Past exam of the quantum Condensed Matter Field Theory course of the University of Cambridge / 2012 / Paper 1 (Quantum Condensed Matter Field Theory, Paper 160)

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.

Use dimensionless spin operators and nearest-neighbour exchange $J_H>0$ on a regular bipartite graph with N sites and coordination number of a lattice z. Count each bond once and define ${\gamma }_{\mathbf{k}}=z^{-1}{\sum }_{\delta }{e}^{i\mathbf{k}\cdot \delta }$, real for the inversion-symmetric examples considered here.

For the ferromagnetic Heisenberg model, $H_F=-J_H{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$. Expanding the Holstein–Primakoff transformation about all spins along +z givesup to interactions of higher order in boson density, with $E_{\mathrm{cl}}=-Nz{J}_H{S}^2/2$. The isotropic spin-wave vacuum energy comparison starts with this normal ordering. The linear spin-wave approximation then givesEach ferromagnetic magnon lowers total $S^z$ by one. The fully aligned state is an exact eigenstate and ground state; there is no harmonic vacuum depletion or added zero-point energy in this normal-ordered Hamiltonian. The global spin-rotation multiplet remains degenerate. On a hypercubic lattice of spacing a, $z=2d$ and ${\gamma }_{\mathbf{k}}=d^{-1}{\sum }_j\cos k_{j}a$, so $E_F\simeq J_{H}S{a}^2{k}^2$. The exact one-magnon sector agrees with this result even at finite S; interactions enter for multiple magnons.

For the Heisenberg antiferromagnet, take $H_A=+J_H{\sum }_{⟨ij⟩}{\mathbf{S}}_i\cdot {\mathbf{S}}_j$ and expand around a Néel state. On A, $S^z=S-a^{†}a$ and $S^{+}\simeq \sqrt{2S}a$; on B, $S^z=-S+b^{†}b$ and $S^{+}\simeq \sqrt{2S}{b}^{†}$. A bond becomesWith N/2 momenta in the magnetic Brillouin zone,The bosonic Bogoliubov transformation has $u_k^2-v_k^2=1$ and $\tanh 2{\theta }_k={\gamma }_k$. For example $a_k=u_k{\alpha }_k-v_k{\beta }_{-k}^{†}$ and $b_{-k}=u_k{\beta }_{-k}-v_k{\alpha }_k^{†}$. It gives two branches in this reduced zone,The reduced-zone counting of bipartite spin waves gives N oscillators in total, not 2N. The negative vacuum correction and quantum depletion of Néel order, $\delta S=(2/N){\sum }_{\mathrm{MBZ}}{v}_k^2$, distinguish the antiferromagnet from the ferromagnet. At nonzero temperature replace $v_k^2$ by $v_k^2+(u_k^2+v_k^2)n_B(E_k)$. Hypercubic small-momentum modes have $E_A\simeq 2J_{H}Sa\sqrt{d}|k|$, hence linear physical frequency $E_A/\hslash$.

Self-consistency requires depletion much smaller than S and respects the Holstein–Primakoff occupation constraint. In one dimension the antiferromagnetic vacuum depletion diverges logarithmically, invalidating ordered spin-wave theory; it does not alone prove an exact gap or its absence. At positive temperature, isotropic short-range models in d≤2 cannot sustain such magnetic order by the Mermin-Wagner theorem. In three dimensions the quadratic ferromagnetic modes give the low-temperature magnetization reduction proportional to $T^{3/2}$, whereas the antiferromagnetic thermal reduction scales as $T^2$. Anisotropy, fields, frustration and magnon interactions alter these conclusions. ^[ref]

Insert position resolutions between short-time evolution factors for $H=p^2/(2M)+V(q)$. Integrating the momenta gives the path integral, with real-time slice factor $[M/(2\pi i\hslash \Delta t){]}^{1/2}$ and weight $e^{iS[q]/\hslash }$. A real classical trajectory cannot pass through a barrier with insufficient energy, but the path integral includes nonclassical paths.

Continue $t=-is$, with s physical imaginary time. The damped kernel isThe Euclidean action saddle obeys $M\ddot{q}=V^{\prime }(q)$: motion in the inverted potential. For degenerate minima at $\pm a$, subtract their common potential energy. A trajectory joining them with zero Euclidean energy satisfiesIt is an instanton; reversing its direction gives an anti-instanton. Thus a crossing amplitude is nonperturbative, proportional to $e^{-S_I/\hslash }$. For the representative quartic well $V=M{\omega }_0^2(q^2-a^2{)}^2/(8a^2)$, direct integration givesThe arbitrary center $s_0$ is an instanton translation zero mode. Expanding $q=q_I+\eta$ gives a Gaussian path integral involving $L_I=-{\partial }_s^2+V^{\prime \prime }(q_I)/M$. Treating the zero mode as a collective coordinate rather than a zero Gaussian denominator produces the instanton fluctuation prefactorCommon endpoint regularization defines the functional determinant ratio; K and kappa have inverse-time units. The prime removes only the zero eigenvalue, while integrating the center supplies the total available time. Large $S_I/\hslash$ and $\kappa \ll {\omega }_0$ justify a dilute instanton gas.

