Quantum fluctuations disorder the 1D Heisenberg AF; summary table

We previously used the Holstein-Primakoff boson representation of spin operators
$$\begin{eqnarray} S^+ &=& (\sqrt{2 S - n_b}) b,\cr S^- &=& b^\dagger (\sqrt{2 S - n_b}),\cr S^z &=& S - n_b. \end{eqnarray}$$
in order to understand spin waves around both ferromagnetic and antiferromagnetic ground states. We complete the antiferromagnetic case from last time, where the ground state of the original HP bosons is not the ground state of the new operators created by a bosonic Bogoliubov transformation.

Our result for the spin-wave spectrum of the antiferromagnet was
$$\begin{eqnarray} H_1 &=& \sum_{\bf k} \omega_{\bf k} (\alpha^\dagger _{\bf k} \alpha_{\bf k}+\frac{1}{2}) - {J S z N \over 2},\cr \omega_{\bf k} &=& |J| S z \sqrt{1 - \gamma_{{\bf k}}^2}. \end{eqnarray}$$
These operators $\alpha_{\bf k}$ were the result of a Bogoliubov transformation to eliminate anomalous products like $bb$, $b^\dagger b^\dagger$:
$$\begin{eqnarray} \alpha_{\bf k} &=& \cosh \theta_{\bf k} b_{\bf k} - \sinh \theta_{\bf k} b^\dagger_{-{\bf k}} \cr b_{\bf k} &=& \cosh \theta_{\bf k} \alpha_{\bf k} + \sinh \theta_{\bf k} \alpha^\dagger_{-{\bf k}}. \end{eqnarray}$$
Here
$$\begin{equation} \tanh 2 \theta_{\bf k} = - \gamma_k, \end{equation}$$
and $\gamma_k$, introduced in the last lecture, was essentially the Fourier transform of the nearest-neighbor points (e.g., $\cos(ka)$ in one dimension).

Clearly the lowest-energy state corresponds to zero occupancy of all the $\alpha$ bosonic states. Hence it may seem that at zero temperature our spin-wave theory is justified since one of our requirements was that the number of bosons satisfy $\langle n \rangle \ll 2S$. But remember that this statement was about the number of the original $b$ bosons, not the rotated $\alpha$ bosons! So we need to find out what $\langle n_b \rangle$ is in the state with $n_\alpha=0$. The reduction of the staggered moment per site is just, from the same calculation as in the ferromagnetic case,
$$\begin{eqnarray} \Delta {\tilde m}_0 &=& - {1 \over N} \sum_i \langle b^\dagger_i b_i \rangle = - \sum_{\bf k} \langle b^\dagger_{\bf k} b_{\bf k} \rangle\cr &=& -{1 \over N} \sum_{\bf k} \langle (\cosh \theta_{\bf k} \alpha^\dagger_{\bf k} + \sinh \theta_{\bf k} \alpha_{-{\bf k}}) (\cosh \theta_{\bf k} \alpha_{\bf k} + \sinh \theta_{\bf k} \alpha^\dagger_{-{\bf k}}) \rangle \cr &=& -{1 \over N} \langle \cosh^2 \theta_{\bf k} \alpha^\dagger_{\bf k} \alpha_{\bf k} + \sinh^2 \theta_{\bf k} \alpha_{-{\bf k}} \alpha^\dagger_{-{\bf k}} \rangle. \end{eqnarray}$$
At zero temperature the only contribution from this is from the second term $\alpha \alpha^\dagger$. So at $T=0$,
$$\begin{eqnarray} \Delta {\tilde m}_0 &=& -{1 \over N} \sum_{\bf k} \sinh^2 \theta_{\bf k} = - {1 \over N} \sum_{\bf k} {\cosh 2 \theta_{\bf k} - 1 \over 2} = {1 \over 2} - {1 \over N} \sum_{\bf k} {1 \over 2 \sqrt{1 - \tanh^2 2 \theta_{\bf k}}} \cr &=& {1 \over 2} - {1 \over N} \sum_{\bf k}{1 \over 2 \sqrt{1 - \gamma_{\bf k}^2}}. \end{eqnarray}$$

Now consider the above sum in one dimension, where $\gamma_{\bf k} = \cos(k a)$. Then the sum is logarithmically divergent at small $|k|$ (can anyone explain to me why Auerbach has it diverging as $1/k$?), and there is no long-range order in the ground state. In two dimensions, as before, we have order at $T=0$ but no order for finite temperature.
However, the moment is reduced by quantum fluctuations even in the ground state relative to the maximum moment in the N\'eel state, unlike in the ferromagnet.

Finally we can make a table to summarize what spin-wave theory has taught us about the ferromagnet and antiferromagnet in various dimensions on a bipartite lattice. We write $???$ for the 1D antiferromagnet at zero temperature because, as we will see in the next few lectures, its behavior is quite complicated: for integer spin it is gapped and truly disordered (correlations fall off exponentially), while for half-integer spin it is critical (correlations fall off algebraically).

\begin{array}{l || c | c} &{\rm Ferromagnet}&{\rm Antiferromagnet}\cr \hline \hline d=1,T=0&{\rm Ordered}&???\cr d=1,T>0&{\rm Disordered}&{\rm Disordered}\cr \hline d=2,T=0&{\rm Ordered}&{\rm Ordered}\cr d=2,T>0&{\rm Disordered}&{\rm Disordered}\cr \hline d=3,T=0&{\rm Ordered}&{\rm Ordered}\cr d=3,T>0&{\rm Ordered}\ ({\rm low}\ T)&{\rm Ordered}\ ({\rm low}\ T)\cr \end{array}