum, the rim at radius r is exponentially longer than on a flat drum (2π sinh r rather than 2πr). This means an oscillation with particular (ℓ) angular nodes is stretched over an exponentially larger distance, so its gradient — and hence its kinetic energy cost — is exponentially suppressed (the centrifugal barrier ℓ²/sinh²r replaces the flat-space ℓ²/r²). Angular excitations thus become cheap compared to radial ones, and the high-angular-momentum modes appear earlier (i.e. lower in energy) in the spectrum — a hyperbolic drum literally sounds different from a flat one! Even Bloch’s theorem, the workhorse of solid-state physics, must be generalized: the translation group of a hyperbolic lattice is non-Abelian (a Fuchsian group), so alongside the usual U(1) Bloch phases the geometry also demands Bloch states where the Bloch factor is a U(N) matrix rather than a single phase.

Together with P. M. Lenggenhager, A. Stegmaier, R. Thomale, I. Boettcher, T. Bzdušek, and others, I have contributed to both the theory and the experimental realization of hyperbolic lattices in topolectric circuits. We demonstrated that the spectral ordering of Laplacian eigenstates on hyperbolic vs. flat lattices has a universally different structure (Nat. Commun. 13, 4373, 2022).

A second thread grew out of my original question: how is the cyclotron motion of an electron in a magnetic field influenced by negative curvature? In flat space, the answer is immediate: the moment the magnetic field is non-zero, the Lorentz force bends the electron into closed cyclotron orbits, and the spectrum collapses into discrete Landau levels. In hyperbolic space this is no longer guaranteed! The negative curvature makes neighbouring trajectories spread apart exponentially fast, and the magnetic field now has to compete against this spreading. Only above a critical magnetic field (set by the curvature radius) does the Lorentz force win and the electron undergo cyclotronic localization (or quantum mechanically Landau quantization) — below it, the orbits stay open and the electron simply escapes. On the lattice, a second competition enters: the lattice length scale interferes with the magnetic length scale (cyclotron radius, semi-classically), and this interference famously produces a fractal energy spectrum, the Hofstadter butterfly, as the field is tuned. On a hyperbolic lattice both competitions act together, and the butterfly acquires a curvature-dependent structure. We computed this hyperbolic Hofstadter butterfly using periodic boundary conditions to access the true bulk spectrum (Phys. Rev. Lett. 128, 166402, 2022).

One insight from this program is a decoupling of geometry from algebra. In all current platforms, the translation group is either Abelian (flat, Euclidean space) or non-Abelian but tied to the curvature of the underlying manifold (hyperbolic lattices). But the non-commutativity itself is an algebraic property — why should it be held hostage by curvature? By grafting the Cayley graph of a non-Abelian group directly onto a regular array of nodes, one obtains planar, curvature-free lattices whose translations do not commute. This decoupling eliminates the exponential scaling of connections that limits hyperbolic lattice implementations, making non-Abelian physics experimentally accessible at arbitrary scales — and it is precisely what motivated us (Marco and me) the Cayley-Schreier program above.

Topolectric Circuits

A topolectric circuit is an electrical network of inductors and capacitors whose admittance spectrum (the eigenvalues of its circuit Laplacian) mirrors the band structure of a quantum tight-binding model. The key insight: Kirchhoff’s laws for LC circuits mathematically mirror the Schrödinger equation. This means the circuit dynamics exactly reproduce tight-binding evolution — inductors implement the hopping terms, while capacitors provide the essential time scale. Just as electrons in a crystal feel the lattice potential, voltage signals in the circuit feel the arrangement of components, and topological boundary modes manifest as impedance peaks localized on the circuit’s edges. The platform is precise, low-cost, and operates at room temperature!

My main contribution to this platform is the realization of a topological temporal pump in electrical circuits (Phys. Rev. Res. 6, 023010, 2024) — a project I supervised and actively contributed to, in experimental collaboration with Prof. Tobias Kiessling. A topological pump transports charge from one end of a lattice to the other by adiabatically varying the underlying potential, with the transport topologically protected. Previous implementations suffered from unwanted dissipation and non-adiabatic effects that disrupted transport over extended pumping periods. We overcame this by incorporating active circuit elements that function as time-variable, voltage-controlled inductors — and the resulting 80-site circuit (significantly larger than previously explored) validated dissipationless transport over extended distances. This marked the first demonstration of topological Floquet effects in electrical circuits.

I have also contributed to the circuit realizations of hyperbolic matter described above (Nat. Commun. 13, 4373, 2022; Nat. Commun. 14, 622, 2023).

Floquet Topological Phases in Photonics

When a system is driven periodically in time, its band structure is governed by Floquet theory — the temporal analog of Bloch’s theorem for spatial periodicity. What makes driven systems special is that topology can emerge explicitly due to the time dependence, even in the absence of a spectral gap! The pedagogical picture: think of the particle as moving on a temporal lattice — it acquires a phase when hopping from one unit cell to the next, but now in time, in complete analogy with the usual Bloch picture in space. This lets us construct the whole band-theory description in the time domain. But most importantly, time is the generator of energy itself — so the quasienergy axis becomes periodic, giving rise to phases with no analogous picture in space anymore. The most striking case: the usual topological invariants vanish, and yet the system remains in a topological regime! This was the central theme of my PhD.

My key contribution was identifying two such new regimes. The first is analogous to a semi-metallic phase: it hosts topological degeneracies that can be selectively manipulated, allowing topologically protected degeneracies and chiral edge states to coexist at the same quasienergy (Phys. Rev. A 102, 023520, 2020). These mutually exclusive topologies were previously thought impossible — the missing ingredient was a mechanism for selective gap opening, which I identified and demonstrated. The second regime involves a winding of the quasienergy spectrum with no static analog (arXiv:1907.09914; Phys. Rev. Lett. 125, 186804, 2020). The winding manifests directly in real space: wave packets undergo Bloch oscillations whose stationary points reveal the winding number of the bands. Notably, Bloch oscillations are meaningless in disordered systems — yet we demonstrated that here they remain topologically protected and are captured by a topological invariant. This theoretical framework was later experimentally verified by my collaborator Alberto Amo in Lille, in a photonic lattice built from coupled fiber loops (Phys. Rev. Lett. 130, 056901, 2023).

The methods behind these results: a 1+1 dimensional framework where time serves as an additional dimension, with phase introduced as a further synthetic dimension, all encoded in an oriented scattering matrix (time-ordered unitary matrices). The winding regime stands out by offering robustness derived from bulk energy winding rather than the conventional Chern number — opening new avenues for robust optical transmission in communication and quantum optics.

Most recently, I asked the reverse question: the topological classification of a system depends on the discrete symmetries of its Hamiltonian (chiral, particle-hole, time-reversal) — but can these abstract symmetries be seen directly from the window of crystalline symmetries? In Floquet waveguide arrays, the answer is yes: I showed that structural properties of the lattice alone — bipartiteness and z-reflection — determine the entire symmetry class (Phys. Rev. B selected, 2026). This beautiful connection between discrete abstract symmetries and crystalline ones allows us to engineer and detect topologically protected boundary states merely by geometry — with no knowledge of the Hamiltonian, the traditional way! It also led me to a new, overlooked symmetry: shifted particle-hole symmetry, which protects boundary states at quasienergy π even in non-bipartite networks lacking every conventional protecting symmetry. This result provides far more flexibility in understanding how non-trivial states emerge.