# Heisenberg-Kitaev Models on Hyperhoneycomb and Stripyhoneycomb Lattices: 3D-2D Equivalence of Ordered States and Phase Diagrams

## TL;DR

This paper demonstrates that many magnetic ordered states in 3D hyperhoneycomb lattices can be mapped onto 2D honeycomb lattices, revealing identical classical energetics and phase diagrams, thus unifying the understanding of 2D and 3D Kitaev materials.

## Contribution

It establishes a 3D-2D mapping for ordered states in Heisenberg-Kitaev models on hyperhoneycomb lattices, including phase diagrams with additional interactions, and shows their equivalence to 2D counterparts.

## Key findings

- 3D and 2D ordered states have identical classical energetics.
- Phase diagrams of 3D hyperhoneycomb models match 2D honeycomb models.
- Explicit demonstration of adiabatic equivalence between spiral orders in different Li$_2$IrO$_3$ phases.

## Abstract

We discuss magnetically ordered states, arising in Heisenberg-Kitaev and related spin models, on three-dimensional (3D) harmonic honeycomb lattices. For large classes of ordered states, we show that they can be mapped onto two-dimensional (2D) counterparts on the honeycomb lattice, with the classical energetics being identical in the 2D and 3D cases. As an example, we determine the phase diagram of the classical nearest-neighbor Heisenberg-Kitaev model on the hyperhoneycomb lattice in a magnetic field: This displays rich and complex behavior akin to its 2D counterpart, with most phases and phase boundaries coinciding exactly. To make contact with the physics of the hyperhoneycomb iridate $\beta$-Li$_2$IrO$_3$, we also include a symmetric off-diagonal $\Gamma$ interaction, discuss its 3D-2D mapping, and determine the relevant phase diagrams. In particular, we demonstrate explicitly the adiabatic equivalence of the spiral magnetic orders in $\alpha$- and $\beta$-Li$_2$IrO$_3$. Our results pave the way to a systematic common understanding of 2D and 3D Kitaev materials.