Topological non-Abelian Gauge Structures in Cayley-Schreier Lattices

Abstract

Recently, novel 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. We show 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 the symmetry sectors can, moreover, be interpreted as blocks of spin models that can effectively realize a wealth of different topological invariants in a single setup. We underpin these general results with concrete models and show how they can be implemented in current experimental platforms. Our work sets the stage for a systematic investigation of topological insulators and metals with non-Abelian gauge structures.

Figures (5)

• Figure 1: Synthetic gauge fields in Cayley-Schreier lattices.a Illustration of a CSL over two-dimensional lattice with $G=Z_2$. Two choices of connection are possible on each bond, represented by the horizontal vs. swapped dashed couplings. b Schematics of a CSL over a one-dimensional chain with gauge group $G=Q_8$. There are $\abs{Q_8}=8$ choices of connection possible on each bond, of which four (namely $a_{j+1,j}\in\{+1,+\textrm{i},+\textrm{j},-\textrm{j}\}$) are illustrated. The components highlighted with orange indicate that the orbital $g_n=+\textrm{j}$ at site $n$ is coupled by connection $a_{n+1,n} = +\textrm{i}$ to the orbital $g_{n+1} = a_{n+1,n}\cdot g_n = (+\textrm{i}) \cdot (+\textrm{j}) = +\textrm{k}$. c Alternating arrangement of trivial and non-trivial connections in a one-dimensional CSL with gauge group $Z_2$ (top). Translation by one pillar flips the connection pattern (middle). Applying gauge transformation $\chi_i = -1$ to every second site (red rectangles) restores the original connection (bottom). The colored orbitals and hoppings serve as a guide to follow the application of the translation and of the subsequent gauge transformation.
• Figure 2: Triangular ladders with quaternion fluxes.a,b Translationally symmetric triangular ladders, both exhibiting flux $W_\triangle=\{\pm\textrm{i}\}$ on the lower and $W_\triangledown = \{\pm\textrm{j}\}$ on the upper set of triangles, with unit cells higlighted in green. Hopping amplitude is $t_1$ on the dashed ($t_2$ on the solid) bonds. The coloring of the bonds indicates the connection $a_{ij}$ according to the legend below panel $\textsf{\smaller b}$. The two models are distinguished by the flux $W_\gamma$ on path $\gamma$ (orange). c,d Energy spectra $\varepsilon(k)$ of the model in $\textsf{\smaller a}$ (black) and in b (green) with parameters $t_1 = 1$ and $t_2 = 2$ 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).
• Figure 3: Honeycomb model with quarternion fluxes.a Model on the honeycomb lattice with nearest-neighbor (amplitude $t_1$, dashed lines) and next-nearest-neighbor hoppings (amplitude $t_2$, solid lines; only a few are explicitly shown). Unit cell is displayed in green. Orange arrows ${\boldsymbol{b}}_{1,2,3}$ indicate nearest-neighbor displacement vectors from sublattice '$1$' to sublattice '$2$'. The model exhibits quaternion-valued connection according to the color legend below Fig. \ref{['fig:nA-SSH']} b. b Spectrum $\varepsilon(k_\parallel)$ of a zigzag-terminated ribbon for parameters $t_1 = t_2 = m =1$ with helical edge modes inside the bulk energy gap.
• Figure 4: Circuit implementation of a CSL with quaternion fluxes.a Every orbital of each pillar is replaced by a circuit node, and every hopping amplitude $t_{nm}$ between pillars $n$ and $m$ is replaced with a capacitor $C_{nm}$ while maintaining the connectivity dictated by the connection. Compare also the connection $a_{m,n}=+\textrm{i}$ here to $a_{n+1,n}=+\textrm{i}$ in Fig. \ref{['fig:CSL+GT']} b. In addition, all nodes are coupled to ground with an inductor $L_{0}$. b States transforming in the spinful irrep $E$ can be selectively excited with the knowledge of the Peter-Weyl decomposition in Eq. (\ref{['eqn:Peter-Weyl-decomp']}) [also Eq. (\ref{['eqn:rotate-to-irrep-basis']}) in Methods]. To excite the state $\ket{\uparrow}$ (state $\ket{\downarrow}$), one needs to inject AC current to the nodes marked with black arrows (with orange arrow) with relative phase shifts indicated to the left.
• Figure 5: Detailed band structures of the triangular ladder models.a Bulk energy bands of the triangular ladder model in Fig. \ref{['fig:nA-SSH']} a for $t_1 = t_2 = 1$, colored according to the phase of the eigenvalue of the glide operator $\mathsf{G}_y$ in Eq. (\ref{['eqn:ladder-a-Gy']}). b,c Bulk energy bands of the triangular ladder model in Fig. \ref{['fig:nA-SSH']} b after Brillouin zone folding per Eq. (\ref{['eqn:ladder-b-Ham-doubled']}), colored according to the phase of the eigenvalue of the translation operator $\mathsf{T}_x$ in Eq. (\ref{['eqn:ladder-b-Tx']}), corresponding to parameter values $t_1 = 1,t_2 = 2$ (panel $\textsf{\smaller b}$) and to $t_1 = t_2 =1$ (panel $\textsf{\smaller c}$).