# Is there any hidden symmetry in the stripe structure of perovskite high temperature superconductors?

## TL;DR

This paper explores the hidden symmetries in stripe structures of doped perovskite high-temperature superconductors, proposing a mathematical model based on superpositions of stripe pairs and four-color theorem mappings to understand their complex lattice modulations.

## Contribution

It introduces a novel theoretical framework representing stripe modulations as superpositions of tilted stripe pairs and maps their arrangements using planar graphs and the four-color theorem.

## Key findings

- Stripe structures can be modeled as superpositions of two tilted stripe types.
- The spatial distribution of stripes relates to four-color theorem mappings.
- Atomic sharpness of stripe boundaries is consistent with the model.

## Abstract

Local and fast structural probes using synchrotron radiation have shown nanoscale striped puddles and nanoscale phase separation in doped perovskites.It is known that the striped phases in doped perovskites are due to competing interactions involving charge, spin and lattice degrees of freedom,but while many theoretical models for spin and charge stripes have been proposed we are missing theoretical models describing the complex lattice striped modulations in doped perovskites. In this work we show that two different stripes can be represented as a superposition of a pair of stripes, U(theta_n) and D(theta_n), characterized by perovskite tilts where one of the pair is rotated in relation to the other partner by an angle Delta(theta _n)=pi/2. The spatial distribution of the U and D stripes is reduced to all possible maps in the well-known mathematical four-color theorem. Both the periodic striped puddles and random structures can be represented by using $planar$ graphs with a chromatic number. To observe the colors in mapping experiments, it is necessary to recover variously oriented tilting effects from the replica. It is established that there is an interplay between the annihilation/creation of new stripes and ordering/disordering tilts in relation to the theta _n angle in the CuO2 plane, where the characteristic shape of the stripes coincides with the tilting-ordered regions. By their origin, the boundaries between the stripes should be atomically sharp.