Modern Physics · Advanced Topics

Many-Body Quantum Physics

Interacting quantum systems of many particles exhibit emergent phenomena — superconductivity, the Mott transition, the fractional quantum Hall effect — that cannot be understood from single-particle physics. Second quantization, Green's functions, and diagrammatic methods are the essential tools.

PrerequisitesQuantummechanics(Ch.QM)⋅Statisticalmechanics(Ch.SM)⋅Solid−statephysics(Ch.SSQuantum mechanics (Ch. QM) \cdot Statistical mechanics (Ch. SM) \cdot Solid-state physics (Ch. SS ⋅QFTbasics(Ch.QFT\cdot QFT basics (Ch. QFT
Learning Goals
  • Write the electron-electron Coulomb interaction in second-quantized form and identify Hartree and Fock contributions.
  • Derive the RPA dielectric function and the Thomas-Fermi screening length from the static Lindhard function.
  • Explain the Hubbard model and the Mott insulator transition in terms of U/t ratio.
  • InterpretthespectralfunctionA(k,ω)andthequasiparticleweightZkinFermiliquidthInterpret the spectral function A(k,\omega) and the quasiparticle weight Z_{k} in Fermi liquid theory.
  • StatetheLaughlinwavefunctionforfillingν=1/mandexplainfractionalchargeusingthState the Laughlin wavefunction for filling \nu = 1/m and explain fractional charge using the plasma analogy.

MB.1 Second Quantization

For N identical particles, the Fock space formalism is more natural than first quantization. Creation and annihilation operators:

Bosons: [b_k, b†_(k')] = δ_(kk'), [b_k, b_(k')] = 0.Fermions: {c_k, c†_(k')} = δ_(kk'), {c_k, c_(k')}= 0 (anticommutators). The Pauli exclusion principle is automatic: (c†_k)² = 0 (can't create two fermions in same state).

The Hamiltonian in second quantization:

H=∑kσεkckσ†ckσ+12∑k,k′,q,σ,σ′Vqck+q,σ†ck′−q,σ′†ck′,σ′ck,σH = \sum_{k\sigma} \varepsilon_k c^\dagger_{k\sigma} c_{k\sigma} + \frac{1}{2} \sum_{k,k',q,\sigma,\sigma'} V_{q} c^\dagger_{k+q,\sigma} c^\dagger_{k'-q,\sigma'} c_{k',\sigma'} c_{k,\sigma}(MB.1)

The first term is kinetic energy; the second is the two-body interaction in momentum space. For the Coulomb interaction V_q = e²/(ε₀ q²) in 3D.

MB.2 Hartree-Fock Theory

The simplest variational ansatz: a Slater determinant |ΦHF⟩ = Π_(k∈occ) c†_k |0⟩. Variation gives the Hartree-Fock equations:

[−ℏ2∇2/(2m)+VH(r)+VX]ψk(r)=εkψk(r)[-\hbar^{2}\nabla^{2}/(2m) + V_{H}(r) + V_{X}] \psi_k(r) = \varepsilon_k \psi_k(r)(MB.2)

where V_H = ∫ n(r') e²/(4πε₀|r−r'|) d³r' (Hartree, electrostatic) and V_X is the non-local Fock exchange operator. For the homogeneous electron gas: exchange energy per electron E_X/N = −(3e²)/(4π)(3/π)^(1/3) r_s^(−1) (in Rydberg units), where r_s = (3/(4πn))^(1/3) a₀ is the Wigner-Seitz radius.

Definition MB.1 — Hubbard Model
The single-band Hubbard model captures thecompetitionbetweenkineticenergy(hoppingt)andon−siteCoulombrepulsion(U):H=the competition between kinetic energy (hopping t) and on-site Coulomb repulsion (U): H =i↑ni↓Athalf−filling(oneelectronpersite):forU/t≫1,hoppingissuppressed→_{i↑} n_{i↓} At half-filling (one electron per site): for U/t ≫ 1, hopping is suppressed \to Mott insulator. For U/t ≪ 1: metallic (Fermi liquid). The Mott transition at intermediate U/t is a paradigmatic strongly correlated problem — not captured by HF.

MB.3 Green's Functions and Self-Energy

The single-particle Green's function encodes excitation properties:

G(k,ω)=1/(ω−εk−∑(k,ω)+iη)(Dysonequation:G=G0+G0∑G)G(k, \omega) = 1/(\omega - \varepsilon_k - \sum(k,\omega) + i\eta) \qquad (Dyson equation: G = G_{0} + G_{0} \sum G)(MB.3)

The self-energy Σ(k, ω) encodes all interaction effects. The spectral function A(k, ω) = −(1/π) Im G(k, ω+iη) gives the probability of creating an excitation with momentum k and energy ω. For a Fermi liquid: A(k, ω) = Z_k δ(ω − ε_k*) + incoherent background, where Z_k is the quasiparticle weight.

