At which magnetic field, exactly, does the Kondo resonance begin to split? A Fermi liquid description of the low-energy properties of the Anderson model

We extend a recently developed Fermi liquid (FL) theory for the asymmetric single-impurity Anderson model [C. Mora et al., Phys. Rev. B 92, 075120 (2015)] to the case of an arbitrary local magnetic field. To describe the system's low-lying quasiparticle excitations for arbitrary values of the bare Hamiltonian's model parameters, we construct an effective low-energy FL Hamiltonian whose FL parameters are expressed in terms of the local level's spin-dependent ground-state occupations and their derivatives with respect to level energy and local magnetic field. These quantities are calculable with excellent accuracy from the Bethe ansatz solution of the Anderson model. Applying this effective model to a quantum dot in a nonequilibrium setting, we obtain exact results for the curvature of the spectral function, $c_A$, describing its leading $\sim {\varepsilon }^2$ term, and the transport coefficients $c_V$ and $c_T$, describing the leading $\sim V^2$ and $\sim T^2$ terms in the nonlinear differential conductance. A sign change in $c_A$ or $c_V$ is indicative of a change from a local maximum to a local minimum in the spectral function or nonlinear conductance, respectively, as is expected to occur when an increasing magnetic field causes the Kondo resonance to split into two subpeaks. Surprisingly, we find that the fields $B_A$ and $B_V$ at which $c_A$ and $c_V$ change sign are parametrically different, with $B_A$ of order $T_K$ but $B_V$ much larger. In fact, in the Kondo limit $c_V$ never vanishes, implying that the conductance retains a (very weak) zero-bias maximum even for strong magnetic field and that the two pronounced finite-bias conductance side peaks caused by the Zeeman splitting of the local level do not emerge from zero-bias voltage.

Figure 1

Low-energy properties of the spectral function at particle-hole symmetry. (a) The normalized height coefficient ${\tilde{c}}_A/{\tilde{c}}_A^K$ and (b) curvature coefficient $c_A/c_A^K$ of the local spectral function at particle-hole symmetry, plotted as functions of magnetic field [in units of $T_K$, as defined in Eq. (20)], for three values of the interaction parameter $U/\mathrm{\Delta }$, including the Kondo limit $U/\mathrm{\Delta }=\infty$ [given by Eq. (35)]. The insets of (a) and (b), respectively, show ${\tilde{C}}_A/{\tilde{C}}_A^K$ and $C_A/C_A^K$ as functions of $B/T_K$, calculated from our FL predictions [Eq. (32)] (solid lines), and by computing the spectral function via NRG and extracting its zero-energy height and curvature (open circles). The sign change for $C_A$ signals the splitting of the Kondo resonance into two resonances due the breaking of the Kondo singlet by the magnetic field. (c) The characteristic field $B_A$ where $C_A$ vanishes, plotted in units of $T_K$ as function of $U/\mathrm{\Delta }$. In the Kondo limit $U/\mathrm{\Delta }\to \infty$ it approaches the value $B_A/T_K=0.75073$.

Figure 2

FL transport properties at particle-hole symmetry and $U/\mathrm{\Delta }=5$, plotted as functions of $B/T_K$. Left axis: Normalized FL transport coefficients $c_V/c_V^K$ (thick solid line) and $c_T/c_T^K$ (thick dashed line). The field $B_T$ where $c_T$ changes sign is of order $T_K$ (slightly larger but comparable to $B_A$), whereas the field $B_V$ where $c_V$ changes sign is much larger, of order $\sqrt{U\mathrm{\Delta }}$. Right axis: Normalized zero-temperature linear conductance $\tilde{G}/G_0={cos}^2\left(\pi m_d\right)$ [from Eq. (32a)] (thin solid line).

Figure 3

Interaction dependence of splitting field properties at particle-hole symmetry, plotted as functions of $U/\mathrm{\Delta }$. (a) Splitting field $B_T$ at which $c_T$ vanishes (left axis, thick line), shown in units of $T_K$, and the normalized zero-temperature linear conductance at that field, $\tilde{G}\left(B_T\right)/G_0$ (right axis, thin line). (b) The splitting field $B_V$ at which $c_V$ vanishes, shown in units of both $T_K$ (left axis, thick line, log scale) and $\sqrt{U\mathrm{\Delta }}$ (right axis, thin line, linear scale). For strong interactions $U\gg \mathrm{\Delta }$, we find $B_V=0.679\sqrt{U\mathrm{\Delta }}$, implying that $B_V\gg B_T$. In the absence of interaction $U=0$, the Kondo temperature extracted from Eq. (20) is $T_K=\pi \mathrm{\Delta }/2$, hence $B_V/T_K=\frac{4}{\pi \sqrt{3}}\simeq 0.735$.

