Walking on Archimedean Lattices: Insights from Bloch Band Theory

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.

32 pages

References in corpus (29)

• Electric Field Effect in Atomically Thin Carbon Films
• Superconductivity in the ${\mathbb{Z}}_2$ kagome metal KV${}_3$Sb${}_5$
• Simulating hyperbolic space on a circuit board
• Crystallography of Hyperbolic Lattices
• From Anderson localization on Random Regular Graphs to Many-Body localization
• Observation of novel topological states in hyperbolic lattices
• Automorphic Bloch theorems for hyperbolic lattices
• Universality of Hofstadter butterflies on hyperbolic lattices
• Holography on tessellations of hyperbolic space
• Unifying Kitaev magnets, kagome dimer models and ruby Rydberg spin liquids
• Non-Abelian hyperbolic band theory from supercells
• Quantum phase transitions of interacting bosons on hyperbolic lattices
• Density of states of tight-binding models in the hyperbolic plane
• Anderson localization transition in disordered hyperbolic lattices
• Simulating Holographic Conformal Field Theories on Hyperbolic Lattices
• Anomalous statistics of laser-cooled atoms in dissipative optical lattices
• Discrete path integral approach to the trace formula for regular graphs
• Quantum spin liquid ground states of the Heisenberg-Kitaev model on the triangular lattice
• Corner transfer matrix renormalization group approach in the zoo of Archimedean lattices
• Dynamic mass generation on two-dimensional electronic hyperbolic lattices
• Candidate quantum spin liquids on the maple-leaf lattice
• Hyperbolic lattices and two-dimensional Yang-Mills theory
• Universal spectral moment theorem and its applications in non-Hermitian systems
• Algebraic area enumeration for open lattice walks
• Algebraic area enumeration of random walks on the honeycomb lattice
• Derivation of the free energy, entropy and specific heat for planar Ising models: Application to Archimedean lattices and their duals
• On the algebraic area of cubic lattice walks
• Magnetism of kagome metals $\left({\text{Fe}}_{1-x}{\text{Co}}_x\right)\text{Sn}$ studied by $\mu$SR
• Lattice random walks and quantum A-period conjecture

Cited by in corpus (1)

• Exact critical-temperature bounds for two-dimensional Ising models