Two-Impurity Fano-Anderson Model

• The two-impurity Fano-Anderson model is a framework describing two localized impurity levels hybridized with continua that leads to interference effects like Fano antiresonance and bonding–antibonding splitting.
• It details how direct inter-impurity tunneling and background reservoir coupling shape transport regimes, producing conductance evolutions from antiresonance to split-peak patterns.
• In correlated realizations, competing Kondo screening and inter-impurity singlet formation induce quantum critical behavior and emergent many-body phenomena.

The two-impurity Fano-Anderson model denotes a family of impurity Hamiltonians in which two localized levels are hybridized with one or more continua, while the measured spectra or transport coefficients are governed not only by resonance formation but also by interference between distinct propagation channels. In its noninteracting realizations the model reduces to localized orbitals coupled to continuum channels with diagonal and off-diagonal hybridization amplitudes; in correlated realizations it becomes a two-impurity Anderson system with onsite repulsion, direct inter-impurity tunneling or exchange, and, in transport geometries, an additional background transmission path. Across these variants, the characteristic phenomena are parity-resolved hybridization, Fano antiresonances, bonding–antibonding splitting, bound states in the continuum, underscreened Kondo behavior, and several forms of impurity quantum criticality (Hamad et al., 2012, Grez et al., 2022).

1. Hamiltonian structure and parity representation

A transport-oriented correlated realization is given by

$H=H_{\rm imp}+H_{\rm hyb}+H_{\rm leads},$

with

$$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$

$$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$

and semi-infinite tight-binding leads

$$H_{\rm leads}=t\sum_{j=L,R}\sum_{i=1,\sigma}^\infty \left(c^\dagger_{j,i\sigma}c_{j,i+1\sigma}+\text{H.c.}\right).$$

In this formulation, $t_{\alpha\beta}$ is direct inter-impurity hopping and $t_{LR}$ is direct lead-lead tunneling. The two relevant current paths are therefore an impurity-assisted resonant path and a direct reservoir-to-reservoir path, which is the defining Fano-interference structure in the two-Co-atom STM interpretation of Bork et al. (Hamad et al., 2012).

A more standard two-impurity Anderson formulation writes

$H_{\mathrm{TIAM}}=H_c+H_D+H_I,$

with two localized orbitals $d_{1,2}$, local Coulomb repulsion $U$, direct tunneling $t$, and bath coupling $$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$0. The natural reorganization is into even and odd parity orbitals,

$$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$1

which diagonalize the impurity subspace and render the bath coupling channel dependent (Lechtenberg et al., 2016).

In the even/odd basis, the low-energy structure is controlled by the hybridization functions

$$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$2

$$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$3

This parity decomposition is central because the entire impurity problem can then be re-expressed as two effective channels with different low-energy spectral weights, phase shifts, and screening tendencies (Eickhoff et al., 2018).

2. Fano interference and transport line shapes

In the STM-motivated two-path geometry, the transport evolution with decreasing interatomic distance is a sequence of three regimes: a Fano antiresonance at large separation, a narrow-to-broadened resonance as the dip is suppressed, and a split conductance peak at small separation. The model attributes this evolution to the simultaneous growth of direct reservoir tunneling and direct impurity-impurity hopping,

$$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$4

with fitted parameters $$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$5, $$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$6, $$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$7, and $$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$8. Because $$H_{\rm imp}=\sum_{i=\alpha,\beta;\sigma}\left(\epsilon_i n_{i\sigma}+\frac{U}{2}n_{i\sigma}n_{i\bar\sigma}\right),$$9, the impurity-impurity channel strengthens faster than the reservoir-reservoir channel as the atoms approach, and this reweights the interference pattern from an antiresonant regime toward a split-resonance regime (Hamad et al., 2012).

Within the finite-$$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$0 slave-boson mean-field approximation, the differential conductance under electron-hole symmetry is written as

$$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$1

where $$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$2 is the many-body propagator between leads in the presence of bias. For the low-bias range of the experiment, $$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$3 is approximated by its equilibrium form because it is nearly bias independent for $$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$4 smaller than a few $$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$5. The same calculations reproduce the observed increase of total conductance $$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$6 as the atoms approach and yield a phase difference of zero between the two current paths for all $$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$7, consistent with constructive interference in the closed geometry considered (Hamad et al., 2012).

