Haldane's open questions on the asymmetric Anderson impurity model: universal mixed-valence crossover functions, the crossover constants, and crossover-region physics beyond the scaling equations

In his 1978 scaling theory of the asymmetric Anderson impurity model, F. D. M. Haldane showed that for temperatures well below the on-site repulsion ($T\ll U$) the model's properties are universal functions of the scaling invariants $\Delta$ (the hybridization width) and $E_d^*=E_d+(\Delta/\pi)\ln(W_0/\Delta)$ (the renormalized level position, with $W_0$ the smaller of the conduction-band width and $U$), identified $|E_d^*|\lesssim\Delta$ as the criterion for a mixed-valence ground state, and described the crossovers between the empty-orbital, mixed-valence, and local-moment regimes — including a Kondo scale $T_K$ of order $\Delta\exp(\pi E_d^*/2\Delta)$ whose preexponential factor is left unevaluated. He left open: (i) determination of the universal two-parameter crossover functions for the impurity susceptibility, linear specific-heat coefficient, and level occupation $\langle n_d\rangle$ in the low-temperature mixed-valence regime as functions of $E_d^*/\Delta$ and $T/\Delta$, which he called the main candidate for a detailed numerical study; (ii) numerical evaluation of the universal crossover constants $\alpha=\bar\alpha$ entering the implicit equations for the charge-freezing scale $T^*$ and the empty-state decoupling scale $\tilde T$; (iii) a controlled treatment of the crossover regions $\omega\simeq|E_d|$ and $\omega\simeq T$ where the scaling equations are ill defined and generated retardation and irrelevant couplings become significant; and (iv) benchmarking of the analytic approximations, including the preexponential factors and the model-dependent scale $\omega_0$ entering $T_K$, against controlled numerical data. Numerical renormalization group (NRG) studies and the exact Bethe-ansatz solution have confirmed parts of the scaling theory, but the two-parameter universality in Haldane's variables has been neither confirmed nor refuted, no numerical value of the crossover constants is known, and the crossover-region physics has been treated only numerically, without quantifying the generated irrelevant couplings. The open core is the establishment (or refutation) of the universal mixed-valence crossover functions in the variables $(E_d^*/\Delta,\,T/\Delta)$, the numerical values of the associated crossover constants, a controlled account of the crossover regions, and the benchmarking of the analytic approximations against accurate numerical data.

Background

The single-impurity Anderson model describes a localized impurity $d$ orbital of energy $E_d$ with on-site Coulomb repulsion $U>0$, hybridized with a conduction band of half-width $D$ through matrix elements $V_{kd}$; the hybridization gives the level a resonance width $\Delta=\pi\rho V^2$, where $\rho$ is the conduction-electron density of states at the Fermi level. At particle-hole symmetry ($E_d=-U/2$) the model exhibits Kondo physics: below a low scale $T_K$ the impurity magnetic moment is asymptotically screened by the conduction electrons into a singlet. Away from symmetry — the asymmetric case, relevant to mixed-valence (valence-fluctuating) impurities in metals and to quantum dots gated away from the symmetric point — no controlled analytic description existed before 1978: the local-moment (Kondo-limit) perturbation theory does not apply when charge fluctuations are strong, and expansions in $U$ are not controlled in this regime.