Figure 4

Evolution of $c_V$ during the crossover to the Kondo limit. (a) The normalized FL transport coefficient $c_V/c_V^K$, plotted as a function of $B/T_K$ for several values of $U/\mathrm{\Delta }$, including the Kondo limit (thick solid line). For the latter, $c_V$ is strictly positive for all magnetic fields. The direct integration of the coupled Bethe ansatz equations, Eq. (S3a) and (S4b) [65], performed here for $U/\mathrm{\Delta }=20$ (BAE, open dots), is in very good agreement with the corresponding Wiener-Hopf solution (dashed line). All curves with large $U/\mathrm{\Delta }$ ($\gtrsim 10$) initially collapse onto a universal scaling curve as function of increasing $B/T_K$, but for large $B$ eventually bend downward and change sign. (b) Same as (a), but plotted as function of $B/\sqrt{U\mathrm{\Delta }}$. Now a scaling collapse sets in at magnetic fields larger than $\sqrt{U\mathrm{\Delta }}$, which is thus seen to be the field scale beyond which $c_V$ changes sign. Together, these two panels show that as long as $B$ is smaller than the high-energy scale $\sqrt{U\mathrm{\Delta }}$ of the Anderson model, $c_V$ shows universal behavior, governed by a single scale $T_K$ and characteristic of the Kondo model, and in this Kondo regime, $c_V$ remains strictly positive.

Figure 5

The coefficient $C_V=c_V/E_{*}^2$, plotted as a function of $B/T_{\mathrm{K}}$ for different interaction strengths $U/\mathrm{\Delta }$, in units of $C_V^{\mathrm{K}}=c_V^{\mathrm{K}}/T_K^2$, with $c_V^{\mathrm{K}}$ defined in Eq. (41a). In contrast to $c_V$ from Fig. 4, $C_V$ is strongly suppressed in the regime $B>T_K$ (even for $U/\mathrm{\Delta }\gg 1$), because for large fields the spin susceptibility becomes very small and hence $E_{*}$ very large [cf. Eq. (21)]. Inset: the normalization factor $C_V^{\mathrm{K}}$, plotted as function of $U/\mathrm{\Delta }$ in units of $1/{\mathrm{\Delta }}^2$ (black points indicate the $U/\mathrm{\Delta }$ values from the main plot). $C_V^{\mathrm{K}}$ shows an exponential increase with $U/T_K$, caused by an exponential decrease in $T_K$. The growth in $C_V^{\mathrm{K}}$ is counteracted by the fact that the voltage window in which our FL analysis applies decreases exponentially, since the FL expansion (29) of the conductance requires $V\ll T_K$.

Figure 6

(a) Transition from the local-moment regime to the empty-orbital regime for the transport coefficient $c_V/c_V^K$, shown using a color scale, as a function of the magnetic field $B$ and the level energy ${\varepsilon }_d$, at $U/\mathrm{\Delta }=20$, a convenient value to highlight features related to Kondo physics. The solid line shows the prediction for the Kondo scale of the analytic formula (22) for $T_K^{\left(\mathrm{af}\right)}$, the grey triangles the numerical evaluation of $T_K$ as defined in Eq. (20) and the grey squares the numerical evaluation of $E_{*}$ in Eq. (19) for $B=0$. All these quantities show a nice agreement as long as ${\varepsilon }_d<0$. The black points signal when $c_V=0$ and changes sign. The light-colored regions correspond to positive values of $c_V$. (b) Same quantity $c_V/c_V^K$ as in (a), now shown along the cuts marked in (a) by grey dashed lines, and plotted as a function of $B/E_{*}^{B=0}$ on a logarithmic scale. The numbers above the data points give the corresponding values of ${\varepsilon }_d/U$ (increasing as colors turn from light to dark). The solid line corresponds to the analytical result for $c_V$ at particle-hole symmetry derived from the Wiener-Hopf solution [65] [see also Fig. 4]. Throughout the local-moment regime in which Kondo correlations occur (${\varepsilon }_d\lesssim -\mathrm{\Delta }$), $c_V$ shows a local minimum around fields of order $T_K$, and changes sign only at a much larger field, $B_V\gg T_K$. For ${\varepsilon }_d\gtrsim 0$, $c_V$ develops a double sign change with a peak in between, which reflects a field-induced resonance between the empty- and singly occupied dot states (see Fig. 7).

Figure 7

Magnetic-field dependence of (a) $c_V$ in units of $c_V^{\mathrm{K}}$ and (b) $C_V=c_V/E_{*}^2$ in units of $1/\mathrm{\Delta }$, in the mixed-valence and empty-orbital regimes. Both are plotted as functions of $B/\mathrm{\Delta }$ on a linear scale, for several values of ${\varepsilon }_d/\mathrm{\Delta }$ (given by numbers above the data points, increasing as colors turn from light to dark). Black solid curves display analytical predictions derived in perturbation theory, showing good agreement with the numerical results (symbols). The peaks in $c_V$ and $C_V$ reflect the field-induced resonance between the empty and singly occupied dot states. Vertical solid lines indicate the predicted values of the resonance field, $B=2{\tilde{\varepsilon }}_d$, where the renormalized level position ${\tilde{\varepsilon }}_d$ is given by Eq. (42) (and $\alpha \simeq 1.62$ therein); for comparison, vertical dashed lines indicate the bare values, $B=2{\varepsilon }_d$. Note that the nontrivial $B$ dependence exhibited by $C_V$ in (b) is as pronounced as that of $c_V$ in (a). The reason is that near the resonance field $B\simeq 2{\tilde{\varepsilon }}_d$, both the spin and charge susceptibilities are large, ensuring that $E_{*}$ remains small.