Frustrated Heisenberg models and order by disorder

5 problems

• Quantum order by disorder or virtual crystal-field selection in $\mathrm{Er_2Ti_2O_7}$?

  Partially resolved

  $\mathrm{Er_2Ti_2O_7}$ orders into one particular spin pattern although its main interactions treat a whole continuous family equally. Both quantum fluctuations and virtual excitations to higher atomic energy levels can pick that pattern, and their relative share is unsettled.

• Ground state of the spin-$1/2$ pyrochlore Heisenberg antiferromagnet

  Contested

  The pyrochlore lattice, a three-dimensional network of corner-sharing tetrahedra, is the most frustrated common lattice in three dimensions. For spin-$1/2$ Heisenberg coupling it is unknown whether the ground state is a spin liquid or breaks a lattice symmetry.

• Spin liquid between plaquette and Neel phases of Shastry-Sutherland model

  Contested

  The Shastry-Sutherland model, which describes the layered magnet SrCu2(BO3)2, changes from a state of spin singlets on squares (plaquettes) to Neel magnetic order as a coupling ratio grows. Calculations disagree on whether a narrow spin-liquid phase lies between the two or they meet directly.

• Direct observation of a spin-nematic phase in a magnet

  Open

  In a spin-nematic phase the spins pair up so that no single spin points anywhere on average, yet the pairs share a common axis. Theory predicts it near magnetic saturation in several frustrated magnets, but no experiment has measured this hidden pair order directly.

• Intermediate phase of the square-lattice $J1-J2$ Heisenberg model

  Contested

  On the square lattice, a second-neighbor coupling competes with the nearest-neighbor one and destroys magnetic order over a window of ratios. What replaces it, a spin liquid, a pattern of spin singlets or both in sequence, has been debated for decades.