In his 1978 Physical Review Letters paper, Haldane treated the regime $U\gg\Delta$, $T\ll U$ by a perturbative renormalization-group scaling analysis (in the spirit of Anderson's 'poor man's' scaling for the Kondo problem, i.e. progressively integrating out the highest-energy band states): reducing the effective bandwidth $W$ generates the flow $dE_d/d\ln W=-\Delta/\pi+\ldots$, whose invariant is the renormalized level position $E_d^*=E_d+(\Delta/\pi)\ln(W_0/\Delta)$, with $W_0$ the conduction-band width $D$ or $U$, whichever is smaller. He concluded that for $T\ll U$ all properties are universal functions of the scaling invariants $\Delta$ and $E_d^*$ — i.e. of the two dimensionless variables $E_d^*/\Delta$ and $T/\Delta$ — identified $|E_d^*|\lesssim\Delta$ (not $|E_d|\lesssim\Delta$, which he explicitly warned against) as the criterion for a mixed-valence ground state, and described the crossovers: scaling stops at the charge-freezing scale $T^*$ or the empty-state decoupling scale $\tilde T$, defined by the implicit equations $T^*+(\Delta/\pi)\ln(\alpha T^*/\Delta)=E_d^*$ and $\tilde T-(\Delta/\pi)\ln(\bar\alpha\tilde T/\Delta)=-E_d^*$ with universal constants $\alpha,\bar\alpha$ of order 1 (footnote 11 records $\bar\alpha=\alpha$), and in the local-moment regime the moment is Kondo-quenched below a scale of order $T_K=\Delta\exp(\pi E_d^*/2\Delta)$, i.e. $T_K=(\omega_0\Delta)^{1/2}\exp(\pi E_d/2\Delta)$ in bare parameters, with an unevaluated model-dependent scale $\omega_0\simeq W_0$ fixed by the perturbation expansion. In the final paragraph he proposed that in the low-temperature mixed-valence region the susceptibility, the linear specific-heat coefficient, and the level occupation $\langle n_d\rangle$, expressed in units of $\Delta$, 'should smoothly change from their Kondo to $E_d^*\gg\Delta$ values as universal functions of $E_d^*/\Delta$', calling this the main candidate for a detailed numerical study by Wilson's numerical renormalization group (NRG — Wilson's nonperturbative iterative method for quantum impurity models); in footnotes he recorded that the scaling equations are not well defined in the crossover regions $\omega\simeq|E_d|$ or $\omega\simeq T$ (they depend on the cutoff structure of the hybridization function $\Delta(\omega)$), where the retardation (frequency dependence of the effective interaction) and nominally irrelevant couplings generated by integrating out high-energy states become significant.

Subsequent work developed parts of this program. Krishna-murthy, Wilkins and Wilson (1980) performed the NRG study for $\chi(T)$: they confirmed and refined Haldane's scaling formulas, identified the four fixed points (free-orbital, valence-fluctuation, local-moment, frozen-impurity), and mapped the local-moment regime onto Wilson's universal Kondo curve — but reported 'nonuniversal properties ... observed for $-E_d(T)\lesssim\Gamma$' in the valence-fluctuation regime ($\Gamma$ their notation for $\Delta$), and did not tabulate two-parameter crossover functions of $(E_d^*/\Delta, T/\Delta)$ or evaluate the constants. The Bethe ansatz (an exact integrability-based solution method) supplied exact thermodynamics for the asymmetric model (Tsvelick–Wiegmann 1982; Kawakami–Okiji 1983), from which the universal functions could in principle be constructed. Costi, Hewson and Zlatić (1994) found the low-temperature coefficients in the mixed-valent and empty-orbital regimes to be non-universal, depending strongly on the level occupation. Merker, Weichselbaum and Costi (2012) computed $\chi_{\mathrm{imp}}(T)$, $C_{\mathrm{imp}}(T)$ and $S_{\mathrm{imp}}(T)$ across the Kondo, mixed-valence, and empty-orbital regimes using full density-matrix NRG (an NRG variant that constructs the complete density matrix from all kept and discarded states, giving smooth finite-temperature thermodynamics), validated at the 2–3% level against the Bethe ansatz — but used the symmetric-model $T_K$ as the single scaling variable and made no collapse claim in Haldane's variables. Whether Haldane's two-parameter universality holds, what the crossover constants ($\alpha=\bar\alpha$, $\omega_0$) are numerically, and how to treat analytically the crossover regions where the scaling equations break down remain unsettled.

