Two-channel Kondo phases and frustration-induced transitions in triple quantum dots

We study theoretically a ring of three quantum dots mutually coupled by antiferromagnetic exchange interactions and tunnel-coupled to two metallic leads: the simplest model in which the consequences of local frustration arising from internal degrees of freedom may be studied within a two-channel environment. Two-channel Kondo (2CK) physics is found to predominate at low energies in the mirror-symmetric models considered, with a residual spin-$\frac{1}{2}$ overscreened by coupling to both leads. It is however shown that two distinct 2CK phases, with different ground-state parities, arise on tuning the interdot exchange couplings. In consequence a frustration-induced quantum phase transition occurs, the 2CK phases being separated by a quantum critical point for which an effective low-energy model is derived. Precisely at the transition, parity mixing of the quasidegenerate local trimer states acts to destabilize the 2CK fixed points; and the critical fixed point is shown to consist of a free pseudospin together with effective one-channel spin quenching, itself reflecting underlying channel anisotropy in the inherently two-channel system. Numerical renormalization group techniques and physical arguments are used to obtain a detailed understanding of the problem, including study of both thermodynamic and dynamical properties of the system.

Figure 1

Schematic of the two-channel quantum dot trimer.

Figure 2

Lowest five NRG energy levels of the left-charge/right-charge/spin subspaces $\left(Q_L,Q_R,S^z\right)\equiv \left(0,0,\frac{1}{2}\right)$ [solid lines] and (0,1,0)/(1,0,0) [dashed], versus (even) iteration number, $N$. Shown for $\rho J=0.1$, $\tilde{U}=U/\left(\pi \Gamma \right)=7$, with $\rho J^{\prime }=0$ [panel (a)] and $\rho J^{\prime }=0.105$ [(b)]. For comparison, (c) shows corresponding results for a pure (single-spin) 2CK model with $\rho J_K=0.055$.

Figure 3

Thermodynamics of the $J^{\prime }<J$ phase. Shown for fixed $\tilde{U}=7$ and $\rho J=0.1$, varying $\rho J^{\prime }=0$ (solid lines), 0.0375 (dotted), and 0.075 (dashed). Panel (a) shows $S_{\text{imp}}\left(T\right)$, (b) shows $T{\chi }_{\text{imp}}\left(T\right)$, both vs $T/\Gamma$. Insets show the universality arising on rescaling in terms of the 2CK Kondo temperature, $T_K$, all curves collapsing to a common form.

Figure 4

Spin susceptibility ${\chi }_{\text{imp}}\left(T\right)$ vs $T/\Gamma$ in the $J^{\prime }<J$ regime, for same parameters as Fig. 3. Inset: scaling collapse onto the universal form $T_K{\chi }_{\text{imp}}\left(T\right)=\alpha \text{ln}\left(T_K/T\right)$.

Figure 5

Thermodynamics of the $J^{\prime }>J$ phase. Shown for fixed $\tilde{U}=7$, $\rho J=0.1$, varying $\rho J^{\prime }=0.105$ (solid lines), 0.115 (dotted), and 0.125 (dashed). Panel (a) again shows $S_{\text{imp}}\left(T\right)$ and (b) shows $T{\chi }_{\text{imp}}\left(T\right)$, both vs $T/\Gamma$. Insets show rescaling in terms of the Kondo temperature, $T_K$, yielding the same universal 2CK scaling curves as in Fig. 3.

Figure 6

$T=0$ local spectrum $\pi \Gamma D_1\left(\omega \right)$ vs $\omega /\Gamma$ in the $J^{\prime }<J$ regime. Shown (as in Fig. 3) for fixed $\tilde{U}=7$ and $\rho J=0.1$, varying $J^{\prime }/J=0$ (solid line), 0.375 (dotted), and 0.75 (dashed). The dot-dashed line is the spectral density $D{\rho }_K\left(\omega \right)$ [Eq. (26)] for a single-spin 2CK model [Eq. (23)], with $\rho J_K=0.07$. Inset: results shown on a linear frequency scale.

Figure 7

$T=0$ scaling spectrum for the 2CK phases. $\pi \Gamma D_1\left(\omega \right)$ vs $\omega /T_K$ is shown both in the $J^{\prime }<J$ regime (for the bare parameter sets used in Fig. 3) and for the $J^{\prime }>J$ regime (with the parameters used in Fig. 5). $D{\rho }_K\left(\omega \right)$ vs $\omega /T_K$ for the single-spin 2CK model is also shown. All spectra collapse to the same universal form. The dotted line shows the low-$\left|\omega \right|/T_K$ asymptotic behavior $\pi \Gamma D_1\left(\omega \right)=\frac{1}{2}\left[1-a{\left(\left|\omega \right|/T_K\right)}^{1/2}\right]$, while the dashed line describes the high-$\left|\omega \right|/T_K$ scaling behavior, $\pi \Gamma D_1\left(\omega \right)=A/\left[{\text{ln}}^2\left(\left|\omega \right|/T_K\right)+B\right]$.

