A Computer Code for Topological Quantum Spin Systems over Triangulated Surfaces

TL;DR

This paper presents a computer code that efficiently computes spectral properties of topological quantum spin systems on triangulated surfaces, facilitating analysis of Hamiltonians in quantum topology.

Contribution

It introduces explicit matrix representations for quantum Hamiltonians, enabling systematic and automated spectral analysis of topological quantum systems on various surfaces.

Findings

Code successfully computes spectral data for Kitaev's toric code.

Supports analysis on surfaces of genus 0, 1, and 2.

Provides a versatile tool for quantum topological research.

Abstract

We derive explicit closed-form matrix representations of Hamiltonians drawn from tensored algebras, such as quantum spin Hamiltonians. These formulas enable us to soft-code generic Hamiltonian systems and to systematize the input data for uniformly structured as well as for un-structured Hamiltonians. The result is an optimal computer code that can be used as a black box that takes in certain input files and returns spectral information about the Hamiltonian. The code is tested on Kitaev's toric code deployed on triangulated surfaces of genus 0 and 1. The input file corresponding to the minimal triangulation of genus 2 is also supplied.