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 lattices (or Cayley crystals) 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.

This is exciting because gauge fields are at the heart of modern physics — from electromagnetism to the Standard Model. Cayley lattices offer a way to engineer non-Abelian gauge structures in simple, flat-space lattice models with purely real hopping amplitudes, without the need for magnetic fields or synthetic spin–orbit coupling. My current research focuses on determining the topological phases that emerge in families of Z_n Cayley lattices, and on characterizing their physical observables — such as Bloch oscillation dynamics and impedance signatures — that can serve as experimental 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 regular pentagons. In flat (Euclidean) space this is impossible — 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 number of sites grows exponentially with radius rather than polynomially, Bloch’s theorem must be generalized, and familiar results like the Hofstadter butterfly acquire a curvature-dependent structure.

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), and we computed the Hofstadter butterfly on hyperbolic tilings using periodic boundary conditions to access the true bulk spectrum (Phys. Rev. Lett. 128, 166402, 2022).

One insight from this program is a decomposition strategy that separates curvature from non-commutative geometry: hyperbolic lattices can be thought of as “curved Euclidean lattices” (addressable via strain engineering in graphene) combined with simpler non-Abelian Z₂ lattices. This factorization makes each ingredient independently experimentally accessible.

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. 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 dramatic impedance peaks localized on the circuit’s edges.

I have contributed to a comprehensive theory of topolectric circuit construction, showing how arbitrary Hermitian tight-binding models can be mapped onto LC networks by supplementing each node with subnodes whose phase factors encode hopping amplitudes. Using this framework, we have realized SSH chains, Chern insulators, quantum spin Hall insulators, Weyl semimetals, and higher-order topological insulators on a circuit board.

More recently, we demonstrated topological temporal pumping in electrical circuits (Phys. Rev. Res. 6, 023010, 2024), and I am now interested in extending these platforms to non-linear regimes (incorporating diodes and transistors) and to the realization of Cayley-lattice models on circuit boards.

Floquet Topological Phases in Photonics

When a system is driven periodically in time, its effective band structure is described by Floquet theory — the temporal analog of Bloch’s theorem for spatial periodicity. Periodic driving can create topological phases with no static counterpart: a system with trivial static bands can develop robust chiral edge states simply by shaking it at the right frequency.

During my PhD, I studied these phenomena in photonic quantum walks — fiber-loop setups where photons undergo discrete steps of propagation and coupling. We engineered a multi-topological Floquet metal that simultaneously carries winding bulk bands and chiral edge states (Phys. Rev. Lett. 130, 056901, 2023), and we demonstrated the topological swing of Bloch oscillations — a phenomenon in which a wavepacket subjected to a linear potential traces out a topological winding number (Phys. Rev. Lett. 125, 186804, 2020).

I am also interested in topological chiral interface states beyond insulators — robust spectral flows that persist even when bulk bands overlap (Phys. Rev. A 102, 023520, 2020).

Broader Interests

Beyond the threads above, I maintain active interests in non-Hermitian physics (skin effects, exceptional points), interacting topological systems (Monte Carlo approaches), the topology–quantum-optics interface (ultrastrong coupling), and diabatic topological pumps. I enjoy developing visualization tools for communicating lattice physics using Mathematica, Blender geometry nodes, and TikZ.