Fermi liquid theory (Landau): despite interactions, the low-energy excitations of a metallic system are quasiparticles — dressed electrons with renormalized mass m* and finite lifetime τ ∝ (ε − ε_F)^(−2). This justifies band theory for metals. Fails when: Mott insulator (strong U), non-Fermi liquids (1D Luttinger liquid, strange metal in cuprates, heavy fermions).

Example MB.1 — RPA Screening and the Plasmon

Intherandomphaseapproximation(RPA),computethedielectricfunctionε(q,ω)fortheeIn the random phase approximation (RPA), compute the dielectric function \varepsilon(q, \omega) for the electron gas. Find the plasmon dispersion.

Bare polarization:Lindhardfunction:∏0(q,ω)=2∑k(fk+q−fk)/(ω−εk+q+εk+iη).Atq→0,ω→0Lindhard function: \prod_{0}(q, \omega) = 2\sum_k (f_{k+q} - f_{k})/(\omega - \varepsilon_{k+q} + \varepsilon_k + i\eta). At q\to0, \omega\to0: ∏0→−N(0)(densityofstatesatEF).Atq→0,finiteω:∏0→−neq2/(mω2)(Thomas−Fermi\prod_{0} \to -N(0) (density of states at E_{F}). At q\to0, finite \omega: \prod_{0} \to -n_{e} q^{2}/(m\omega^{2}) (Thomas-Fermiimit).
RPA dielectric function:ε(q,ω)=1−Vq∏0(q,ω)whereVq=e2/(ε0q2).ε(q,ω)=1+(e2/(ε0q2))N(0)atstatic\varepsilon(q,\omega) = 1 - V_{q} \prod_{0}(q,\omega) where V_{q} = e^{2}/(\varepsilon_{0} q^{2}). \varepsilon(q,\omega) = 1 + (e^{2}/(\varepsilon_{0} q^{2})) N(0) at static long−wavelength→Thomas−Fermiscreening:ε=1+qTF2/q2whereqTF2=e2N(0)/ε0long-wavelength \to Thomas-Fermi screening: \varepsilon = 1 + q_{TF}^{2}/q^{2} where q_{TF}^{2} = e^{2} N(0)/\varepsilon_{0}
Plasmon:PolesofGscreenedatε(q,ω)=0.Atq→0:ε(q,ω)=1−ωp2/ω2whereωp2=nee2/(ε0mPoles of G_{screened} at \varepsilon(q,\omega) = 0. At q\to0: \varepsilon(q,\omega) = 1 - \omega_p^{2}/\omega^{2} where \omega_p^{2} = n_{e} e^{2}/(\varepsilon_{0} m. Plasmonfrequency:ω=ωp(independentofqatlongwavelength).Formetals:ωp 10eVPlasmon frequency: \omega = \omega_p (independent of q at long wavelength). For metals: \omega_p ~ 10 eVUVplasmaedge—whymetalsreflectbelowωpandtransmitaboveforω>ωp).DispersionUV plasma edge — why metals reflect below \omega_p and transmit above for \omega > \omega_p). Dispersion atsmallq:ω=ωp+(3vF2/(10ωp))q2+O(q4).Theplasmonisagappedcollectivemode(at small q: \omega = \omega_p + (3v_F^{2}/(10\omega_p))q^{2} + O(q^{4}). The plasmon is a gapped collective mode (oldstone if long-range forces were absent).
Screening length:Screenedpotential:Vscr=Vq/ε(q)→V(r)=(e2/r)e−qTFr(Yukawa).Thomas−FermiscScreened potential: V_{scr} = V_{q}/\varepsilon(q) \to V(r) = (e^{2}/r) e^{-q_{TF} r} (Yukawa). Thomas-Fermi screeninglength:λTF=1/qTF.ForCu:ne=8.4×1028m−3,N(0)=(3/2)n/EF≈1.51×1047Jreening length: \lambda_TF = 1/q_{TF}. For Cu: n_{e} = 8.4\times10^{28} m^{-3}, N(0) = (3/2)n/E_{F} \approx 1.51\times10^{47} J⁻¹m−3.λTF≈0.55A˚—veryshortscreeninginmetalsm^{-3}. \lambda_TF \approx 0.55 Å — very short screening in metals

MB.4 Diagrammatic Perturbation Theory

The self-energy and correlation functions can be expanded in Feynman diagrams. At the Hartree-Fock level: Σ_HF = Σ_direct + Σ_exchange. Beyond HF: the GW approximation Σ = iG W (W = screened Coulomb interaction from RPA) is the standard first-principles method for band gaps in semiconductors.

