# Competing orders in pyrochlore magnets from a $\mathbb{Z}_2$ spin liquid perspective

## TL;DR

This paper classifies $ ext{Z}_2$ spin-liquid states on the pyrochlore lattice, linking them to magnetic orders and predicting experimental signatures, thus advancing understanding of competing quantum phases in pyrochlore magnets.

## Contribution

It provides a complete classification of spin-orbit-coupled $ ext{Z}_2$ spin liquids on the pyrochlore lattice using PSG, and constructs phase diagrams connecting spin liquids to magnetic orders.

## Key findings

- Identifies intertwined magnetic and hidden orders in pyrochlore materials.
- Classifies critical theories into $z=1$ and $z=2$ types with distinct experimental signatures.
- Maps out phase diagrams linking spin liquids and magnetic orders.

## Abstract

The pyrochlore materials have long been predicted to harbor a quantum spin liquid, an intrinsic long-range-entangled state supporting fractionalized excitations. Existing pyrochlore experiments, on the other hand, have discovered several weakly ordered states and a tendency of close competition amongst them. Motivated by these facts, we give a complete classification of spin-orbit-coupled $\mathbb{Z}_2$ spin-liquid states on the pyrochlore lattice by using the projective symmetry group (PSG) approach for bosonic spinons. For each spin liquid, we construct a mean-field Hamiltonian that can be used to describe phase transitions out of the spin liquid via spinon condensation. Studying these phase transitions, we establish phase diagrams for our mean-field Hamiltonians that link magnetic orders to specific spin liquids. In general, we find that seemingly unrelated magnetic orders are intertwined with each other and that the conventional spin orders seen in the experiments are accompanied by more exotic hidden orders. Our critical theories are categorized into $z=1$ and $z=2$ types, based on their spinon dispersion and Hamiltonian diagonalizability, and are shown to give distinct signatures in the heat capacity and the spin structure factor. This study provides a clear map of pyrochlore phases for future experiments and variational Monte Carlo studies in pyrochlore materials.