Nonperturbative serial transport and crossover boundaries of the SU(4) Kondo state in a capacitively coupled double quantum dot

A double quantum dot with strong capacitive interdot coupling, each dot contacted by its own leads, realizes at its charge degeneracy point an SU(4)-symmetric Kondo Fermi-liquid state in which spin and orbital (pseudospin) degrees of freedom are entangled and screened together (Borda, Zaránd, Hofstetter, Halperin, and von Delft, Phys. Rev. Lett. 90, 026602 (2003)). The original analysis was restricted to the fully symmetric dot configuration and to interdot tunneling amplitudes $t$ far below the Kondo scale, where the serial conductance is perturbative in $t$, and the crossover boundaries of the SU(4) regime were argued only qualitatively. The open problem is to complete the quantitative low-energy theory of this device away from those limits: compute the serial conductance in the regime $t\gtrsim T_K(B)$ where perturbation theory in $t$ fails, map quantitatively (including crossover scales and finite-temperature behavior) the boundaries separating the SU(4) Fermi liquid from the spin-only Kondo, orbital-only Kondo, and molecular or local-singlet regimes across the joint parameter space of magnetic field, dot-level detuning, interdot tunneling, and dot-lead coupling asymmetry, and characterize the SU(4) Kondo temperature reliably in general asymmetric configurations. Later literature has mapped several single crossover axes at vanishing interdot tunneling and realized the SU(4) state experimentally, but the nonperturbative finite-$t$ serial-transport regime and a unified finite-temperature crossover map for this device remain open.

Background

A quantum dot is a nanoscale conductor that, when small enough, carries electrons one at a time because of Coulomb blockade (the electrostatic cost of adding charge). When a dot holds a single unpaired spin, repeated virtual tunneling of lead electrons produces the Kondo effect: an exchange coupling $J$ to the leads that grows under renormalization until the spin is screened below a Kondo temperature $T_K\approx D e^{-1/J}$, with $D$ a high-energy cutoff, producing a Fermi-liquid ground state (a state of long-lived quasiparticles) with a scattering phase shift $\pi/2$ by the Friedel sum rule (which fixes phase shifts from the occupied impurity level).

Borda, Zaránd, Hofstetter, Halperin, and von Delft (Phys. Rev. Lett. 90, 026602 (2003); preprint arXiv:cond-mat/0207001) studied a double quantum dot (DD) with large capacitive interdot coupling, each dot attached to its own pair of leads, at the degeneracy of the charge states $(1,0)$ and $(0,1)$. Below the charging energy $\tilde E_C$, the DD dynamics reduces to four states — spin $S^z=\pm1/2$ combined with the orbital pseudospin $T^z=(n_+-n_-)/2=\pm1/2$ that records which dot is occupied. The dot Hamiltonian is $H_{\mathrm{dot}}=-\delta E\,T^z-t\,T^x-B\,S^z$, where $\delta E$ is the detuning between the two charge states, $t$ the interdot tunneling amplitude, and $B$ a Zeeman field. Integrating out virtual charge fluctuations generates an effective Kondo Hamiltonian with spin exchange $J_\pm$, spin-flip-assisted tunneling $Q_\perp$ (which flips spin and pseudospin together), and orbital couplings $V_z,V_\perp$. For $\delta E=t=B=0$ the scaling flow drives all couplings together and the low-energy model becomes an SU(4) single-channel exchange model — SU(4) being the symmetry group acting on the four combined spin-pseudospin states, the 'hyperspin'. The authors verified by Wilson's numerical renormalization group (NRG, a controlled iterative method for quantum impurity models) that the fixed point is a Fermi liquid with phase shift $\delta=\pi/4$ in each of four spinless channels, and that a large field $B$ drives a crossover to a purely orbital (pseudospin) Kondo state with $\delta_\uparrow\to\pi/2$, giving a nearly perfectly spin-polarized current in a four-lead geometry: the device acts as a spin filter.

