Walking on Archimedean Lattices: Insights from Bloch Band Theory

TL;DR

This work establishes a direct link between counting returning walks on Archimedean lattices and Bloch-band theory in tight-binding models. By introducing the Bloch adjacency matrix $A(\boldsymbol{k})$ and its spectrum, the authors derive $S_n$ from Brillouin-zone moments $\int_{\boldsymbol{k}}\varepsilon_\alpha(\boldsymbol{k})^n$, yielding closed forms or generating functions for all eleven two-dimensional Archimedean lattices. They also connect these results to the density of states and provide explicit asymptotics for large $n$, with analytical DOS expressions for several lattices and numerical validation via large flakes/clusters. The methods generalize to other periodic Euclidean lattices and dimensions, offering a powerful bridge between combinatorics, graph theory, and solid-state physics. Overall, the paper delivers a comprehensive, analytically tractable framework to count returning walks, compute DOS, and study asymptotics across a rich family of lattices.

Abstract

Returning walks on a lattice are sequences of moves that start at a given lattice site and return to the same site after $n$ steps. Determining the total number of returning walks of a given length $n$ is a typical graph-theoretical problem with connections to lattice models in statistical and condensed matter physics. We derive analytical expressions for the returning walk numbers on the eleven two-dimensional Archimedean lattices by developing a connection to the theory of Bloch energy bands. We benchmark our results through an alternative method that relies on computing the moments of adjacency matrices of large graphs, whose construction we explain explicitly. As condensed matter physics applications, we use our formulas to compute the density of states of tight-binding models on the Archimedean lattices and analytically determine the asymptotics of the return probability. While the Archimedean lattices provide a sufficiently rich structure and are chosen here for concreteness, our techniques can be generalized straightforwardly to other two- or higher-dimensional Euclidean lattices.

Figures (5)

• Figure 1: We show the eleven two-dimensional Archimedean lattices. Some frequently studied members of the family include the Square, Triangular, Honeycomb, and Kagome lattices, while the names of the other lattices follow common conventions in the literature, often named after chemical compounds related to the lattice FarnellBetts1995. Note that the Ruby lattice PhysRevB.84.155116Verresen2022PhysRevE.109.045305 is also referred to as Bounce lattice Farnell. The numbers in brackets indicate the Grünbaum--Shephard notation: For this, note that every vertex or site is surrounded by the same sequence of regular polygons such that, for instance, for the Square and Triangular lattice, at each vertex $4$ squares ($4^4$) or $6$ triangles ($3^6$) meet, respectively. Also note that every site has the same coordination number, defined as the number of nearest neighbors of any vertex.
• Figure 2: The density of states for the Square, Triangular, Honeycomb, Trellis, and Kagome lattices is shown alongside their respective unit cells (in golden, filled). $E$ denotes the energy in units of $t$. The lattice translation vectors $\boldsymbol{e_1}$ and $\boldsymbol{e_2}$ are illustrated in turquoise. FB represents a flat band in the Brillouin Zone.
• Figure 3: Same setting as in Fig. \ref{['FigPanel1']} but for lattices with $N_{\rm{u}}\geq 4$. The density of states has new features like partial flat bands (PFs). A PF is a band that is non-dispersive along a high symmetry direction in the Brillouin zone. Note the striking resemblance of features from the Honeycomb or Triangular lattice DOSs in small neighborhoods of $E$ for these lattices.
• Figure 4: Left. CaVO lattice shown together with its unit cell $A_{\rm u}$. Note the numbering scheme within the unit cell, which is applied consistently among the repeated unit cells. Right. The quotient graph underlying the CaVO lattice, consisting of black dots and purple lines, is of Type S. Each quotient vertex (black dot) corresponds to a unit cell $A_{\rm u}$ and each squiggly line (repeated for the entire quotient graph) corresponds to connections $\Gamma_{(i,j)}$ between sites from different unit cells.
• Figure 5: Same as in Fig. \ref{['cavotess']}, but for the Star lattice, where the unit cell now consists of $N_{\rm{u}}=6$ sites. The underlying quotient graph, indicated by the black dots and purple lines, is of Type T. This can be seen from the fact that there exists a diagonal link between the unit cells.