Past exam of the theoretical physics II course of the University of Cambridge / 2019 / 3

For bosonic creation operators and bosonic annihilation operators, the canonical commutation relations areFor fermionic annihilation operators and their adjoints, the canonical anticommutation relations areThe respective number operators are $n_i=a_i^{†}{a}_i$ and $n_i=f_i^{†}{f}_i$. Their allowed eigenvalues are $\boxed{0,1,2,\dots \text{ for bosons};0,1\text{ for fermions}}$. In the fermionic case $n_i^2=n_i$, so its eigenvalues must satisfy $n^2=n$; this is the single-mode Pauli exclusion principle.

In the ordered product basis $(|\uparrow \uparrow ⟩,|\uparrow \downarrow ⟩,|\downarrow \uparrow ⟩,|\downarrow \downarrow ⟩)$, the two-spin XY Hamiltonian isThe transverse term exchanges opposite spins and leaves parallel spins unchanged. Diagonalizing its two-dimensional block gives the complete orthonormal eigenbasisThe antisymmetric state is the singlet state; the other three are triplet states. Accidental equality of energies does not invalidate this eigenbasis.

There is a normalization mismatch in the printed definitions: ${\sigma }^x\pm i{\sigma }^y$ have nonzero entries of magnitude two, whereas a single fermionic creation or annihilation operator has magnitude one. To obtain the intended Jordan–Wigner transformation, use normalized spin ladders $s_j^{\pm }=({\sigma }_j^x\pm i{\sigma }_j^y)/2$. The given fermion expressions represent these $s_j^{\pm }$. The two-site Jordan-Wigner normalization and signs then fix every factor unambiguously.

Put $n_j=f_j^{†}{f}_j$ and $P_1=e^{i\pi n_1}=1-2n_1$. The canonical anticommutation relations give $P_1^2=I$, $P_1{f}_1=-f_1{P}_1$, and $[P_1,f_2]=[P_1,f_2^{†}]=0$. HenceandThese are the spin commutation relations, with distinct sites commuting. If one retains the unscaled symbols ${\sigma }_j^{\pm }=2s_j^{\pm }$, the first and last displayed commutators instead equal $-4{\sigma }_j^z$; the off-site commutators remain zero. Setting the unscaled ${\sigma }_j^{-}$ equal to $f_j$ would contradict the original Pauli matrices.

Using the consistently normalized Jordan–Wigner transformation from part (c), write ${\sigma }_1^x{\sigma }_2^x+{\sigma }_1^y{\sigma }_2^y=2(s_1^{+}{s}_2^{-}+s_1^{-}{s}_2^{+})$. Since $f_1^{†}{P}_1=f_1^{†}$ and $f_1{P}_1=-f_1$, the parity string givesThereforeThe orthogonal transformation $b=(f_1+f_2)/\sqrt{2}$, $c=(f_1-f_2)/\sqrt{2}$ preserves the canonical anticommutation relations and diagonalizes this bilinear Hamiltonian:The Fock vacuum is $|\downarrow \downarrow ⟩$, with energy $2h$. Occupying $b$ alone gives the symmetric opposite-spin state with energy $-2J$; occupying $c$ alone gives the antisymmetric state with energy $2J$; occupying both gives $|\uparrow \uparrow ⟩$, up to a global phase, with energy $-2h$. This reproduces the entire spectrum in part (b).