Nonperturbative low-energy theory of Haldane's boson-coupled Anderson impurity model for mixed valence: the valence-fluctuation scale and long-time dynamics

In 1977 Haldane proposed a microscopic impurity model for the mixed-valence state of rare-earth materials: the Anderson impurity Hamiltonian augmented by a linear coupling of the impurity charge to a bosonic field representing electronic screening of the impurity by the nonresonant conduction channels of the host. The coupling is fixed by the Friedel sum rule, the bath spectrum is Ohmic, and the exchanged bosons generate an effective attraction $C$ between the two impurity electrons. Static mean-field theory of this model exhibits first-order phase boundaries whose neighborhood was identified with the slowly fluctuating mixed-valence state, but the paper itself showed these boundaries to be spurious and left the true low-temperature behavior open: it called for a renormalization-group treatment to evaluate the analogue of the Kondo temperature, questioned the validity of the Tomonaga boson mapping beyond Born order, and posed extensions to lattices, phonons, and temperature-dependent screening. Subsequent work solves the zero-coupling limit exactly; the fate of the mean-field boundaries away from particle-hole symmetry (smooth crossovers) and the strong-coupling quantum-phase-transition structure are established by numerical renormalization-group studies of the charge-coupled Bose-Fermi Anderson model, the direct descendant of the 1977 Hamiltonian; and the lattice, phonon, and temperature-dependent-bath questions have been absorbed into later frameworks. The surviving open core is the characteristic valence-fluctuation temperature and the long-time valence dynamics in the weak-coupling mixed-valence regime of the Friedel-normalized model.

Background

Rare-earth compounds frequently contain ions whose f shell lies near degeneracy between two configurations, $f^n$ and $f^{n-1}$, with the difference electron shared with the conduction band. This is the mixed-valence (or intermediate-valence) state [Varma (1976); Lawrence, Riseborough and Parks (1981)]: because f orbitals are compact and their occupancy strongly perturbs the electrostatic environment, the ionic valence fluctuates in time, and different probes see it differently — fast core-level spectroscopy resolves integral valences, while slower probes such as Mössbauer isomer shifts see a time-averaged valence. A canonical example is CeAl$_3$, whose resistivity passes through a maximum near 37 K and falls into a coherent regime below roughly 10 K. Whether the effect is primarily electronic or requires lattice (phonon) participation was a central question of the 1970s, and the dilute-impurity limit of the question is still experimentally live [Zonno et al. (2024)].

The standard microscopic description of a single valence-fluctuating or magnetic impurity is the Anderson impurity model [Anderson (1961)]: a localized orbital of energy $E$ and on-site Coulomb repulsion $U$ hybridizes with a conduction band, with resonance width $U_c$ (the hybridization-induced breadth of the impurity level). In the local-moment regime the residual exchange interaction screens the impurity spin below the Kondo temperature $T_K$ — an exponentially small scale generated by renormalization — while away from particle-hole symmetry (the condition $E = -U/2$, under which adding and removing an impurity electron are equivalent) charge, i.e. valence, fluctuations dominate. The plain model was solved nonperturbatively by scaling theory [Haldane (1978)], which characterizes its mixed-valence regime by the criterion $|E_d^*| ≲ Δ$ ($E_d^*$ the renormalized level position, Δ the renormalized resonance half-width), and by the numerical renormalization group (NRG) [Krishna-murthy, Wilkins and Wilson (1980); NRG is Wilson's iterative method that resolves a quantum impurity problem down to exponentially small energy scales], which found smooth crossovers and no phase transitions for both the symmetric and asymmetric cases.

The source paper [Haldane (1977)] extends this picture to include electronic screening of the impurity charge. In a host whose point symmetry separates the resonant channel (angular momentum $l = 3$, carrying the f-like impurity orbital) from the screening channels ($l = 2$, the d-like band orbitals), the slow charge response of the screening electrons is represented by Tomonaga bosons — the bosonic representation of the particle-hole excitations of a Fermi gas with linearized spectrum [Tomonaga (1950)]. The Anderson orthogonality catastrophe (the power-law response of a Fermi gas to a static potential switched on at the impurity) is thereby built in, and the coupling constant $g$ is normalized so that the bosons reproduce the Friedel sum rule (the exact statement that the charge displaced by an impurity is counted by the Fermi-level scattering phase shift), giving $2gσ(ε_F) = 1$ in Born approximation, where $σ(ε_F)$ is the conduction-band density of states projected onto the impurity site. The resulting bath spectral function is Ohmic — proportional to $ω$, the hallmark of electron-hole pairs in a metal: $f(ω) = 2εω$ for $|ω| < D$, with slope $ε ≈ 1/(4d_0)$, where $D$ is the effective screening-band bandwidth and $d_0$ the multiplicity (number of degenerate screening orbitals) of that band — for example $d_0 = 5$ for a d band, giving $ε = 0.05$. The exchange of these bosons between the two impurity electrons produces an effective attraction $C = 4εD$. Static (Hartree-Fock, i.e. self-consistent single-particle) mean-field theory of this Hamiltonian [Haldane (1977), companion paper] yields a phase diagram with a magnetic region and first- and second-order phase boundaries and identifies the mixed-valence regime $U_c ≪ C ≪ U$ with slow valence fluctuations concentrated near the first-order boundaries. However, the paper itself shows that all these transitions are spurious (mean-field artifacts) except at the singular point $U_c = 0$, $T = 0$, that perturbation theory is infrared-divergent precisely in this regime, and that the Tomonaga solution of the screening response is a Born-order (lowest-order) truncation whose validity for a scatterer with internal degrees of freedom and a degenerate ground state is questionable — its naive application to the Kondo model eliminates the Kondo effect altogether. The paper therefore called for a renormalization-group treatment of the full boson-coupled model to evaluate the analogue of the Kondo timescale.

