Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2013 / Paper 1 / 1

Use a real, even interaction $V_{-\mathbf{k}}=V_{\mathbf{k}}$. Let $\mathcal{V}=L^3$ and $N_{\mathrm{ex}}={\sum }_{\mathbf{k}\ne 0}{b}_{\mathbf{k}}^{†}{b}_{\mathbf{k}}$. In the Bogoliubov theory expansion, keep terms with zero or two noncondensed operators. The zero-momentum term gives $V_0{N}_0(N_0-1)/(2\mathcal{V})$. The terms with two noncondensed operators giveHere $V_0$ is the direct condensate interaction and $V_{\mathbf{k}}$ is its exchange contribution. At fixed total particle number, $N_0=N-N_{\mathrm{ex}}$, soThe second term cancels the direct normal term. Adding the kinetic energy yields the number-conserving Bogoliubov quadratic HamiltonianThe constant is the condensate mean-field theory interaction energy; the normal term describes moving a particle out of the condensate, while the anomalous term creates or annihilates an opposite-momentum pair by exchanging two particles with the condensate. Total particle number remains fixed in the underlying theory; the condensate amplitudes in this notation are treated as classical amplitudes.

For condensate fluctuation power counting, put $\delta =N_{\mathrm{ex}}/N\ll 1$. Formal interaction-amplitude counting gives $H_2-E_{\mathrm{mf}}\sim E_{\mathrm{mf}}\delta$, cubic terms $\sim E_{\mathrm{mf}}{\delta }^{3/2}$ and quartic terms $\sim E_{\mathrm{mf}}{\delta }^2$; these are not uniform bounds on individual modes. Thus the latter are down by $\sqrt{\delta }$ and $\delta$ relative to the quadratic interaction terms. The $N_0-1$ correction is relative $1/N$. Homogeneity eliminates terms linear in nonzero-momentum operators. Small depletion and weak interactions, not low temperature alone, justify the truncation. ^[ref]

For each opposite-momentum pair use the Bogoliubov transformationEquivalently $u_k=\cosh {\theta }_k$, $v_k=\sinh {\theta }_k$, with $\tanh (2{\theta }_k)=B_k/A_k$. The identity $u_k^2-v_k^2=1$ preserves the canonical commutation relation. Vanishing anomalous terms require $B_k(u_k^2+v_k^2)-2A_k{u}_k{v}_k=0$, and hence the Bogoliubov quasiparticle dispersion isSubstitution and normal ordering giveThis is the requested form after putting each $E_k/2$ inside the oscillator term. By momentum-pair counting in bosonic diagonalization, all sums cover the full nonzero-momentum set; no further factor of two is needed. The bosonic Bogoliubov zero-point shift combines the two vacuum sums before taking a continuum limit.

If $V_k\to V_0>0$ smoothly, the low-energy excitations are gapless phonons:For contact repulsion, the healing length $\xi =\hslash /\sqrt{2mn{V}_0}$ separates this regime from the particle-like regime $E_k\simeq {ϵ}_k+n{V}_0$. Stability requires ${ϵ}_k+2n{V}_k>0$ for every nonzero mode. A general momentum-dependent interaction need not share the contact model's entire curve. An unregularized contact model also makes the vacuum-energy shift ultraviolet divergent; a physical finite-range interaction or scattering-length renormalization is needed for that absolute ground-state energy.

In the Bogoliubov quasiparticle vacuum, ${\alpha }_{\mathbf{k}}|0_{\alpha }⟩=0$, but the inverse Bogoliubov transformation gives $⟨b_{\mathbf{k}}^{†}{b}_{\mathbf{k}}⟩=v_k^2$. Consequently the quantum depletion of a Bose condensate isThis expression assumes zero temperature. At nonzero temperature, thermal Bogoliubov quasiparticles add to the depletion:Thus the printed formula is the quantum contribution, or a low-temperature approximation when the thermal contribution is negligible. It is nonnegative for a stable quadratic Hamiltonian and vanishes in the noninteracting zero-temperature limit.