Magnetic Soft Modes in the Distorted Triangular Antiferromagnet $\alpha$-CaCr$_{2}$O$_{4}$

We have explored the phase diagram and excitations of a distorted triangular lattice antiferromagnet. The unique two-dimensional distortion considered here is very different from the ``isosceles''-type distortion that has been extensively investigated. We show that suprisingly it is able to stabilize the 120$^{\circ}$ spin structure (typical of the undistorted triangular antiferromagnet) for a large range of exchange interaction values, with new structures found only for extreme distortions. A physical realization of this model is $\alpha$-CaCr$_{2}$O$_{4}$. Despite its highly symmetric 120$^{\circ}$ spin structure, the magnetic excitation spectrum of $\alpha$-CaCr$_{2}$O$_{4}$ is very complex. The unique pattern of nearest-neighbor exchange interactions as well as the substantial next-nearest-neighbor interactions place it close to the phase boundary of the 120$^{\circ}$ structure as is clearly revealed by the presence of low energy modes acting as soft modes of the neighboring structure. Indeed, fitting to linear spin-wave theory favors a set of exchange parameters within the nearby multi-$k$ phase in contradiction to the observed 120$^{\circ}$ order, and quantum fluctuations may be necessary to stabilize $\alpha$-CaCr$_{2}$O$_{4}$ within the 120$^{\circ}$ phase.