The model class reappeared in later literature. The Anderson impurity coupled to a single local phonon mode (the Anderson-Holstein model) was solved by NRG across all coupling regimes [Hewson and Meyer (2002)]; the Bose-Fermi Kondo model — a spin coupled to both a fermionic and a bosonic bath — by NRG [Glossop and Ingersent (2005)]; and the charge-coupled Bose-Fermi Anderson model, the direct descendant of the 1977 Hamiltonian in which the impurity charge couples to a bosonic bath of tunable exponent, was studied by NRG [Cheng, Glossop and Ingersent (2009)]. That study found that the boson coupling renormalizes the effective on-site interaction from repulsive toward attractive, driving a crossover from the usual spin-Kondo screening to charge-Kondo screening (the screening of a two-fold charge degeneracy, realized when the effective interaction becomes attractive); genuine zero-temperature quantum phase transitions exist only at particle-hole symmetry — continuous, in the universality class of the spin-boson model (a two-level system coupled to a dissipative bath), for sub-Ohmic bath exponents ($s < 1$, the bath spectrum scaling as $ω^s$), and of Kosterlitz-Thouless character (an infinite-order transition approached through an essential singularity rather than a closing gap) for the Ohmic bath $s = 1$ relevant here — while away from symmetry the transition is replaced by a smooth crossover. The methodological lineage includes poor-man's scaling (the iterative elimination of the highest-energy band states, yielding flow equations for the low-energy couplings) and the first NRG schemes for impurities coupled to bosonic baths [Hewson (1981); Evangelou and Hewson (1982)], the local-polaron (a local electron dressed by the bosonic modes it excites) and local Fermi-liquid treatments of intermediate valence [Hewson and Newns (1979, 1980); Newns and Hewson (1980)], polaronic stabilization of mixed valence [Khomskii (1978)], and the self-consistent reappearance of such impurity models in extended dynamical mean-field theory — a lattice technique that maps an extended Hubbard model onto a self-consistent impurity problem with both fermionic and bosonic baths [Smith and Si (2000)] — with early finite-cell lattice work in [Jullien and Martin (1982)].

Problem Statement

Determine the true low-temperature, long-time behavior of Haldane's electronically screened Anderson impurity model: the spinful Anderson impurity Hamiltonian whose impurity charge couples linearly to an Ohmic bosonic bath representing the screening response of the nonresonant conduction channels, with the coupling normalized by the Friedel sum rule ($2gσ(ε_F) = 1$ in Born approximation, where $g$ is the coupling constant and $σ(ε_F)$ the conduction-band density of states projected onto the impurity site), bath spectrum $f(ω) = 2εω$ for $|ω| < D$ with slope $ε ≈ 1/(4d_0)$ ($D$ the effective screening-band bandwidth, $d_0$ the multiplicity, i.e. the number of degenerate screening orbitals, of that band), and effective attraction $C = 4εD$. Determine, in the weak-coupling mixed-valence regime $U_c ≪ C ≪ U$ with the impurity level near valence degeneracy:

1. The characteristic valence-fluctuation temperature — the analogue of the Kondo temperature for this model — as a function of the microscopic parameters (level position $E$, on-site repulsion $U$, resonance width $U_c$, attraction $C$, cutoff $D$). In particular: does the screening-mediated attraction generate a parametrically small (for example, exponentially small) fluctuation scale, or only power-law renormalization of the plain-Anderson valence-fluctuation scale?

2. The valence-fluctuation correlation function and its long-time decay throughout the mixed-valence regime, including the neighborhoods of the washed-out mean-field first-order boundaries with which the source identified the mixed-valence state — the interplay of charge and spin fluctuations that static mean-field theory cannot reproduce.

The behavior to be determined is the nonperturbative one that static Hartree-Fock theory, Born-order perturbation theory, and the Tomonaga solution of the screening response cannot supply. The lattice-coherence, phonon, and temperature-dependent-bath extensions posed alongside these questions in the source paper lie outside the surviving core.

References

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