Figure 8

Electron self-energy ${\Sigma }_1\left(\omega \right)/\Gamma$ vs $\omega /\Gamma$ for the same parameters as Fig. 6. Panel (a): imaginary part, ${\Sigma }_1^I\left(\omega \right)/\Gamma$; panel (b): real part, ${\Sigma }_1^R\left(\omega \right)/\Gamma$. Insets: universal scaling forms as a function of $\omega /T_K$, with the dot-dashed lines showing the low-frequency asymptotic behavior ${\Sigma }_1^I\left(\omega \right)/\Gamma =1+c{\left|\omega /T_K\right|}^{1/2}$, and ${\Sigma }_1^R\left(\omega \right)/\Gamma =-ϵ/\Gamma -\text{sgn}\left(\omega \right)c^{\prime }{\left|\omega /T_K\right|}^{1/2}$.

Figure 9

Local dynamic susceptibility $\Gamma {\chi }_1\left(\omega \right)$ vs $\omega /\Gamma$ at $T=0$ for the same bare parameters as Fig. 6. Inset: showing universal scaling of $T_K{\chi }_1\left(\omega \right)$ vs $\omega /T_K$.

Figure 10

Kondo temperature $T_K^{\chi }/\Gamma$ (points, with solid line as guide to eye) vs $J^{\prime }/J$ for fixed $\tilde{U}=7$ and $\rho J=0.1$; as defined (Sec. 3A) via $T_K^{\chi }{\chi }_{\text{imp}}\left(T_K^{\chi }\right)=0.07$. Kondo scales obtained from perturbative scaling [Eq. (9)] are also shown, with $\rho J_{K+}$ (for $J^{\prime }/J>1$) from Eq. (8a) (dashed line) and $\rho J_{K-}$ (for $J^{\prime }/J<1$) from Eq. (8b) (dotted); with a trivial constant prefactor adjusted to fit the data.

Figure 11

$T=0$ spin-spin correlation functions $⟨{\hat{\mathbf{S}}}_1\cdot {\hat{\mathbf{S}}}_2⟩$ (solid line) and $⟨{\hat{\mathbf{S}}}_1\cdot {\hat{\mathbf{S}}}_3⟩$ (dashed) vs $J^{\prime }/J$, for fixed $\tilde{U}=7$ and $\rho J=0.1$. Inset: magnification around the point of frustration, where $⟨{\hat{\mathbf{S}}}_1\cdot {\hat{\mathbf{S}}}_2⟩=⟨{\hat{\mathbf{S}}}_1\cdot {\hat{\mathbf{S}}}_3⟩$, occurring at $J^{\prime }/J=1.032\dots$

Figure 12

NRG energy levels for even iteration number, $N$. The lowest four levels are shown in each charge/spin subspace for: the full TQD model precisely at the transition [left panel], a 1CK model with $\rho J_K=+0.056$ [middle] and $\rho J_K=-0.056$ [right]. For the full model, subspaces shown are $\left(Q_L,Q_R,S^z\right)\equiv \left(0,1,0\right)/\left(1,0,0\right)$ [solid lines], (0,1,1)/(1,0,1) [dashed], and (0,3,0)/(3,0,0) [dotted]. For the 1CK models, the subspaces are $\left(Q,S^z\right)\equiv \left(0,0\right)$ [solid lines], $\left(1,\frac{1}{2}\right)$ [dashed], and (0,1) [dotted].

Figure 13

Entropy $S_{\text{imp}}\left(T\right)$ vs $T/\Gamma$ on progressively approaching the QPT as: ${\tilde{E}}_{\Delta }=\left(J_c^{\prime }-J^{\prime }\right)=\pm \lambda T_K^c$ with $T_K^c/\Gamma ={10}^{-4}$ (as discussed in text) and results shown for $\lambda ={10}^n$ with $n=0,-1,-2,-3,-4$ [lines (a)–(e), respectively]. The QCP itself $\left(\lambda =0\right)$ is shown as line (f), while the dotted line shows $\text{ln}2+S_{\text{imp}}^{1CK}\left(T\right)$ calculated for a single-channel Kondo model with $T_K^{1CK}=T_K^c$; the two coinciding for all $T/\Gamma \lesssim 1$. Inset: results in (b)–(e) rescaled (solid line) in terms of the vanishing Kondo scale $T_K\equiv T_K^S$, showing universal crossover from the CFP to the stable 2CK FP; and contrasted to that arising in the single-spin 2CK model (dashed line).