Linked-cluster theorem: only connected diagrams contribute to extensive quantities (free energy, self-energy). Unlinked (vacuum bubble) diagrams cancel between numerator and denominator in perturbation theory — this is what makes many-body perturbation theory tractable.

Quantum Monte Carlo (QMC): stochastic evaluation of the many-body path integral. Variational MC (VMC): optimize a trial wavefunction (Jastrow factor × Slater determinant). Diffusion MC (DMC): project to ground state via e^(−τH). Sign problem for fermions (cancellations between ± signs) is the fundamental bottleneck. Fixed-node approximation: constrain to same nodal surface as trial function.

MB.5 Correlated Phases

Mott insulator → superfluid transition(Bose-Hubbard model in optical lattice): at integer filling, for U/t > z×5.83: Mott lobes in the phase diagram. Jaksch et al. (1998) predicted; Greiner et al. (2002) observed with ultracold bosons in an optical lattice — quantum phase transition at T = 0.

Fractional quantum Hall effect(FQHE): at ν = 1/3 (and other fractions), the ground state is the Laughlin wavefunction Ψ = Π_(i<j) (z_i − z_j)^m e^(−Σ|z_k|²/4ℓ²) (m = 3 for ν=1/3). Excitations carry fractional charge e/3 and obey fractional statistics (anyons). Topological order — no local order parameter.

High-T_c superconductors: cuprates (La₂CuO₄ doped with Sr) have T_c up to 138 K. Undoped: Mott insulator (half-filled Cu 3d band, strong U). Doped: d-wave superconductivity (gap Δ_k = Δ₀(cos k_x − cos k_y)), pseudogap phase, strange metal. Not fully explained.

Definition MB.2 — Common Traps
  • Mean field replaces interactions with averages: it can miss correlations and fluctuations.
  • Quasiparticles are emergent excitations: they are not always bare particles.
  • Green's functions encode response: poles, residues, and spectral weight all matter.
  • Strong coupling often needs new methods: weak perturbation can fail qualitatively.
Exercises — MB.1–MB.5 Many-Body Physics
1.
Write the electron-electron Coulomb interaction in second-quantized form. Identify the Hartree and Fock (exchange) contributions and give their physical interpretations.
eV
Straightforward
2.
Derive the Thomas-Fermi screening length from the static Lindhard function at long wavelengths. What are Friedel oscillations and why do they have wavevector 2k_
Å
Intermediate
3.ExplainquasiparticleweightZkandtheFermiliquidlifetimeτ∝(ε−εF)−2.HowdoesExplain quasiparticle weight Z_{k} and the Fermi liquid lifetime \tau \propto (\varepsilon-\varepsilon_F)^{-2}. How does ARPES measure the spectral function, and what signals a non-Fermi liquid?
Intermediate
4.WritedowntheLaughlinwavefunctionforfillingν=1/m.ShowitdescribesfractionalchaWrite down the Laughlin wavefunction for filling \nu = 1/m. Show it describes fractional charge e/m quasiparticles using the plasma analogy. What is the energy gap?
Challenging
Key Takeaways
  • Fermionoperators:ck,ck′†=δkk′(anticommutators).(c†)2=0encodesPauliexcFermion operators: {c_{k}, c^\dagger_{k'}} = \delta_{kk'} (anticommutators). (c^\dagger)^{2} = 0 encodes Pauli exclusion.
  • HF theory: Slater determinant ansatz. Hartree (direct Coulomb) + Fock (exchange). Misses correlations.
  • Hubbardmodel:kinetictvson−siteU.U/t≫1→Mottinsulator.Half−filling+largeU→Hubbard model: kinetic t vs on-site U. U/t ≫ 1 \to Mott insulator. Half-filling + large U \to antiferromagnetism.
  • Self−energy∑(k,ω):DysoneqG=G0+G0∑G.Fermiliquid:sharpquasiparticle,lifetimeSelf-energy \sum(k,\omega): Dyson eq G = G_{0} + G_{0} \sum G. Fermi liquid: sharp quasiparticle, lifetime ~ (ε−εF)2\varepsilon-\varepsilon_F)^{2}
  • RPAplasmon:collectiveoscillationatωp=(ne2/(ε0m)).Thomas−FermiscreeningλTF=1RPA plasmon: collective oscillation at \omega_p = \sqrt(ne^{2}/(\varepsilon_{0}m)). Thomas-Fermi screening \lambda_TF = 1/qTFq_{TF}
  • FQHELaughlinstateν=1/m:fractionalchargee/m,anyonicstatistics.Topological—nolFQHE Laughlin state \nu = 1/m: fractional charge e/m, anyonic statistics. Topological — no local order parameter.