Kondo Screening Cloud Around a Quantum Dot: Large-Scale Numerical Results

Measurements of the persistent current in a ring containing a quantum dot would afford a unique opportunity to finally detect the elusive Kondo screening cloud. We present the first large-scale numerical results on this controversial subject using exact diagonalization and density matrix renormalization group (RG). These extremely challenging numerical calculations confirm RG arguments for weak to strong coupling crossover with varying ring length and give results on the universal scaling functions. We also study, analytically and numerically, the important and surprising effects of particle-hole symmetry breaking.

Figure 1

The current at $1/2$-filling for $N=4p-1$ (a) and $N=4p$ (b) for the EQD. Results are shown for a number of system sizes with $J_K=1$ as a function of $\alpha /\pi$. The lines are ED, the symbols DMRG results.

Figure 2

The current $jL/e{v}_F$ for $J_K=1$ and $\alpha /\pi =0.05$ vs $L$ for the EQD. Shown are ED results ($•$), DMRG results with ($m_L=1024$, $m_R=512$) ($\circ$) and ($m_L=1024$, $m_R=2048$) ($\square$). The dashed lines indicate the free ring result.

Figure 3

The current at $1/2$-filling $jL/e{v}_F$, at $\alpha /\pi =0.4$ ($N=4p$ $\circ$) and at $\alpha /\pi =0.8$ ($N$ odd $△$), plotted vs $L/{\xi }_K$ obtained by combining our results for various values of $J_K$ for the EQD. We fix ${\xi }_K(J_K=0.3)\equiv 1$ and obtain the remaining ${\xi }_K$ relatively to ${\xi }_K(J_K=0.3)$ by rescaling. The inset shows the obtained ${\xi }_K(J_K)$ as a function of $1/J_K$ for even ($\circ$) and odd ($△$) $N$.

Figure 4

(a) The current $jL/e$ at $1/2$-filling for the SCQD for $\tilde{\alpha }/\pi =0.15$ and $J_K=0.75$, 1.60, 4.00 vs $L$. Solid (open) symbols are ED (DMRG with $m_L=1024$, $m_R=2048$) results. A function with pure $1/L$ behavior is shown. (b) The current $jL/e$ at $1/4$-filling for the SCQD for $\tilde{\alpha }/\pi =0.75$ and $J_K=4$ vs $L$. ED results are shown for even $N$ ($\circ$) and odd $N$ ($•$). The dashed line is a power-law fit to the points with $N=5-11$, of $\mathrm{\text{slope}}=-1$. The symmetry reduced Hilbert space at $N=12(L=22)$ is 3, 929, 717, 484.