Figure 14

Variation in the Kondo scale $T_K\equiv T_K^S$: $T_K/\Gamma$ vs $\left(J^{\prime }-J_c^{\prime }\right)/T_K^c$ (with fixed $\tilde{U}=7$ and $\rho J=0.1$, for which $T_K^c/\Gamma ={10}^{-4}$). Inset (b) shows an expanded view in the vicinity of the transition (solid line); inset (c) shows the same results (solid) on a log-log scale, together with the power-law decay $T_K\propto {\left|J_c^{\prime }-J^{\prime }\right|}^2$ (dashed line) as the transition is approached. In both insets, $T_K^{\chi }$ is also shown (dotted lines) and remains finite at the transition.

Figure 15

Uniform spin susceptibility $T_K^c{\chi }_{\text{imp}}\left(T\right)$ vs $T/T_K^c$ for fixed $\tilde{U}=7$ and $\rho J=0.1$, varying $J^{\prime }$ as: ${\tilde{E}}_{\Delta }=\left(J_c^{\prime }-J^{\prime }\right)=\lambda T_K^c$ with $\lambda ={10}^{n/4}$ and $n=1,0,-1,-2,-3$ (solid lines, in order of decreasing slope). The QCP itself $\left(\lambda =0\right)$ is the bottom solid line. Also shown (dotted) is a 1CK model chosen such that $T_K^{1CK}=T_K^c$, which describes the QCP; and a single-spin 2CK model with $T_K=T_K^c$ (dashed). Near the transition ${\chi }_{\text{imp}}\left(T\right)$ diverges as $T_K^c{\chi }_{\text{imp}}\left(T\right)\sim x\text{ln}\left(T_K/T\right)$, with amplitude $x\sim T_K/T_K^c$ as shown in the inset (see also text).

Figure 16

$\pi \Gamma D_1\left(\omega \right)$ vs $\omega /\Gamma$, varying ${\tilde{E}}_{\Delta }=J_c^{\prime }-J^{\prime }$ as the transition is approached from the $(+)$ parity 2CK phase: ${\tilde{E}}_{\Delta }=-\lambda T_K^c$ with $\lambda ={10}^{n/2}$ and $n=+4\to -4$ ($\text{a}\to \text{i}$, respectively). The QCP spectrum $\left(\lambda =0\right)$ is indistinguishable from case (i). Inset: QCP scaling spectrum vs $\omega /T_K^c$ (solid line, in practice coinciding with all spectra for $\lambda \lesssim {10}^{-2}$). The QCP spectrum is seen to be the scaling spectrum $\frac{1}{2}\pi \Gamma D^{1CK}\left(\omega \right)$ (dashed line) for a single-channel Anderson impurity model. The dotted line shows the distinct scaling spectrum of a single-spin 2CK model [onto which cases (a)–(c) scale].

Figure 17

Local dynamic susceptibility $T_K^c{\chi }_1\left(\omega \right)$ vs $\omega /T_K^c$, varying ${\tilde{E}}_{\Delta }=J_c^{\prime }-J^{\prime }$ on approaching the transition from the $(-)$-parity 2CK phase: ${\tilde{E}}_{\Delta }=\lambda T_K^c$ with $\lambda ={10}^{n/2}$ and $n=+4\to -4$. $T_K^c{\chi }_1\left(0\right)$ vanishes as the transition is approached. The dashed line, $T_K^c{\chi }_1\left(\omega \right)\propto \omega$, shows the low-$\omega$ Fermi liquid behavior of the single-channel Anderson model arising at the QCP $\left(\lambda =0\right)$. Inset: $X_1\left(\omega \right)={\chi }_1\left(\omega \right)/\left(2\omega {\left[\text{Re}{⟪S_1^z;S_1^z⟫}_{\omega =0}\right]}^2\right)$ vs $\omega /T_K^c$ at the transition $\left(\lambda =0\right)$, showing recovery of the Korringa-Shiba relation $X_1\left(\omega =0\right)=1$.

Figure 18

Evolution of the 2CK scale for the reduced model Eq. (36): $T_K/D$ vs ${\tilde{E}}_{\Delta }/D$ (with $D$ the conduction bandwidth), for fixed $\rho J_{\text{mix}}=0.075$. Inset: $T_K/D$ vs $\left|{\tilde{E}}_{\Delta }\right|/T_K^c$ on logarithmic axes, with $T_K\sim {\left|{\tilde{E}}_{\Delta }\right|}^2$ on approaching the transition.

Figure 19

Schematic illustration of RG flow between the FPs (circles) of the reduced model. See text for discussion.