# The origin of hour-glass magnetic dispersion in underdoped cuprate superconductors

## TL;DR

This paper explains the hour-glass magnetic dispersion in underdoped cuprates as arising from Lifshitz-type magnetic criticality, providing a unified theory that accounts for various experimental observations and proposing a new sum rule.

## Contribution

It introduces a Lifshitz spin liquid framework to explain magnetic dispersions and confirms a new sum rule with experimental data.

## Key findings

- The hour-glass dispersion is due to Lifshitz magnetic criticality.
- The theory explains all major experimental observations.
- The Lifshitz spin liquid sum rule is validated by data.

## Abstract

In the present work we explain the hour-glass magnetic dispersion in underdoped cuprates. The dispersion arises due to the Lifshitz-type magnetic criticality. Superconductivity also plays a role, but the role is secondary. We list six major experimental observations related to the hour-glass and explain all of them. The theory provides a unified picture of the evolution of magnetic excitations in various cuprate families, including "hour-glass" and "wine-glass" dispersions and an emergent static incommensurate order. We propose the Lifshitz spin liquid "fingerprint" sum rule, and show that the latest data confirm the validity of the sum rule.