Duality analysis on random planar lattice

TL;DR

This paper extends duality analysis to random planar lattices with bond modifications, successfully estimating critical points for classical models and revealing implications for quantum error correction thresholds.

Contribution

It introduces a modern duality approach with real-space renormalization for random lattices, enabling critical point estimation across a broad randomness spectrum.

Findings

Accurately estimates critical points for Ising and Potts models.

Determines bond-percolation thresholds on random lattices.

Provides insights into quantum error correction thresholds.

Abstract

The conventional duality analysis is employed to identify a location of a critical point on a uniform lattice without any disorder in its structure. In the present study, we deal with the random planar lattice, which consists of the randomized structure based on the square lattice. We introduce the uniformly random modification by the bond dilution and contraction on a part of the unit square. The random planar lattice includes the triangular and hexagonal lattices in extreme cases of a parameter to control the structure. The duality analysis in a modern fashion with real-space renormalization is found to be available for estimating the location of the critical points with wide range of the randomness parameter. As a simple testbed, we demonstrate that our method indeed gives several critical points for the cases of the Ising and Potts models, and the bond-percolation thresholds on the random planar lattice. Our method leads to not only such an extension of the duality analyses on the classical statistical mechanics but also a fascinating result associated with optimal error thresholds for a class of quantum error correction code, the surface code on the random planar lattice, which known as a skillful technique to protect the quantum state.