# Orbital and spin ordering physics of the Mn$_3$O$_4$ spinel

## TL;DR

This paper provides a comprehensive theoretical analysis of Mn$_3$O$_4$ spinel, exploring how orbital and spin orderings emerge from lattice, spin, and orbital interactions, and predicts a possible orbital-spin liquid phase under pressure.

## Contribution

It introduces a detailed microscopic model for Mn$_3$O$_4$, linking structural transitions to orbital and spin orderings, and predicts a pressure-induced orbital-spin liquid phase.

## Key findings

- Orbital ordering relieves geometric frustration of spins.
- Low-temperature ferrimagnetic Yafet-Kittel order observed.
- Potential stabilization of orbital-spin liquid phase under pressure.

## Abstract

Motivated by recent experiments, we present a comprehensive theoretical study of the geometrically frustrated strongly correlated magnetic insulator Mn$_3$O$_4$ spinel oxide based on a microscopic Hamiltonian involving lattice, spin and orbital degrees of freedom. Possessing the physics of degenerate e$_g$ orbitals, this system shows a strong Jahn-Teller effect at high temperatures. Further, careful attention is paid to the special nature of the superexchange physics arising from the 90$^o$ Mn-O-Mn bonding angle. The Jahn-Teller and superexchange-based orbital-spin Hamiltonians are then analysed in order to track the dynamics of orbital and spin ordering. We find that a high-temperature structural transition results in orbital ordering whose nature is mixed with respect to the two originally degenerate $e_{g}$ orbitals. This ordering of orbitals is shown to relieve the intrinsic geometric frustration of the spins on the spinel lattice, leading to ferrimagnetic Yafet-Kittel ordering at low-temperatures. Finally, we develop a model for a magnetoelastic coupling in Mn$_3$O$_4$, enabling a systematic understanding of the experimentally observed complexity in the low-temperature structural and magnetic phenomenology of this spinel. Our analysis predicts that a quantum fluctuation-driven orbital-spin liquid phase may be stabilised at low temperatures upon the application of pressure.