Past exam of the advanced quantum theory: path integrals and many-particle physics course of the University of Oxford / 2023 / 2 / b

For the specified positive-exponent discrete Fourier transform, the inverse discrete Fourier transforms have negative exponents:Every sum runs over $q=2\pi j/L$, $j=0,\dots ,L-1$. Indeed, orthogonality of roots of unity givesso substituting the forward discrete Fourier transform into either proposed inverse recovers the original operator. Taking Hermitian conjugates supplies the inverse formulas for the creation operators.

Multiplying the inverse discrete Fourier transforms gives the local number operator combinationThis local expression includes off-diagonal momentum terms. Only after summing over sites does orthogonality of roots of unity giveIn a physical fixed-$s$ state the first boxed operator acts as $s$ at every site, by the local spin constraint in the Schwinger boson representation.

Take $L\ge 2$, so neighbouring operators act at distinct sites. The identityexpresses the Hamiltonian operator of this quantum XY chain in spin ladder operators. Substituting the Schwinger boson representation gives the requested quartic expression:Operators at different sites commute, allowing normal ordering without a correction here. For the first quartic term, the inverse discrete Fourier transforms giveHere ${\delta }^{(L)}$ is one when its two momenta agree modulo $2\pi$ and zero otherwise; it expresses lattice momentum conservation. The other quartic term has the same ordered operators with phase $e^{i(p-r)}$. On the support of this Kronecker delta, $p-r=t-q$ modulo $2\pi$, so the two phases sum to $2\cos (q-t)$. Thus the Fourier vertex of a Schwinger boson chain isAn equivalent three-sum expression sets $t=p+q-r$ modulo $2\pi$, so its coefficient is $\cos (r-p)$. The factor $1/L$ comes from four factors $L^{-1/2}$ and one site sum, while the factor of two from the two exchange terms cancels the original $1/2$. The formula is Hermitian: conjugation exchanges $p$ with $t$ and $q$ with $r$, and the coefficient is unchanged after imposing lattice momentum conservation. It also preserves each local total boson number in its real-space form, hence preserves the physical constrained subspace.