Finite-state automata for exact matrix product operators of tight-binding Hamiltonians: fractals, quasicrystals, trees and hyperbolic lattices
Abstract
Inspired by the recent progress in the simulation of tight-binding Hamiltonians on large lattices using tensor networks, we introduce a systematic matrix product operator (MPO) construction for single-particle Hamiltonians on recursively structured lattices. Taking advantage of this recursive structure, we encode the lattice geometry in a finite-state automaton and, adapting ideas from Abelian-symmetric tensor networks, obtain an exact and analytical MPO representation of the Hamiltonian, where the number of tensors grows logarithmically with the system size and the bond dimension is set by the number of automaton states. Several examples demonstrate the generality of the framework: regular lattices, fractals, Cayley trees, hyperbolic lattices, and one- and two-dimensional Fibonacci quasicrystals. Combined with the kernel polynomial method, these MPO representations enable large-scale calculations of spectral properties without explicitly constructing the full Hamiltonian. This provides a unified route to the exploration of electronic properties across a broad class of lattices at exponentially large system sizes.
Related
- One-dimensional quasicrystals with tensor-network finite-state automata
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BibTeX
@misc{brzezinska2026finitestate,
title = {{Finite-state automata for exact matrix product operators of tight-binding Hamiltonians: fractals, quasicrystals, trees and hyperbolic lattices}},
author = {Marta Brzezińska and Jeanne Colbois and Loïc Herviou},
howpublished = {HAL},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25276},
eprint = {2609.25276},
archivePrefix = {arXiv},
url = {https://hal.science/hal-05774818v1},
note = {CC BY-NC 4.0},
}
RIS
TY - UNPB
TI - Finite-state automata for exact matrix product operators of tight-binding Hamiltonians: fractals, quasicrystals, trees and hyperbolic lattices
AU - Marta Brzezińska
AU - Jeanne Colbois
AU - Loïc Herviou
PY - 2026
DA - 2026/09/21
PB - HAL
DO - 10.48550/arxiv.2609.25276
UR - https://hal.science/hal-05774818v1
AB - Inspired by the recent progress in the simulation of tight-binding Hamiltonians on large lattices using tensor networks, we introduce a systematic matrix product operator (MPO) construction for single-particle Hamiltonians on recursively structured lattices. Taking advantage of this recursive structure, we encode the lattice geometry in a finite-state automaton and, adapting ideas from Abelian-symmetric tensor networks, obtain an exact and analytical MPO representation of the Hamiltonian, where the number of tensors grows logarithmically with the system size and the bond dimension is set by the number of automaton states. Several examples demonstrate the generality of the framework: regular lattices, fractals, Cayley trees, hyperbolic lattices, and one- and two-dimensional Fibonacci quasicrystals. Combined with the kernel polynomial method, these MPO representations enable large-scale calculations of spectral properties without explicitly constructing the full Hamiltonian. This provides a unified route to the exploration of electronic properties across a broad class of lattices at exponentially large system sizes.
ER -