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Quantum Impurity Models and Local Moments

A quantum-impurity problem consists of a finite local Hilbert space coupled to one or more gapless or structured baths. Its physics is fixed not by the word “impurity” but by the local states, bath spectral functions, channel and symmetry content, couplings, and hierarchy among charge, spin, temperature, and bandwidth scales. A local moment exists when charge-changing states are energetically suppressed while a degenerate spin or multiplet remains active; it is not guaranteed by a large instantaneous value of ⟨S2⟩\langle\mathbf S^2\rangle alone.

Required background. Fermi-surface kinematics supplies the bath density of states. Dyson equations supplies the hybridization self-energy.

The impurity contract

Section titled “The impurity contract”

Write every model as

H=Hbath+Hloc+Hmix+Hint.H=H_{\mathrm{bath}}+H_{\mathrm{loc}}+H_{\mathrm{mix}}+H_{\mathrm{int}}.

HlocH_{\mathrm{loc}} declares the finite set of impurity states and their charges or representations. HbathH_{\mathrm{bath}} declares dispersion, chemical potential, density of states, gap or pseudogap, and ultraviolet cutoff. HmixH_{\mathrm{mix}} changes local charge by exchanging particles with the bath; exchange or density couplings may instead preserve it.

For a single orbital hybridized linearly with bath modes,

Δ(z)=∑k∣Vk∣2z−εk,Γ(ω)=−Im⁡Δ(ω+i0).\Delta(z)=\sum_k\frac{\lvert V_k\rvert^2}{z-\varepsilon_k}, \qquad \Gamma(\omega)=-\operatorname{Im}\Delta(\omega+i0).

The function Δ(z)\Delta(z), not a bare number VV, is the bath seen by the impurity. A flat metallic bath has Γ(0)>0\Gamma(0)>0; a pseudogap bath may have Γ(ω)∝∣ω∣r\Gamma(\omega)\propto\lvert\omega\rvert^r and a different phase diagram. Two microscopic leads couple as two screening channels only if two orthogonal bath combinations remain coupled after the lead rotation. Often an even combination couples and the odd one decouples, leaving a single channel.

Hewson 1993, chs. 2–3 develops the metallic Anderson and Kondo impurity definitions; Vojta 2006, §§2–4 shows how structured baths and relevant perturbations change the fixed-point problem.

Local-moment criteria

Section titled “Local-moment criteria”

Suppose a charge-QQ multiplet is the atomic ground sector. Let

E+=E(Q+1)−E(Q),E−=E(Q−1)−E(Q).E_+=E(Q+1)-E(Q), \qquad E_-=E(Q-1)-E(Q).

A controlled local-moment regime requires

E+,E−≫Γ,T,ω,E_+,E_-\gg \Gamma,T,\omega,

over the observables of interest, together with a protected degeneracy within the QQ sector. Virtual charge fluctuations then generate exchange and potential scattering. If a charge gap becomes comparable to Γ\Gamma, the system is in mixed valence and a pure spin model is not a faithful reduction.

Useful impurity observables subtract the bath without the impurity. For example,

Simp=Stotal−Sbath,χimp=χtotal−χbath.S_{\mathrm{imp}}=S_{\mathrm{total}}-S_{\mathrm{bath}}, \qquad \chi_{\mathrm{imp}}=\chi_{\mathrm{total}}-\chi_{\mathrm{bath}}.

The subtraction can include displaced conduction charge and need not equal the expectation value of a strictly local operator. At temperatures between the charge gaps and the eventual screening scale, a spin-SS moment contributes approximately log⁡(2S+1)\log(2S+1) entropy and a Curie susceptibility, with convention-dependent factors of gμBg\mu_B.

Symmetry and channel classification

Section titled “Symmetry and channel classification”

Before selecting a solver, record:

  • impurity representation and degeneracy, including crystal-field and spin–orbit splittings;
  • the number of conserved screening channels after all basis rotations;
  • spin, charge, time-reversal, particle–hole, and point-group symmetries;
  • metallic, gapped, pseudogapped, superconducting, Luttinger-liquid, or bosonic bath structure;
  • relevant perturbations such as channel anisotropy, local field, potential scattering, and dissipation; and
  • the observable and scale window.

This information distinguishes Kondo screening, level broadening, dissipative localization, and boundary criticality. It also prevents a single-impurity crossover from being promoted to lattice coherence or magnetic order.

The chapter structure follows these declared inputs through matching, flow, fixed point, solver, and observable.

A declared local Hilbert space and bath hybridization determine charge versus moment regimes, after which Anderson-to-Kondo matching, RG, strong-coupling phase shifts, solvers, and observables become meaningful.

Impurity physics begins with the local state space, bath function, channels, symmetry, and scale hierarchy. Solver or Kondo language enters only after those inputs establish the relevant regime. Original schematic, not to scale.

The complete impurity claim test matrix lists the required observable and failure test for each regime.

Exercise

Section titled “Exercise”

Count channels. A dot couples to two identical noninteracting leads through amplitudes VLV_L and VRV_R to the same orbital. Show that only one lead combination hybridizes.

Solution

For the convention Hmix=∑ak(Vacak†d+h.c.)H_{\mathrm{mix}}=\sum_{ak}(V_a c_{ak}^\dagger d+\mathrm{h.c.}), define cek=(VL∗cLk+VR∗cRk)/Vc_{ek}=(V_L^\ast c_{Lk}+V_R^\ast c_{Rk})/V and cok=(−VRcLk+VLcRk)/Vc_{ok}=(-V_Rc_{Lk}+V_Lc_{Rk})/V, where V2=∣VL∣2+∣VR∣2V^2=\lvert V_L\rvert^2+\lvert V_R\rvert^2. This is a unitary rotation, Hmix=V∑k(cek†d+h.c.)H_{\mathrm{mix}}=V\sum_k(c_{ek}^\dagger d+\mathrm{h.c.}), and the odd field cancels. The model therefore has one screening channel despite two physical leads, unless another conserved orbital or lead structure prevents this rotation.

References

Section titled “References”