Order-by-disorder and magnetic field response in the Heisenberg-Kitaev model on a hyperhoneycomb lattice

We study the finite-temperature phase diagram of the Heisenberg-Kitaev model on a three-dimensional hyperhoneycomb lattice. Using semiclassical analysis and classical Monte Carlo simulations, we investigate quantum and thermal order-by-disorder, as well as the magnetic-ordering temperature. We find the parameter regime where quantum and thermal fluctuations favor different magnetic orders, which leads to an additional finite-temperature phase transition within the ordered phase. This transition, however, occurs at a relatively low temperature and the entropic effects may dominate most of the finite-temperature region below the ordering temperature. In addition, we explore the magnetization process in the presence of a magnetic field and discover spin-flop transitions which are sensitive to the applied-field direction. We discuss implications of our results for future experiments on a hyperhoneycomb lattice system such as the recently discovered $\ensuremath{\beta}$-Li${}_{2}$IrO${}_{3}$.