Particle-hole Symmetric Slave Boson Method for the Mixed Valence Problem

TL;DR

This work develops a particle-hole symmetric slave-boson framework for the finite-$U$ Anderson impurity model by introducing two bosons that track valence fluctuations, which are ultimately represented by a single symmetric $s$-boson in an effective action. The method preserves the low-energy physics across finite-$U$, infinite-$U$, and Kondo limits, recovers established limits, and reveals that Kondo coherence begins to emerge in the normal state before $s$-boson condensation, via Gaussian fluctuations around a mean-field solution. The authors connect the $s$-boson construction to the slave-rotor method within a unified functional-integral approach, and provide analytical expressions for spectral functions and charge susceptibility, showing how Hubbard bands and central resonances arise from the symmetric boson kinetics and the fermionic bubbles. They discuss limitations in strongly mixed-valent regimes due to an enlarged Hilbert space and outline directions for normalization, multiorbital extensions, and DMFT applications.

Abstract

We introduce an analytic slave boson method for treating the finite $U$ Anderson impurity model. Our approach introduces two bosons to track both $Q\rightleftharpoons Q\pm1$ valence fluctuations and reduces to a single symmetric $s$-boson in the effective action, encoding all the high energy atomic physics information in the boson's kinematics, while the low energy part of the action remains unchanged across finite $U$, infinite $U$, and Kondo limits. We recover the infinite $U$ and Kondo limit actions from our approach and show that the Kondo resonance already develops in the normal state when the slave boson has yet to condense. We show that the slave rotor and $s$-boson have the same algebraic structure, and we establish a unified functional integral framework connecting the $s$-boson and slave rotor representations for the single impurity Anderson model.

Figures (9)

• Figure 1: Schematic of the atomic energy levels of the single impurity Anderson model shown in (a) the physical $d$-electron Hilbert space and (b) the enlarged slave boson Hilbert space. In the physical Hilbert space, $\Delta E_-$ and $\Delta E_+$ represent the excitation energies to remove or add a $d$-electron to the singly occupied state $d^{\dagger}_\sigma \vert \emptyset \rangle$, respectively. The slave boson representation accurately captures the physical atomic energy levels when the excitation energies $\Delta E_{\pm}$ are sufficiently large to suppress transitions to fictitious states (dashed lines).
• Figure 2: Mean-field transition temperature $T_c$in units of $\Delta$ as a function of $d$-level position $\tilde{\epsilon}_d$ for a half-filled impurity $q = Q/N = 0.5$, $\Delta = 7.5$, $U = \textcolor{black}{4 \Delta}$, and conduction bath with bandwidth $2D$ and cutoff $D = 8\Delta / 3$. For reference, the Kondo temperature at particle-hole symmetry $\tilde{\epsilon}_d = 0$ is $T_K(0, \tilde{\epsilon}_d = 0) = \textcolor{black}{0.11 \Delta }$, growing with increasing asymmetry such that the ratio of the transition temperature to the Kondo temperature $T_c^{q = 1/2}/T_K(0) = 1.13$.
• Figure 3: Mean-field transition temperature $T_c$in units of $\Delta$ plotted versus electron filling $x$ for $\Delta = 25$, $U = \textcolor{black}{4 \Delta}$, and a conduction bath with bandwidth $2D$ and cutoff $D = 4\Delta/5$ for different integer atomic fillings $Q$ of the $N = 6$ impurity ($Q = 1$, blue, $Q = 2$, orange, $Q = 3$, green, $Q = 4$, red, and $Q = 5$, purple). For each $Q$, the $d$-level position from half filling $\tilde{\epsilon}_d$ is taken to vary linearly from $U/2$ to $-U/2$ between half-integer fillings. $x$ is restricted for each $Q$ to the range where there is at least one solution for $T_c$. We only plot the lower calculated $T_c$ solutions.
• Figure 4: Mean-field impurity valence at zero temperature as a function of electron filling $x$ for $\Delta = 25$, $U = \textcolor{black}{4\Delta}$, and a conduction bath with bandwidth $2D$ and cutoff $D = 4\Delta/5$ for different atomic fillings $Q$ of the $N=6$ impurity ($Q = 1$, blue, $Q = 2$, orange, $Q = 3$, green, $Q = 4$, red, and $Q = 5$, purple). For each $Q$, the $d$-level position from half filling $\tilde{\epsilon}_d$ is taken to vary linearly from $U/2$ to $-U/2$ between half-integer fillings. $x$ is restricted for each $Q$ to the range where there is at least one solution for $T_c$.
• Figure 5: The full physical $d$-electron spectral function for the symmetric limit (multiplied by $\Delta$ so that the y-axis is dimensionless), $q = 0.5$, $\Delta = 7.5$, $\tilde{\epsilon}_d = 0.0$, $U = 4 \Delta$, and a conduction band with bandwidth $2D$ and cutoff $D = 8\Delta/3$. The Kondo mean-field transition temperature $T_c = \textcolor{black}{0.13 \Delta}$ for these parameters. We plot the $d$-electron spectral function for different temperatures: high temperature ($T \gg T_c$, green); intermediate temperature ($T > T_c$, but where a central resonance begins to form, orange), and low temperature (with temperature just below $T_c$, blue).