Quantum Fluctuations along Symmetry Crossover in a Kondo-Correlated Quantum Dot

Universal properties of entangled many-body states are controlled by their symmetry and quantum fluctuations. By the magnetic-field tuning of the spin-orbital degeneracy in a Kondo-correlated quantum dot, we have modified quantum fluctuations to directly measure their influence on the many-body properties along the crossover from SU(4) to SU(2) symmetry of the ground state. High-sensitive current noise measurements combined with the nonequilibrium Fermi liquid theory clarify that the Kondo resonance and electron correlations are enhanced as the fluctuations, measured by the Wilson ratio, increase along the symmetry crossover. Our achievement demonstrates that nonlinear noise constitutes a measure of quantum fluctuations that can be used to tackle quantum phase transitions.

Figure 1

Central inset: the spin and orbital (valley) degrees of freedom of an electron in a CNT are shown by the straight and circle arrows, respectively. The magnetic field is applied making an angle $\theta$ with the CNT axis. The figure is partially drawn by VESTA [25]. (a) Top panel: representation of the four degenerate ground states for a dot containing one particle ($N=1$ and $N=3$). Middle panel: single-particle energy spectrum as a function of $B$ for the case of $g_{\text{orb}}{\mu }_B\cos \theta =g_S{\mu }_{B}S$. Bottom panel: at $B=0$, four states are degenerate forming an SU(4) Kondo state. At finite field, the Kondo effect disappears. (b) Top panel: representation of the six degenerate ground states for a dot with two particles ($N=2$). Middle panel: the total magnetic energy is $E=({\sigma }_1+{\tau }_1+{\sigma }_2+{\tau }_2){\mu }_{B}B$. Each line is twice degenerate and corresponds to the states labeled on the graph. The shade around the ground state represents $k_B{T}_K$. Bottom panel: at $B=0$, an SU(4) Kondo state is formed due to the sixfold degeneracy. At high field it evolves continuously to SU(2) (see text).

Figure 2

(a) Stability diagrams for $B=0$, 4, 8, and 10 T at $T=16\mathrm{mK}$ as contour plots of the conductance $G$ as a function of $V_g$ and $V_{\text{sd}}$. The Kondo resonance produces the bright broad vertical lines at $V_{\text{sd}}=0$. This ridge disappears at high field for $N=1$ and 3. It is split into two satellite peaks at high $V_{\text{sd}}$ separated by $\mathrm{\Delta }e{V}_{\text{sd}}\approx 4{\mu }_{B}B$. At $N=2$, $G$ decreases but it remains maximum at $V_{\text{sd}}=0$. (b) Comparison of zero-bias conductance between the experiment (solid lines) and the NRG calculation (dashed lines) for several magnetic fields. (c) Field dependence of $T_K$ at $N=2$. The dashed line is the result of the NRG calculation multiplied by a factor of 1.3 to fit the experimental value at $B=0$. The error bars result from the scaling procedure.

Figure 3

(a) Wilson ratio $R$ as a function of $B$. This ratio measures the strength of dominant quantum fluctuations between the two states represented on the graph, which yield the crossover [26]. $R$ is computed by NRG calculations with parameters yielding the successful comparison with experiments for $G$ and $F$ at the filling $N=2$. (b) Conductance $G$ and Fano factor $F$ as a function of $B$. The $F$’s are obtained at each field by a linear fit of the current noise at very low current using the formula $S_I=2eF{I}_{\text{sd}}$. The dashed lines are the corresponding NRG results. (c) Magnetic field dependence of the two transport channels $T_1$ and $T_2$ (symbols), which are calculated from $G$ and $F$ as explained in the text. The dashed lines are the corresponding NRG results.

Figure 4

(a) Nonlinear noise $S_K$ as a function of the backscattered current $I_K$ at $B=0\mathrm{T}$ [SU(4) state] and $B=13\mathrm{T}$ [SU(2) state]. The solid and dashed lines are the result of the linear fit yielding $e^{*}/e=1.4\pm 0.1$ at $B=0\mathrm{T}$ and $e^{*}/e=1.7\pm 0.1$ at $B=13\mathrm{T}$, respectively. (b) The filled circles show the effective charge $e^{*}/e$ as a function of $R$, which quantifies the strength of fluctuations (the error bars for $R$ originate from the uncertainty of $g_{\text{orb}}\cos \theta$). The three square symbols represent the theoretical prediction for SU(4), SU(2), and noninteracting particles. The dashed line is the extended theoretical prediction given in the text [Eq. (2)].