Multiple crossings alternate direction. Integrating n ordered centers gives ${\mathcal{T}}^n/n!$. Return paths contain even n, and transfer paths contain odd n, so their common local-well factor multiplies $\cosh (\kappa \mathcal{T})$ and $\sinh (\kappa \mathcal{T})$ respectively. The two lowest levels are consequentlyThe lower state is the even combination of the two localized states. Local anharmonic corrections modify $E_{\mathrm{well}}\simeq \hslash {\omega }_0/2$ and higher semiclassical corrections modify K. The thermal trace as a closed Euclidean path integral gives $Z\simeq 2e^{-\beta E_{\mathrm{well}}}\cosh (\beta \hslash \kappa )$ when higher local levels are negligible. This describes coherent level splitting in a stable well, not irreversible decay.

For a scattering barrier at energy E, the leading WKB tunnelling exponent gives an amplitude $\exp [-{\int }_{q_1}^{q_2}\sqrt{2M(V-E)}dq/\hslash ]$ and a transmission probability with twice that exponent. Endpoint matching fixes prefactors. Metastable escape instead uses a Euclidean bounce, with a negative fluctuation mode and a decay rate. Neither its negative mode nor the probability/amplitude factor should be transplanted into the stable double-well splitting. ^[ref]

Use the reduced attractive reduced BCS pairing Hamiltonian with volume V, ${\xi }_k={\epsilon }_k-\mu$ and attraction $\lambda >0$:A specified pairing shell or an ultraviolet renormalization is part of the model. The trace is grand canonical. Insert fermionic coherent states into its time-sliced trace. The resulting Grassmann variables are antiperiodic, and with inverse-energy imaginary time $0\le \tau \le \beta$ the dimensionless action isThis boundary condition is essential for Fermi-Dirac statistics; replacing it by a bosonic periodic trace changes the answer. A complex Hubbard–Stratonovich transformation uses the local identitywith time-slice factors understood. The pairing field renders the fermion action quadratic. For a uniform static field the independent Nambu spinor block $({\psi }_{k\uparrow },{\bar{\psi }}_{-k\downarrow }{)}^T$ hasIts determinant is $-({\omega }_n^2+E_k^2)$. The phase-independent determinant sign cancels under consistent reference normalization. Integrating the fermions gives the BCS pairing-field effective action, ${\mathcal{S}}_{\mathrm{eff}}={\int }_0^{\beta }V|\Delta {|}^2/\lambda d\tau -\mathrm{Tr}\ln {\mathcal{G}}^{-1}$, with the normal-ordering/reference term retained. The fermionic pairing-block determinant normalization uses one independent pairing block for every k; a fully doubled spin-particle-hole trace instead needs one half. The subtracted grand potential of the reduced BCS model is particularly unambiguous:Performing the Matsubara frequency sum yieldsup to a common constant. At zero gap this reduces exactly to the two-spin free-gas result. The Matsubara derivation of the BCS saddle follows by differentiating with respect to ${\Delta }^{\ast }$ and using ${\beta }^{-1}{\sum }_n({\omega }_n^2+E^2{)}^{-1}=\tanh (\beta E/2)/(2E)$ givesThe second equation fixes the chemical potential at specified number. This stationary action is BCS mean-field theory, not an exact integration of pairing-field fluctuations. With constant single-spin density of states ${\nu }_F$ per volume and a symmetric energy cutoff $\Lambda$, weak coupling gives $\Delta (0)\simeq 2\Lambda e^{-1/(\lambda {\nu }_F)}$ and $k_B{T}_c\simeq (2e^{{\gamma }_E}/\pi )\Lambda e^{-1/(\lambda {\nu }_F)}$. The zero-temperature condensation grand-potential density is $-{\nu }_F{\Delta }^2/2$ to leading weak-coupling order; using a spin-summed density would change these displayed factors. The BCS coherence factors are $u_k^2=(1+{\xi }_k/E_k)/2$ and $v_k^2=(1-{\xi }_k/E_k)/2$, and the associated BCS ground state is the normalized product of empty and occupied pair states. ^[ref]

Allowing a slowly varying complex pairing field generates phase stiffness and, near the transition, the Ginzburg-Landau theory of superconductivity. A neutral paired system has a gapless phase sound mode; in a charged bulk superconductor electromagnetic coupling produces the Anderson-Higgs mechanism in a superconductor and the Meissner effect. These collective fluctuations are distinct from the gapped fermionic excitations. A microscopic dynamical calculation is needed to fix collective-mode dispersion and damping; the static saddle alone does not determine them. ^[ref]

For the untwisted quantum rotor, periodicity on the angle circle requires $\psi (\varphi +2\pi )=\psi (\varphi )$. The normalized energy eigenstates and their angular momentum areTaking the trace in this orthonormal basis therefore givesThe n=0 state is nondegenerate and each nonzero $|n|$ has two opposite-angular-momentum states. No flux or twisted angular boundary condition is present.