Three restrictions of that analysis, all acknowledged in the paper, define the present problem. First, a closed expression for the SU(4) Kondo scale was obtained only for the symmetric dot configuration, $T_K^{(0)}\approx D e^{-1/4J}$ with $J\sim\Gamma/2\pi\tilde E_C$ (the paper's symmetric-dot estimate, valid when fluctuations to the $(0,0)$ charge state dominate) ('We could not find a closed expression for $T_K^{(0)}$ in the general case'). Second, the serial conductance across the DD (from the left leads through dot $-$ to the right leads) was computed only to leading order in the interdot tunneling: $G=(2\pi^2 e^2/h)t^2\lim_{\omega\to0}\varrho_T^y(\omega)/\omega$, where $\varrho_T^y(\omega)$ is the $T^y$-pseudospin spectral function of the (separately equilibrated) dots; this requires $t\ll T_K(B)$, and the paper states that 'for larger values of $t$ a more complete calculation is needed'. Third, the stability of the SU(4) fixed point to small detuning, field, tunneling, and coupling anisotropy was argued via the marginal irrelevance of the perturbing operators, and the crossovers to spin-only or orbital-only Kondo screening were described qualitatively, but no quantitative boundaries (crossover scales, finite-temperature signatures) were mapped in the general parameter space. These three items form one coherent objective: a complete quantitative theory of the SU(4) Kondo double-dot device away from the symmetric, weak-tunneling limit. Related prior and later settings should be distinguished from this one: the serial double dot (both dots between the same two leads) studied by Pohjola, Schoeller, and Schön (Europhys. Lett. 54, 241 (2001)) has a different lead geometry and no SU(4) hyperspin.

Problem Statement

For the capacitively coupled double quantum dot in its SU(4) Kondo regime — each dot attached to its own leads, dot Hamiltonian $H_{\mathrm{dot}}=-\delta E\,T^z-t\,T^x-B\,S^z$ with the dynamics restricted to the four spin-pseudospin states, and effective exchange couplings $J_\pm$, $Q_\perp$, $V_z$, $V_\perp$ allowed to be asymmetric between the two dots — construct the complete quantitative low-energy theory away from the symmetric, weak-interdot-tunneling limit. Concretely: (1) compute the serial conductance across the double dot (transport from the left leads through one dot to the right leads) in the regime $t\gtrsim T_K(B)$, where the perturbative result $G=(2\pi^2e^2/h)t^2\lim_{\omega\to0}\varrho_T^y(\omega)/\omega$ and the scaling $G\sim(t/T_K(B))^2$ break down, including the crossover of this conductance as $t$ grows through the Kondo scale and the competition with molecular or local-singlet formation between the dots; (2) determine the quantitative boundaries — crossover scales and finite-temperature signatures — separating the observable SU(4) Fermi-liquid regime from the regimes dominated by spin-only or orbital-only (pseudospin) Kondo screening, and from molecular/local-singlet regimes, over the general parameter space of magnetic field $B$, detuning $\delta E$, interdot tunneling $t$, and dot-lead coupling asymmetry, with the crossover boundaries fixed along each of these four axes and the regime realized on either side identified, beyond the qualitative stability argument that small $\delta E,B,t\ll T_K^{(0)}$ only weakly perturb the SU(4) fixed point; and (3) characterize the SU(4) Kondo scale reliably in the general asymmetric case, where the symmetric-dot estimate $T_K^{(0)}\approx De^{-1/4J}$ and $T_K^{(0)}(B=\infty)\approx\mathrm{cst.}[T_K^{(0)}]^2/D$ do not apply. These three deliverables jointly support one objective: a quantitative, experimentally testable map of when and how the SU(4) Kondo state, its spin-filter response, and its magnetoconductance survive in realistic (asymmetric, finite-tunneling, finite-temperature) double-dot devices.

References

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