How to understand the elementary excitation of spin density wave?

Question

From Ch.4 of Interacting electrons and quantum magnetism by Auerbach, the elementary excitation of spin density wave can expressed as: $$\alpha^\dagger_{k+}=\cos\theta_kc^\dagger_{k\uparrow}+\sin\theta_kc^\dagger_{k+q\downarrow}$$ $$\alpha^\dagger_{k-}=-\sin\theta_kc^\dagger_{k\uparrow}+\cos\theta_kc^\dagger_{k+q\downarrow}$$

But I cannot understand the motivation of such transformation and cannot relate this excitation operator with its boson operator: $$\rho_{sq\alpha}=\sum_ka^\dagger_{s(k+q)\alpha}a_{sk\alpha},\alpha=\uparrow,\downarrow$$.

What's more, I know the state after above transformation has a "wave" behavior in x-y plane, i.e. $\langle S^+_i\rangle=m_qe^{-iqx_i}$: But the picture of spin wave is also similar: So, I am confused that what the difference between this two picture?

Answer 1

You are misreading Auerbach: these expressions are not for elementary excitations of the spin density wave, but a variational ansatz for the (spiral) spin density wave ground state. This is actually the essential differences between spin waves and spin density waves - spin waves are excitations above a magnetically ordered ground state characterized by a modulation of the spin, whereas the spin density wave is a ground state characterized by a periodic modulation of the spin density. So a spin density wave is a state of matter, much like an antiferromagnet is one.

That said, Auerbach introduces that transformation without much ado, so you'll have to look elsewhere in the literature for a motivation. Personally, I think it's described quite clearly in chapter 2 of the book "Quantum Theory of the Electron Liquid" by Giuliani and Vignale. The basic idea is to to do a Hartree-Fock decoupling of the Hubbard model (or a more general theory), and consider a simplified, non-interacting mean-field problem. The resulting Hamiltonian can be rewritten $$\hat{H}_{HF} = \sum_\vec{k} \epsilon_{\vec{k}+\vec{q},\downarrow} c^\dagger_{\vec{k}+\vec{q},\downarrow}c_{\vec{k}+\vec{q},\downarrow} + \sum_\vec{k} \epsilon_{\vec{k},\uparrow} c^\dagger_{\vec{k},\uparrow}c_{\vec{k},\uparrow} + \sum_\vec{k} g_\vec{k} \left[ c^\dagger_{\vec{k}+\vec{q},\downarrow}c_{\vec{k},\uparrow} + c^\dagger_{\vec{k},\uparrow}c_{\vec{k}+\vec{q},\downarrow}\right]$$ The transformation is then introduced to diagonalize this Hamiltonian.