Problem Statement

For the asymmetric Anderson impurity model — a single impurity level of energy $E_d$ with on-site repulsion $U>0$, hybridized with a conduction band (density of states $\rho$ at the Fermi level, hybridization matrix element $V$, conduction-band half-width $D$) so that the level acquires the width $\Delta=\pi\rho V^2$ — in the regime $U\gg\Delta$ and $T\ll U$, with the renormalized level position $E_d^*=E_d+(\Delta/\pi)\ln(W_0/\Delta)$ (where $W_0=\min(U,D)$), establish or refute the four linked questions Haldane's 1978 scaling theory left open: (1) Are the impurity susceptibility $\chi_{\mathrm{imp}}$, the linear specific-heat coefficient $\gamma_{\mathrm{imp}}=C_{\mathrm{imp}}/T$ (with the impurity entropy $S_{\mathrm{imp}}$ following by integrating $\gamma_{\mathrm{imp}}/T$), and the impurity level occupation $\langle n_d\rangle$, expressed in units of $\Delta$, universal functions of the two variables $E_d^*/\Delta$ and $T/\Delta$ in the mixed-valence regime $|E_d^*|\lesssim\Delta$? Compute these crossover functions and exhibit data collapse for distinct bare parameter sets $(E_d,U,\Delta,D)$ — or, where the collapse fails (as the non-universality reported for $-E_d(T)\lesssim\Delta$ by Krishna-murthy, Wilkins and Wilson suggests may occur in part of the regime), identify the correct scaling variables and the correspondingly corrected universality statement. (2) Evaluate numerically the universal crossover constants $\alpha=\bar\alpha$ of order 1 that enter the implicit equations $T^*+(\Delta/\pi)\ln(\alpha T^*/\Delta)=E_d^*$ and $\tilde T-(\Delta/\pi)\ln(\bar\alpha\tilde T/\Delta)=-E_d^*$ and fix the relation between the low-temperature scales $T^*$, $\tilde T$ and the effective level position above the crossover. (3) Provide a controlled quantitative account of the crossover regions $\omega\simeq|E_d|$ and $\omega\simeq T$, where the leading-order scaling equations are ill defined because they depend on the cutoff structure — quantify the corrections from the generated retardation (frequency-dependent effective interaction) and from couplings nominally irrelevant at the fixed points, delimit the parameter range in which leading-order scaling is quantitatively trustworthy, and give the resulting corrections to the crossover functions of (1), stating the order in the generated retardation and in the leading irrelevant coupling to which these corrections are computed, and the range of $E_d^*/\Delta$ and $T/\Delta$ over which leading-order scaling is validated. (4) Benchmark the analytic approximations of the scaling theory — in particular the Kondo scale, of order $T_K=\Delta\exp(\pi E_d^*/2\Delta)$, equivalently $T_K=(\omega_0\Delta)^{1/2}\exp(\pi E_d/2\Delta)$ in bare parameters, with its unevaluated preexponential factor and model-dependent scale $\omega_0\simeq W_0$, together with the crossover scales $T^*$ and $\tilde T$ — against accurate numerical data, fixing each constant to a numerical value with its stated uncertainty, so that the scaling theory yields fully quantitative, model-independent predictions.

References

1. F. D. M. Haldane, 'Scaling Theory of the Asymmetric Anderson Model', Physical Review Letters 40, 416–419 (1978), doi:10.1103/PhysRevLett.40.416, https://doi.org/10.1103/PhysRevLett.40.416
2. F. D. M. Haldane, 'Scaling Theory of the Asymmetric Anderson Model (Erratum)', Physical Review Letters 40, 911 (1978), doi:10.1103/PhysRevLett.40.911.2, https://doi.org/10.1103/PhysRevLett.40.911.2
3. H. R. Krishna-murthy, J. W. Wilkins, K. G. Wilson, 'Renormalization-group approach to the Anderson model of dilute magnetic alloys. II. Static properties for the asymmetric case', Physical Review B 21, 1044–1083 (1980), doi:10.1103/PhysRevB.21.1044, https://doi.org/10.1103/PhysRevB.21.1044
4. F. D. M. Haldane, 'Theory of the atomic limit of the Anderson model. I. Perturbation expansions re-examined', Journal of Physics C: Solid State Physics 11, 5015–5034 (1978), doi:10.1088/0022-3719/11/24/030, https://doi.org/10.1088/0022-3719/11/24/030
5. A. M. Tsvelick, P. B. Wiegmann, 'Low-temperature properties of the asymmetric Anderson model (exact solution) II', Physics Letters A 89, 368–372 (1982), doi:10.1016/0375-9601(82)90195-5, https://doi.org/10.1016/0375-9601(82)90195-5
6. Norio Kawakami, Ayao Okiji, 'Temperature-dependent susceptibility for the asymmetric Anderson model', Physics Letters A 98, 54–56 (1983), doi:10.1016/0375-9601(83)90544-3, https://doi.org/10.1016/0375-9601(83)90544-3
7. T. A. Costi, A. C. Hewson, V. Zlatić, 'Transport coefficients of the Anderson model via the numerical renormalization group', Journal of Physics: Condensed Matter 6, 2519–2558 (1994), doi:10.1088/0953-8984/6/13/013, https://arxiv.org/abs/cond-mat/9310032
8. L. Merker, A. Weichselbaum, T. A. Costi, 'Full density-matrix numerical renormalization group calculation of impurity susceptibility and specific heat of the Anderson impurity model', Physical Review B 86, 075153 (2012), doi:10.1103/PhysRevB.86.075153, https://arxiv.org/abs/1207.2631