1 TQM TODOs

Compiled from a review of the Notes at tqm.tripos.org and the past papers (2017–2023, 2025). Organised as: site/migration bugs → physics/math errors → typos by chapter → content updates → exam ideas.

1.1 Site & migration bugs (Jekyll → Quarto debris)

• Unconverted Liquid cite tags render literally: {% cite dirac1926 %} (Many Body Wavefunctions), {% cite Fisher:1989aa %} (Lattice Models)
• Broken {{< ref >}} shortcodes render as raw markup in: Spin Models (Appendix link), A is for Annihilation (two “Lecture 1” links), Lattice Models (Lecture 4, Problem Set 2), Bose Gas (Lecture 2), Superconductivity (“above figure”), Response and Correlation (Lecture 1)
• Dead links to excised chapters: notes/lieb-liniger.qmd (Spin Models “Lecture 15”), notes/jellium.qmd (B is for Bunching “Lecture 11”, Superconductivity “Lecture 12”) — purge or restore; also reword passages that assume Lieb–Liniger was covered
• Un-rendered math: “the phase of the $ _k$” (Elastic Chain §2.6, presumably ρ_k); “$ H_ = - f(t)y, $” (Response §2.6); raw \braket macro in Response Eq. 27; empty inline “$ $” in Superconductivity (“for small $ $”)
• Empty section: Superconductivity §2.2 “Discussion: Heisenberg’s Problem” — write it or delete the heading
• Truncated sentence in Elastic Chain §2.6: “To find the dependence on Δq” stops mid-thought; text missing before “This leads to the conclusion…”
• Past-papers page: embedded PDFs may not render for all users (page shows bare year headings without them) — consider adding plain download links alongside the embeds

1.2 Physics / math errors

1.2.1 Many Body Wavefunctions

• Integral limits: ∫_{k_F}^{k_F} → ∫_{−k_F}^{k_F} (density-matrix Check box)
• δ-function cusp condition off by factor 2 (integration over x_i misses the δ contribution from ∂²/∂x_j²): jump in ∂Ψ/∂x_i should be mcΨ

1.2.2 Quantum Hall Effect (Appendix)

• Schrödinger equation missing ψ on kinetic term
• Drift missing 1/m: v = (∇R + ∇S − qA)/m
• Many-body drift: sum over j ≠ i (not j ≠ 1); X_i/Y_i vs X_t/Y_t subscripts; “m” doubles as mass and Laughlin exponent — remark or relabel

1.2.3 Elastic Chain

• Appendix 3.2: sum should be n = 0 … j−1 (starts at 1/(1/2)), not n = 1 … j
• a_n definition: √(ZkL) needs |k_n|; “k” clashes with the spring constant
• [u_i, p_j] = iδ_ij stated in the classical section before quantization

1.2.4 Spin Models

• Sign slip: Δs = −⟨(1/N)Σ a†a⟩ combined with ⟨sᶻ⟩ = (−1)ʲ(s − Δs) gives depletion the wrong sign; also η_n left inside the integral
• Anderson tower: ΔS_A² = −𝒩/2, not −𝒩/8
• “goes to zero as 𝒩 → 0” → 𝒩 → ∞
• Rotor energies: l(l+1), not L(L+1)
• “zero quanta on the odd sites, but 2s quanta on the odd sites” — one should be even
• Projector: P_η = (1/N)Σ_n e^{−iηn}Tⁿ (exponent and 1/N)
• “des Cloiseaux” → des Cloizeaux (multiple places incl. figure caption; caption L=26 vs text N=26)
• Two-magnon appendix: missing “= 0” in the c₁/c₂ consistency equation; clarify units in “−4 not −8” remark

1.2.5 Lattice Models

• Validity condition inverted: N_sites ≳ (U/t)², not (t/U)²
• Fermi–Hubbard: U Σ_j N_{j,↑}N_{j,↓} (site index missing)
• i/j index mismatch in the opening Hamiltonian (recurs in Response: H_pert and T)
• t–J aside: unclosed parenthesis; “constant term in Equation 9” points at the wrong equation

