Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2012 / Paper 1 / 1

For the Transverse-field Ising model, use dimensionless spin one-half operators, so $S^x$ has eigenvalues $\pm 1/2$. At zero transverse field every aligned bond contributes $-J$. The two ground states areThey span an exactly two-dimensional ground space on the finite periodic ferromagnetic chain. Any linear combination is also a ground state; parity eigenstates are the symmetric and antisymmetric combinations. Choosing one ordered product state in the thermodynamic limit is spontaneous symmetry breaking.

For large positive field the leading term is $-2Jg{\sum }_n{S}_n^z$. Its unique minimizing state isThe exchange has zero first-order expectation in this state and produces virtual spin-pair fluctuations at finite g. Thus the polarized product is the limiting wavefunction, not an exact finite-field ground state. Bulk perturbation theory gives $E_0/N=-Jg-J/(4g)+O(J/g^3)$. The field-polarized phase has no twofold ordered-state degeneracy.

Let $n_n=c_n^{†}{c}_n$ and $P_n={\prod }_{m<n}(1-2n_m)$. The Jordan–Wigner transformation gives $2S_n^x=P_n(c_n+c_n^{†})$ and $P_{n+1}=P_n(1-2n_n)$. The canonical anticommutation relations implyConsequently the bondwise cancellation of a Jordan-Wigner string yieldsSubstitution therefore givesThe finite-ring parity qualification of an Ising quasiparticle gap matters here. For an exact periodic spin chain, the closing bond retains total fermion parity. The parity-dependent boundary condition of the Jordan-Wigner chain is $c_{N+1}=-P{c}_1$, with $P=(-1{)}^{N_f}$. Fixing that parity determines the allowed Fourier grid, and the resulting many-body states must still have the chosen parity. Dropping this boundary issue preserves the thermodynamic energy density, but can lose a degeneracy or the correct finite-ring excitation gap. ^[ref]

With the specified Fourier transform convention, normal ordering giveswhere the possible constant ${\sum }_k{e}^{ik}$ vanishes on the complete lattice momentum grid. Antisymmetry of opposite-momentum pairs similarly givesHenceUse the Ising-chain Nambu spinor ${\Psi }_k=(c_k,-i{c}_{-k}^{†}{)}^T$. Expanding its two diagonal terms uses $c_{-k}{c}_{-k}^{†}=1-c_{-k}^{†}{c}_{-k}$, retaining the constant. The result isThis is a full-zone sum. Its opposite-momentum components are redundant; treating each matrix eigenvalue as an independent physical excitation would miss the factor of two in the quasiparticle energy. Self-paired momenta have zero anomalous term and are ordinary number levels.

Put $d_k=\sqrt{(g-\cos k{)}^2+{\sin }^{2}k}$. The matrix in the Ising-chain Nambu spinor has eigenvalues $\pm d_k$. A canonical Bogoliubov transformation, with paired angles satisfying $\cos {\vartheta }_k=(g-\cos k)/d_k$ and $\sin {\vartheta }_k=-\sin k/d_k$, diagonalizes it. Accounting for full-zone fermionic Nambu energy counting givesThus the bulk Ising-chain ground-state energy isExact finite-chain states additionally obey their fermion parity sector.

The Ising-chain quasiparticle dispersion has bulk gap $2J|1-g|$, closing at the Ising-chain quantum critical point. Finite-system parity splittings and the bulk gap are distinct scales. The three requested limits are illustrated below.

At g=0, ${ϵ}_k=2J$ is flat. A domain wall costs 2J, but walls occur in pairs on a periodic spin ring. A single bulk quasiparticle is therefore not automatically the first physical finite-ring spin excitation.

At large positive g, ${ϵ}_k=2Jg-2J\cos k+O(J/g)$. The gap grows as $2J(g-1)$ while the leading bandwidth remains 4J. The field-polarized vacuum is unique.

At g=1, ${ϵ}_k=4J|\sin (k/2)|\sim 2J|k|$. The gap closes at zero momentum and $E_0/N=-4J/\pi$. With unit lattice spacing the critical velocity is $2J/\hslash$.

The Hellmann–Feynman theorem applied to the source field normalization givesDifferentiating the Ising-chain ground-state energy, using ${\partial }_g{ϵ}_k=4J^2(g-\cos k)/{ϵ}_k$, therefore provesThis is the Ising-chain field derivative and transverse polarization, not its spontaneous x-order parameter. It tends to zero at g=0 and to N/2 at large positive field. The critical transverse polarization of an Ising chain is finite: in the thermodynamic critical limit this gives $⟨S^z⟩=1/\pi$. A zero-energy isolated momentum is treated by the appropriate limiting state, rather than by assigning a value to a literal 0/0 summand.