Frustrated Magnetism: Exact Diagonalization and Spin Wave Studies

In magnetically frustrated compounds the pairwise exchange interactions of spins cannot all be minimized simultaneously in any microscopic moment configuration. This may arise already in the case of nearest neighbor interactions when the lattice has the property of geometric frustration like, e.g. trigonal, Kagome, checkerboard or pyrochlore type lattices.

Frustration can also arise through the competition of longer range interactions even in simple structures like the two-dimensional (2D) square lattice. At low temperatures there are basically two alternatives: quantum fluctuations may select one of the degenerate states as the true magnetic state (`order by disorder') or they may lead to an ordered quantum phase with a new type of order parameter that is of the `hidden order' type, i.e. it does not display a macroscopic modulation of the spin density.

In our research, we focus on the thermodynamic properties of frustrated spin systems. A benchmarking example is the two-dimensional frustrated Heisenberg model on a square lattice, showing, as a function of its exchange parameters, a rich phase diagram of magnetically ordered and disordered phases.

As an example, the figure to the left shows the calculated heat capacity of this model as a function of frustration angle and magnetic field strength. The white lines divide the three classical ordered phases from each other. Quantum disorder effects show up as the double-ridge shape (yellow and orange color coding) around the classical phase transitions. [Clicking onto the figure will lead to a larger version.]

We use both exact diagonalization for finite clusters as well as analytical spin-wave methods. The eigensystems of the model Hamiltonians obtained from the exact-diagonalization data are used to directly evaluate arbitrary thermodynamic expectation values and related cumulants like the heat capacity (displayed above), magnetic susceptibility, or spin correlation functions. This method is called the finite-temperature Lanczos method.