Kondo effect given exactly by density functional theory
Abstract
Transport through an Anderson junction (two macroscopic electrodes coupled to an Anderson impurity) is dominated by a Kondo peak in the spectral function at zero temperature. The exact single-particle Kohn-Sham potential of density functional theory reproduces the linear transport exactly, despite the lack of a Kondo peak in its spectral function. Using Bethe ansatz techniques, we calculate this potential exactly for all coupling strengths, including the cross-over from mean-field behavior to charge quantization caused by the derivative discontinuity. A simple and accurate interpolation formula is also given.
It is a truth universally acknowledged that many-body effects in strongly correlated systems are not reproduced by mean-field theory. Although Kohn-Sham (KS) density functional theory (DFT) is formally exact, it is a non-interacting theory yielding only the ground-state energy and density of a system. No other information about the correlated many-body wavefunction is available. Dynamical properties, such as excitations and response functions, are also not predicted by ground-state DFT, even with the exact functional Dreizler and Gross (1990). The hope is that, for weakly-correlated systems in which ground-state DFT approximations perform well for total energies, geometries, etc., the errors in such calculations are small. Nothing in the theorems of DFT guarantees that a ground-state KS calculation can describe transport correctly Koentopp et al. (2008).
Consider transport through an Anderson junction Anderson (1961); Pustilnik and Glazman (2004), composed of two macroscopic leads coupled to an Anderson impurity. As an integrable system, the Anderson model is a paradigm of many-body physics. It is also an accurate model of the low-energy spectrum of molecular radical-based junctions Bergfield et al. (2011). In Fig. 1, we show the zero-temperature transmission through an Anderson junction as a function of the energy of the resonant level using Bethe ansatz (BA), exact Kohn-Sham (KS) DFT, and Hartree-Fock (HF). In the figure, is the chemical potential (Fermi energy) of the metal electrodes. Remarkably, the KS-DFT treatment of this problem reproduces the transmission exactly, apparently describing the non-perturbative Kondo effect whose spectral peak is the source of the perfect transmission when .
The inability to describe sharp steps in transmission is a well-understood failure of standard density functional approximations. In the limit of weak coupling to the leads, the system is a prototype example where the effects of the infamous derivative discontinuity is seen Perdew et al. (1982). For such a system, the exchange-correlation (XC) energy of the molecule is strictly linear between integer values, and so the XC potential, its functional derivative, jumps discontinuously at such values Perdew et al. (1982). This effect has been implicated in many well-known failures of DFT approximations such as strongly-correlated systems Mori-Sanchez et al. (2009), charge-transfer excitations Maitra (2005), and over-estimation of the current in organic junctions Evers et al. (2004).
In this letter, we (a) solve the Anderson junction using BA and invert the KS equations to derive the exact KS potential, (b) show that the transport is recovered exactly, but only for zero temperature and weak bias, and (c) parametrize the XC potential, providing a unique interpolation formula that works for all correlation strengths. The exact and KS-DFT transport are intimately related via the Friedel sum-rule, a statement equating the occupancy and the transmission phase at the Fermi energy, as has been discussed for nearly isolated resonances Schmitteckert and Evers (2008); Mera and Niquet (2010) or single-mode leads Mera and Niquet (2010). Since our system fits neither of these categories, it reopens the question of just when an accurate ground-state KS calculation will yield accurate transport.
Using BA Wiegmann and Tsvelick (1983); Konik et al. (2001), we calculate the exact KS potential for the Anderson model, and investigate how the derivative discontinuity develops in the limit of weak impurity-lead coupling. From this solution, we extract the exact asymptotic scaling form of the derivative discontinuity and establish an interpolation formula for the KS potential which is accurate for all coupling strengths, computationally simple, and illustrates the crossover from mean-field behavior at strong impurity-lead coupling to charge quantization at weak coupling.
Consider a junction composed of a nanoscale central region (C) connected to two macroscopic electrodes, labeled left (L) and right (R). The Hamiltonian of the system is , where describe Fermi gases in the electrodes and describes tunneling between the central region and the electrodes. The central region has the Hamiltonian Anderson (1961):
(1) |
where is the number operator for spin and the charging energy is given by the Coulomb integral. The exact Green’s function of an Anderson junction may be found using Dyson’s equation in an orthonormal basis:
(2) |
where is the Coulomb self-energy and is the tunneling self-energy Bergfield and Stafford (2009). We consider transport in the broad-band limit where is pure imaginary and independent of energy, and define the mean tunneling-width .
At zero temperature, the linear-response transmission function of an Anderson junction is given by Pustilnik and Glazman (2004); Konik et al. (2001)
(3) |
where is the sum of transmission eigenphases evaluated at the Fermi energy . The total number of electrons on the central impurity is related to at zero temperature by the Friedel sum-rule Langreth (1966); Friedel (1958); Mera et al. (2010)
(4) |
From the Bethe ansatz, one finds Wiegmann and Tsvelick (1983)
(5) |
with Q defined by the condition
(6) |
making use of the following functions
(7) |
, and , where is the Gamma function and . is plotted as a function of in Fig. 2.
The KS ansatz of DFT employs an effective single-particle description, defined to reproduce the ground-state density of the interacting system. By the Hohenberg-Kohn theorem, this potential is unique if it exists (it usually does) Dreizler and Gross (1990). The relationship between potential and density is fixed only in the full basis-set limit, but can be defined for lattice models. The leads of an Anderson model are non-interacting and characterized by a total charge, which remains constant. Thus the definition of the KS system in this extreme case is that of a non-interacting junction () with an on-site potential chosen to reproduce the on-site occupancy of the interacting system.
