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

On a connected bipartite graph, every exchange bond can attain its lower classical energy $-J{S}^2$. Set ${\mathbf{S}}_i=S\mathbf{n}$ on one sublattice and ${\mathbf{S}}_j=-S\mathbf{n}$ on the other, for any unit vector $\mathbf{n}$. This Néel state minimizes all bonds simultaneously and hasThe direction $\mathbf{n}\in S^2$ parametrizes the continuous orientation degeneracy; disconnected components can choose their orientations independently.

A triangular lattice illustrates geometric frustration. For three fixed-length spins on a triangle,The minimum requires their sum to vanish, giving coplanar $120$-degree spins rather than antiparallel alignment on every bond. A three-sublattice pattern realizes this condition on the nearest-neighbour triangular lattice. Its energy per bond is $-J{S}^2/2$, and global rotations produce equivalent classical states.

The Holstein–Primakoff transformation acts on the physical Fock states $|n⟩$, $0\le n\le 2S$, with the specified operator ordering. Its matrix elements areThe unavailable endpoint states have zero coefficient. These formulas directly give $[S^z,S^{\pm }]|n⟩=\pm S^{\pm }|n⟩$. MoreoverThus the spin commutation relations hold in units $\hslash =1$. The Holstein–Primakoff occupation constraint is essential: the spin Hilbert space has dimension $2S+1$, so unrestricted boson Fock space is not itself this finite-spin representation. The square root and oscillator operators must retain their order for the displayed matrix elements and endpoint conditions to agree.

Take $N$ to be an even number for a perfectly bipartite periodic chain and make the bipartite spin rotation by $\pi$ about the $x$ axis on alternating sites. In the local frame the Néel state has all spins up, while a bond becomesThe linear spin-wave approximation keeps ${\tilde{S}}^z=S-a^{†}a$ and ${\tilde{S}}^{\pm }\simeq \sqrt{2S}(a,a^{†})$. ThereforeEvery site has two neighbours. Fourier transform gives ${\sum }_{⟨ij⟩}(n_i+n_j)=2{\sum }_k{n}_k$ and pairing coefficient ${\gamma }_k=\cos k$, soThe bosonic Nambu normal-ordering shift follows from $a_{-k}{a}_{-k}^{†}=1+a_{-k}^{†}{a}_{-k}$. It adds $NJS$ inside the matrix expression, which must be subtracted in its constant. HenceA periodic chain with an odd number of sites is frustrated at the boundary and lacks this exact two-sublattice reference; the bulk thermodynamic calculation uses the even-chain sequence.

Use bosonic Bogoliubov diagonalization with $A_k=2JS$, $B_k=2JS{\gamma }_k$. Away from the zero modes, letHere $u_k^2-v_k^2=1$ preserves the canonical commutation relations, while the sign of $v_k$ cancels anomalous pairing. In hyperbolic notation $u_k=\cosh {\theta }_k$, $v_k=\sinh {\theta }_k$, this condition is $\tanh (2{\theta }_k)=-B_k/A_k$. Combining the normal ordering constants givesThe dispersion vanishes linearly near $k=0$ and $\pi$, with spin-wave velocity $2JS$ in unit lattice spacing. The exact zero modes make the Bogoliubov coefficients singular; use a small infrared regulator or a symmetry-selected reference and treat the global rotations separately. A finite transformation at ${\omega }_k=0$ is not asserted.

On a hypercubic lattice of coordination $z=2d$, the same hypercubic antiferromagnetic spin-wave dispersion usesThe corresponding constant is $-dNJS(S+1)$, and ${\omega }_{\mathbf{k}}\sim 2JS\sqrt{d}|\mathbf{k}|$ near a Goldstone point. Restoring lattice spacing $a$ multiplies this velocity by $a$.

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.