Kitaev, perturbed

No, Alexei is not upset.  Only his model is being perturbed.  In 2006, Kitaev introduced his “honeycomb model” – an exactly soluble Hamiltonian describing spin-1/2 spins interacting via anisotropic exchange on the nearest neighbor bonds of a honeycomb lattice.  He showed that it is a beautiful example of a quantum spin liquid: a highly entangled very non-trivial zero temperature phase of matter.  Most remarkably, its elementary excitations are not spin waves as in a usual magnet, but instead they are Majorana fermions and some exotic “vortices”!  In the past few years, following a beautiful proposal by Jackeli and Khaliullin, it has been recognized that despite the apparently artificial appearance of the model, it might be a good first approximation to a number of real materials.  So one can hope maybe to find Kitaev’s quantum spin liquid in the laboratory!

This hope is reasonable because of stability: Kitaev showed that any small perturbation to his model which preserves time-reversal symmetry leaves the system in the same spin liquid phase.  So an experimental material doesn’t need to be extremely finely tuned to land in the state.  How would one look for it?  The most powerful probe in quantum magnetism is inelastic neutron scattering, which measures a scattering amplitude proportional to the number of excitations of a given momentum and energy created in response to flipping a spin.   It can be calculated exactly for Kitaev’s soluble model.

That is a pretty straightforward calculation, which yields a surprise: there are no excitations created below some minimum energy, or “gap”.  This appears to be a “spin gap”.  It is surprising because Kitaev’s solution shows that there is no true gap: the Majorana fermions actually have a massless relativistic dispersion, like light, so exist at arbitrarily small energies when the wavelength is arbitrarily long.


Here is the calculated structure factor – actually just the new gapless contribution.  The “obvious” part obtained from the Kitaev point adds to this, and contributes above the dashed line.

It’s a bit of a weird result, but it is so straightforward it can’t be wrong…or can it?  In fact, in a paper that just appeared on the arXiv, Xue-Yang Song, Yi-Zhuang You and I showed that the “spin gap” is only a feature of the exactly soluble model.  For any generic Hamiltonian in Kitaev’s spin liquid phase, there is not even an apparent spin gap.  By combining quantum mechanical perturbation theory and field-theoretic arguments, we were able to work out precisely how the gap fills in, and what the low energy structure of spin excitations looks like in a generic system.

This work was spearheaded by Xue-Yang Song, who is a third year undergraduate at Peking University!  Well done Xue-Yang!  You can read about it in the preprint.

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