1.2.6 Bose Gas

• LHY appendix factor of 2: prefactor should be V/(8π²m); E_reg = (8/15π²)(V/m)(mnU₀)^{5/2}, consistent with the (correct) 128/15√π in the E₀ formula below it
• Vortex energy functional: kinetic pieces should carry n (not n²); “U” → U₀
• “difference between N and N+1” → N and N−1
• “Onsager–Feynmann” → Feynman

1.2.7 Fermi Gas

• Galilean appendix: δk = mv, not v/m
• “f_{↑↑} = f_{↑↑}” → f_{↑↑} = f_{↓↓}
• Missing “+” between the two sums in the f_{↑↓} display
• z formula: U → U₀ for consistency

1.2.8 Superconductivity

• T_c formula: k_B T_c = (e^γ/π)Δ_BCS ≈ 0.567Δ with γ the Euler–Mascheroni constant (or redefine γ ≡ e^C); as written it reads as 0.18Δ
• Eq. 15 and two reprises: sum over p, summand indexed p′
• Eq. 14: product over i only with j = P(i); say φ(r) is the FT of v_p/u_p
• Appendix Eq. 20: U / U₀ / Ũ / U₀(p) notation inconsistent
• “|BCS⟩⟩” doubled ket

1.2.9 Response and Correlation

• Lamb model closed equation missing m’s (or γ = τ/(mc))
• e^{−γ/2mt} → e^{−γt/2m} (twice)
• m used as both mass and level index in the oscillator spectral representation
• β used for both inverse temperature and compressibility in one chapter — use κ

1.3 Typos & small style fixes (by chapter)

• Intro: “decribing”; “at the the Curie temperature”; “the end of story”
• Many Body Wavefunctions: “provide we ignore”; “verions”; “explict”; “single particle state |φ_α⟩ are”
• QHE: “obvserved”; “electon”; “antianaltyic”; “the m=1 an m>1”; “distrubution”; “A Laughlin Laughlin state” (caption); unclosed parenthesis after “∂_z̄“;”ρq₀” → qρ₀; “is then be given”; “intepret”
• Elastic Chain: “or original Hamiltonian”; “has lead to”; “way to to see”; “For a vector x_j = 1,…N” → f_j, j = 1,…,N; “∫₀^∞ φ(x)V(x−y)φ(y)” limits → [0, L]; backtick quote artifact in “the `size’ of the system”
• Spin Models: “we will at each state end up” → stage; “read of the dispersion” → off; “vanishes both η=0 and at”; “eigenergies”; missing full stop after “lower energy”
• A is for Annihilation: “movement of of”; “a_a” subscript; “on N−1 particle wavefunctions” → are; undefined shorthand B_{αβ} and Latin/Greek index mixing in the two-particle check; “Here we give simpler derivation”
• B is for Bunching: “from with Bose condensates”; “orginating”; “succintly”; “φ_r” → φ_R; bra ⟨N̄_R,N̄_L| has L/R swapped; bold ρ for scalar density; E_K and E_V used but never defined; “is the ground state is”; unclosed parenthesis at “(as we did in Lecture 4”
• Lattice Models: “in that in involves”; “neighouring”; “ocurrence”; “antiferromangetic”; missing full stop after “U/t→∞”; “the boundaries that we have have found”
• Bose Gas: “trignometric”; “digaonal”; “quantiatively”; “in erms of”; “There is far-reaching analogy”; unclosed parenthesis at Problem Set 2 link
• Fermi Gas: “Hamilonian”; “modifed”; “Discontinutity” (caption); “Thus we have Finally”; “In this limit we can take the |k|=|k′|=k_F”; unclosed parenthesis at Problem Set 3 link; stray period inside the 2×2 matrix
• Superconductivity: “which are are all”; “their theory become”; “though the BCS-BEC crossover” → through; “accross” (caption); “inhomogenous”; “as the form” → has; “isolated pair” → an isolated pair; lowercase fermi/bose/fermi-dirac scattered throughout; unclosed parenthesis at Lecture-12 reference; n_p vs n^P_p in Eq. 2
• Response and Correlation: “the dissipative dissipative part”; “far less significant that”; “corrsponding”; “quentity”; “eignestates”; missing full stop after “up or down in energy by ω”; double-bar rendering in “∂u/∂x‖”