The KS Green’s function in the central region may be written as
(8) |
where is the KS potential for an electron on the impurity, and is written
(9) |
where the second term is the Hartree contribution and the last is the correlation potential (there are no exchange contributions). The Anderson model has no internal molecular structure so the KS lead-molecule coupling is Bergfield et al. (2011); Evers and Schmitteckert (2011). For more complex systems, need not be equal to the KS lead-molecule coupling. In a standard DFT calculation, is approximated as a functional of the densityDreizler and Gross (1990). The occupancy of the central region is
(10) |
where the “lesser” Green’s function is found using the Keldysh relation Bergfield and Stafford (2009)
(11) |
where at zero temperature, and is the Heaviside function. Inserting the KS Green’s function and solving Eq. (10) for gives
(12) |
which defines the exact KS potential within the Anderson model. The KS transmission is then
(13) |
Plugging Eq. (12) into Eq. (13), we find that is identical to , as was shown in Fig. 1. Although this identity can be derived using, e.g., local Fermi liquid theory Pustilnik and Glazman (2004), nonetheless its significance is profound: If (and only if) a mean-field theory yields the exact occupation will it yield the exact transmission.
In an Anderson junction, the Friedel sum-rule connects the transmission at the Fermi energy to the occupancy at zero temperature. As shown in Fig. 3, the full transmission spectrum of an Anderson junction exhibits three peaks: Two Coulomb-blockade peaks of width centered about and and a third zero-bias Kondo peak of width pinned at . In contrast, the KS-DFT transmission spectrum is a single Lorentzian of width peaked at . As indicated in the figure, the KS value is a huge overestimate anywhere more than several away from , implying that the ground-state KS potential does not accurately predict transport at temperatures larger than or for bias voltages larger than .
From Eqs. (3), (4), (12) and (13) it is evident that the HF errors in transmission in Fig. 1 stem from errors in the occupancies of Fig. 2. When , HF yields accurate occupancies and transmissions. But for , the HF occupancies lack the distinct steps present in the exact solutions, causing corresponding discrepancies in the transport. Qualitatively similar errors would be found with any local or semilocal approximation for the XC potential, because such approximations are smooth functions of the interaction strength Koentopp et al. (2006). But the exact KS potential of an isolated system, infinitely weakly coupled to a reservoir, displays discontinuous jumps at integer particle number Perdew et al. (1982), just as ours does as . The KS potential is shown as a function of occupancy in Fig. 4a for several values of . The HF potential is linear with a slope of . For large but finite , the KS potential is not discontinuous but has steps (of width ) corresponding to the plateaus in the occupancy, becoming discontinuous in the limit. When is sufficiently small there is no step in the KS potential and the HF approximation is accurate.
We now show how the step develops as . In Fig. 4a, develops a step of height at whose width decreases as . Fig. 4b shows the exact derivative in the vicinity of . The horizontal and vertical axes are rescaled to illustrate the scaling behavior of the step as . From the BA solution, as Liu et al. ():
(14) |
whose derivative yields a Lorentzian. In Fig. 4b, we show this limit and how it is approached as grows, but notice also that the Lorentzian shape is approximately correct for all down to . We thus parametrize the XC potential for by the approximate form
(15) |
where and are functions of which and , respectively, in the limit , and determine the amplitude and width of the correlation contribution as a function of . Varying between 0 and 1 corresponds to “turning on” charge quantization, a nonperturbative interaction effect that would not be described by any local or semilocal approximation.
To determine those parameterizations, we first note the behavior near for . This is not generic, but stems from the restricted Hilbert space of the Anderson model. For , Eqs. (12) and (10) imply that near . However, fits the data most accurately over the range . Precisely the same form is applied at . Then a least-squares fit of Eq. (15) to the exact BA results, subtracting the features at and limiting the fit range to , yields the values of and given in Fig. 4c by the points.
We also found simple fits for these functions, with
(16) |
as shown by the smooth curves in Fig. 4c. The corresponding derivatives are given by dashed curves in Fig. 4d. Eqs. (15)–(16) define an interpolation formula for the XC potential which yields the exact KS potential in both the and limits and is accurate for all intermediate values of , as shown in Fig. 4. To check that our parametrization is sufficiently accurate, we performed self-consistent calculations using Eqs. (15) and (16), finding transmissions indistinguishable from the exact ones for all values of (see Fig. 2). Our interpolation formula shows that the cross-over between weak and strong correlation occurs for .
Our results should prove useful for the development of density functional theory in general, and for its application to transport through molecular junctions. While much is known of the consequences of the derivative discontinuity in the extreme limit of weak coupling, our results describe the entire range from strong to weak coupling, i.e., from weak to strong correlation. For transport through molecular junctions, our results provide an exact limit for which both many-body and DFT approximations can be tested. Nor are these just theorists’ games with toy models. For example, for the archetypal Au-[1,4]benzenedithiol single-molecule junction, . In such a system, any approximate XC functional which fails to account for derivative discontinuity effects is unlikely to yield accurate results.
Note added in proof: After submission of our manuscript, Refs. 18; 25; 26 (chronological order) reporting related results appeared on arXiv. This work was supported by DOE under grant number DE-FG02-08ER46496. KB thanks David Langreth for a decade-long conversation on the subject, and a lifetime of inspiration.
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