Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 75 / 2 / e

The vacuum of the diagonal quasiparticles contains original spin bosons: $⟨a_k^{†}{a}_k⟩=v_k^2$. Thus the quantum depletion of Néel order givesIn one dimension $\sqrt{1-{\gamma }_k^2}=|\sin k|$. Removing the global zero modes with an infrared cutoff, the continuum integral contains $\int dk/|k|$ near both gapless points and diverges logarithmically as the cutoff is removed. This is a breakdown of the assumed Néel order reference, not a physical infinitely negative magnetization. It indicates that the one-dimensional ordered spin-wave expansion cannot maintain a finite staggered moment; it does not determine the exact spectral gap or all properties of the spin chain for arbitrary $S$.

At zero temperature in $d$ dimensions the singular contribution scales as ${\int }_0{k}^{d-2}dk$. It is infrared finite for $d\ge 2$, permitting a finite quantum reduction and a self-consistent ordered spin-wave description in an appropriate regime. This differs from the positive-temperature contribution, where $n_B(\omega )\sim T/\omega$ yields ${\int }_0{k}^{d-3}dk$. Its divergence for $d\le 2$ agrees with the Mermin-Wagner theorem for short-range continuous-symmetry models. Finite-temperature nonordering and the zero-temperature one-dimensional depletion argument are separate statements.