First lift the angle to the real line. The time-sliced configuration-space path integral for a free coordinate with inertia I has kinetic action $\int dtI{\dot{\varphi }}^2/2$. The rotor kernel sums lifted endpoints separated by ${\varphi }_F-{\varphi }_I+2\pi m$, because all such endpoints represent the same final angle on the circle.

Continue real time as $t=-i\hslash \tau$. Then $e^{-iHt/\hslash }=e^{-H\tau }$ and the end of the thermal interval is $\tau =\beta$, in inverse-energy units. ThusThe trace integrates the initial angle over one copy of the circle, not the whole lifted line. It yields the rotor winding-sector path integralThe integer m is temporal winding number, not the angular-momentum label n. Locally the measure has the free-kernel normalization $[I/(2\pi {\hslash }^2\Delta \tau ){]}^{1/2}$ per time slice. Summing topological sectors and retaining this normalization are necessary for the correct thermal trace.

In winding sector m write the lifted path asThe cross term vanishes since ${\int }_0^{\beta }\dot{\eta }d\tau =0$. The action separates into $I(2\pi m{)}^2/(2{\hslash }^2\beta )$ and the same fixed-endpoint Gaussian path integral for every m. Neither term depends on the initial angle, so its integral supplies $2\pi$:The fixed-endpoint rotor fluctuation normalization isIt is the diagonal free-particle thermal kernel on the lifted line, or the amplitude density for a zero-displacement fluctuation bridge. It is not by itself the circle partition function or a separately countable angular zero mode. The integration over ${\varphi }_0$ has already counted that mode. Quadraticity makes the saddle-plus-fluctuation evaluation exact.

Analytically continue the supplied free propagator using $M=I$ and $t=-i\hslash \beta$. This gives $Z_0=\sqrt{I/(2\pi {\hslash }^2\beta )}$ without an arbitrary measure constant.

Apply the Poisson summation formula to $h(x)=e^{-A{x}^2}$, with $A=2{\pi }^{2}I/({\hslash }^2\beta )$. Its Gaussian integral isThe prefactors cancel exactly because $2\pi Z_0\sqrt{\pi /A}=1$. Therefore the momentum-winding duality of the quantum rotor givesin agreement with the spectrum. The thermal winding dominance of a free rotor interchanges between the two representations. At low temperature the momentum sum is dominated by n=0 and $Z\to 1$. At high temperature the winding sum is dominated by m=0 and $Z\sim \sqrt{2\pi I/(\beta {\hslash }^2)}$, the classical angular phase-space integral. Winding and momentum labels are dual descriptions; keeping only m=0 at low temperature would be incorrect.

The coupled quantum rotor chain has a compact phase and conjugate integer number $n_j={\ell }_j/\hslash$. Expanding the cosine givesIn the phase-space Euclidean path integral, using inverse-energy time, each rotor has the action term ${\ell }_j^2/(2I)-i{\ell }_j{\dot{\varphi }}_j/\hslash$. Completing its Gaussian square produces $I{\dot{\varphi }}_j^2/(2{\hslash }^2)$. Equivalently, apply the short-time free-rotor kernel locally and retain winding sectors. The smooth-sector continuum limit at unit spacing is consequentlyThe overall constant is omitted when the reference bond energy is subtracted. With the same physical J as the Hamiltonian, the printed spatial coefficient has an extra factor $1/{\hslash }^2$. The imaginary-time normalization for a coupled rotor chain requires ${\hslash }^{2}J/2$ inside a bracket with common prefactor $1/{\hslash }^2$. Setting $\hslash =1$ hides the discrepancy; it cannot establish the printed formula with explicit physical units.

Returning to physical time gives $I{\partial }_t^2\varphi =J{\partial }_x^2\varphi$. The harmonic phase mode of a quantum rotor chain therefore hasThis is a gapless sound-like collective phase mode, analogous to neutral superfluid sound. In rotor-chain phase-number hydrodynamics, number fluctuations supply the compressibility and the intersite coupling supplies phase stiffness. The mode is a phase-density oscillation, not a free single-rotor transition. The global phase zero mode of a finite rotor chain is a compact rotor with inertia NI, which should not be counted as an independent zero-frequency oscillator.

The compactness qualification of a harmonic rotor chain matters here. The smooth harmonic theory requires small neighbouring phase differences and cannot settle the entire strong-fluctuation phase diagram. In one dimension its zero-temperature phase variance grows logarithmically, so the gapless regime has algebraic phase correlations rather than a true condensate. Thermal fluctuations and compact phase slips require additional care; phase-slip proliferation can produce a gapped phase outside the Gaussian regime. ^[ref]