- #1

conana

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A certain four-spin correlation function for Kitaev's spin-1/2 model on the honeycomb lattice is defined in Eq. 28, and from Eq.'s 29 and 30 one arrives at the final expression for the correlation function as

[tex] C(\mathbf{r}_1, \mathbf{r}_2) = \frac{1}{N^2} \sum_{q,q'} \cos{[(\mathbf{q}-\mathbf{q}')\cdot(\mathbf{r}_1-\mathbf{r}_2)]} \frac{\Delta_q \Delta_{q'} - \epsilon_q \epsilon_{q'}}{E_q E_{q'}},[/tex]

where N is the number of sites in the lattice, the summations are over momenta in the first Brillouin zone, [itex]\mathbf{r}_1, \mathbf{r}_2[/itex] denote unit cell positions, [itex] \Delta_q = \sin{q_1} + \sin{q_2} [/itex], [itex] \epsilon_q = \cos{q_1} + \cos{q_2} + 1 [/itex], and [itex] E_{q} = \sqrt{\epsilon_q^2 + \Delta_q^2} [/itex] with [itex] q_1, q_2 [/itex] being the components of [itex] \mathbf{q} [/itex] in the basis of reciprocal lattice vectors.

Arriving at this expression is not a problem. However, the authors go on to argue that for large [itex] |\mathbf{r}_1 - \mathbf{r}_2| [/itex], the correlation function decays algebraically as

[tex] C(\mathbf{r}_1, \mathbf{r}_2) \sim \frac{1}{|\mathbf{r}_1 - \mathbf{r}_2|^4}. [/tex]

In arriving at this expression, the authors state "[...] the denominator in Eq. (30) has two zero points, which are of order 1/N in the large N limit. Their contribution causes the summation to be finite in the thermodynamic limit. Then using the stationary phase method, we can evaluate the exponents of the correlation function at long distance to be 4." Eq. 30 refers to (essentially) the above expression for the correlation function involving the summations over the Brillouin zone, and the two zero points of the denominator correspond to the two Dirac points in the fermionic quasiparticle spectrum.

It is the calculation of this asymptotic behavior which I am trying to reproduce, but I seem to be unable to do this with the information given. It seems there are no momenta at which the phase is stationary due to the phase being a linear function of momentum. So then I thought, perhaps the authors intend to imply that the phase is never stationary and, thus, the summation would vanish at all points in the Brillouin zone EXCEPT for at the Dirac points where the denominator vanishes, yielding a singularity in the summand. However, the numerator also vanishes at these points and, in fact, the limit of the summand at these points is finite.

Having run into only dead-ends with the stationary phase method, I though about approximating the sums as integrals in the [itex] N \rightarrow \infty [/itex] limit and performing successive integration by parts to extract the asymptotic behavior. (For sake of simplicity, in the following I take [itex]\mathbf{r} = \mathbf{r}_1 - \mathbf{r}_2[/itex] to be parallel to one of the lattice vectors). However, I seem to be getting terms proportional to

[tex] \frac{\sin^2{[\pi r]}}{r^{2n}}, \qquad n\in \mathbb{Z}^+,[/tex]

where the lattice spacing is taken to be unity. These terms all vanish as [itex] r [/itex] is an integer (it denotes a distance along one direction of the lattice).

It seems clear to me that I am not understanding the spirit of the calculation/am making math errors as all of my attempts suggest that the correlation function vanishes. The numerics (of the authors and of my own) show, however, that the power law behavior described above is correct.

Any help would be greatly appreciated. Thank you in advance.