Bounds on cubic Lorentz-violating terms in the fermionic dispersion relation

We study the recently proposed Lorentz-violating dispersion relation for fermions and show that it leads to two distinct cubic operators in the momentum. We compute the leading order terms that modify the nonrelativistic equations of motion and use experimental results for the hyperfine transition in the ground state of the ${}^9\mathrm{B}{e}^{+}$ ion to bound the values of the Lorentz-violating parameters ${\eta }_1$ and ${\eta }_2$ for neutrons. The resulting bounds depend on the value of the Lorenz-violating background four-vector in the laboratory frame.