A conceptually important control case is the single-impurity limit. Varying the same tunneling parameters with only one Co atom does not produce the dip $$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$8 peak $$H_{\rm hyb}=\sum_\sigma \Big( t_L c^\dagger_{L,1\sigma}c_{\alpha\sigma} +t_R c^\dagger_{R,1\sigma}c_{\beta\sigma} +t_{\alpha\beta}c^\dagger_{\alpha\sigma}c_{\beta\sigma} +t_{LR}c^\dagger_{L,1\sigma}c_{R,1\sigma} +\text{H.c.}\Big),$$9 split-peak evolution; instead the conductance remains an asymmetric antiresonance whose width does not shrink into a peak and then split. This establishes that the observed sequence is not a generic Fano effect of one resonant level plus one background path, but a genuinely two-impurity phenomenon (Hamad et al., 2012).

3. Correlations, Kondo screening, and effective inter-impurity coupling

Once onsite repulsion is retained explicitly, the two-impurity Fano-Anderson problem inherits the central competition of the two-impurity Anderson model: Kondo screening of each localized spin by the continuum versus direct or indirect formation of an inter-impurity singlet. In the formulation with explicit antiferromagnetic exchange,

$$H_{\rm leads}=t\sum_{j=L,R}\sum_{i=1,\sigma}^\infty \left(c^\dagger_{j,i\sigma}c_{j,i+1\sigma}+\text{H.c.}\right).$$0

strong $$H_{\rm leads}=t\sum_{j=L,R}\sum_{i=1,\sigma}^\infty \left(c^\dagger_{j,i\sigma}c_{j,i+1\sigma}+\text{H.c.}\right).$$1 suppresses the Kondo effect because the impurity spins bind to each other and are no longer available for individual screening by the bath. At the zero-field quantum critical point $$H_{\rm leads}=t\sum_{j=L,R}\sum_{i=1,\sigma}^\infty \left(c^\dagger_{j,i\sigma}c_{j,i+1\sigma}+\text{H.c.}\right).$$2, the transition between the Kondo-resonant regime and the inter-impurity singlet regime is continuous, the staggered susceptibility diverges, and the low-energy scale vanishes as

$$H_{\rm leads}=t\sum_{j=L,R}\sum_{i=1,\sigma}^\infty \left(c^\dagger_{j,i\sigma}c_{j,i+1\sigma}+\text{H.c.}\right).$$3

The low-energy spectrum in the singlet regime exhibits a pseudogap-like suppression associated with quasiparticle localization (Zhu et al., 2010).

In the two-path transport model, however, a split zero-bias anomaly is not by itself evidence of Kondo destruction. The conductance splitting is interpreted as hybridization of the two Kondo resonances associated with the two impurities, producing bonding and antibonding many-body combinations. The renormalized impurity level $$H_{\rm leads}=t\sum_{j=L,R}\sum_{i=1,\sigma}^\infty \left(c^\dagger_{j,i\sigma}c_{j,i+1\sigma}+\text{H.c.}\right).$$4 remains pinned near the Fermi energy over most of the parameter range where splitting is observed, and the logarithmic-discretization embedded-cluster approximation finds that the spin correlation between each impurity and its adjacent lead remains stronger than the impurity-impurity correlation in those regimes. The theoretical claim is therefore that a split conductance peak can coexist with a Kondo-screened ground state (Hamad et al., 2012).

A complementary low-energy description identifies an emergent local tunneling term generated by the antisymmetric part of the parity-dependent hybridization. In that mapping,

$$H_{\rm leads}=t\sum_{j=L,R}\sum_{i=1,\sigma}^\infty \left(c^\dagger_{j,i\sigma}c_{j,i+1\sigma}+\text{H.c.}\right).$$5

and, for particle-hole symmetric impurities,

$$H_{\rm leads}=t\sum_{j=L,R}\sum_{i=1,\sigma}^\infty \left(c^\dagger_{j,i\sigma}c_{j,i+1\sigma}+\text{H.c.}\right).$$6