1.4 Content updates (2016 → present)

• Add a lecture on topological band theory via the SSH model (Berry phase, winding number, edge states); optional second step: Kitaev chain paired with the Superconductivity lecture (already examined in 2019 — lecture material would close the loop)
• Add a short section on entanglement entropy (reduced density matrices of many-body states, area laws, why MPS/DMRG works) — hook: the density-matrix Note box in Ch. 1 and the N=26 exact-diagonalization section
• Refresh experimental anchors:
  • Fractional statistics measured 2020 (Bartolomei et al. anyon collider; Nakamura et al. interferometry); fractional QAH in moiré materials (2023)
  • Unitary gas: Bertsch parameter ξ = 0.376(4) (experiment) and ≈0.37 (modern QMC), replacing 0.44 ± 0.01
  • Cuprates/Hubbard: sharpen “still controversial” — pure 2D Hubbard at relevant doping is stripe-ordered, superconductivity restored by t′ (Simons-collaboration-era numerics)
  • Bose/Fermi–Hubbard: quantum gas microscope figure (AF correlations imaged 2017)
• Optional closing lecture on dynamics: thermalization/ETH, quantum simulation (Rydberg arrays); MBL only with 2020s caveats
• Note box connecting the QHE Monte-Carlo sampling appendix to neural quantum states / diffusion-model sampling (Carleo–Troyer 2017)
• Candidates for removal/compression to make room:
  • Fermi Gas §2.1 second-order f-function computation → state structure, quote F(φ), G(φ), keep effective-mass argument (~1 lecture freed)
  • Lattice Models WKB “Physical Motivation” → appendix or cut
  • Response damped-oscillator preamble → compress ~1/3 (Langevin detour → problem set)
  • Fourier review → handout
  • Stoner criterion → Check box
  • Keep Anderson tower and two-magnon appendices (already non-examinable)

1.5 Exam bank ideas (never-examined territory: QHE, impenetrable Bose gas, HOM/HBT, Hartree–Fock/Stoner, sum rules)

• LLL & bosonic Laughlin: analytic wavefunctions → harmonic potential in LLL → Ψ₂ zero-energy proof for δ interactions → quasihole degeneracy counting / plasma density
• Beam splitter & N-particle HOM: SU(2) mode rotation → odd-suppression law for N vs N → HBT connection → fermionic antibunching
• Girardeau mapping with a twist: mapping bookwork → n(k) for N=2 → twisted boundary conditions and n_s (boson vs fermion periodicity) → persistent currents
• Stoner meets Zeeman: HF bookwork → susceptibility with Stoner enhancement → d=2 first-order transition → discussion
• Feynman single mode: f-sum rule → S(q,ω) in Bogoliubov theory → Feynman bound ω(q) ≤ q²/2mS(q) → roton phenomenology
• Attractive Hubbard dimer as baby BCS: exact spectrum → Anderson pseudospins → BCS variational vs exact → number fluctuations
• SSH chain: two-band spectrum → winding number → explicit edge-state ansatz → Peierls argument (check overlap with 2018 Q3 part (a))
• Magnons in a field: FM with field bookwork → AFM doublet splitting ω(η) ± h → spin-flop softening → transverse magnetization (magnon BEC)
• Possible “modern classic” synthesis: Affleck–Lieb twist (2019 Q1) applied to Majumdar–Ghosh (2023 Q1) — twist argument consistent with a gap given two-fold degeneracy (hard)