Universality classes in Coulomb blockade conductance peak-height statistics

We investigate, using exact diagonalization techniques, the distribution of conductance peak heights in Coulomb blockade regime of a quantum dot connected to two leads under generic dot conditions. The study reveals a three-parametric dependence of the distribution: (i) two dot-lead contact characteristics and (ii) the complexity parameter, mimicking the combined effect of all dot conditions. This also indicates the presence of an infinite range of universality classes of conductance statistics, dominantly characterized just by the complexity parameter and global symmetry constraints.

Figure 1

Size dependence of ${\Lambda }_g$ for AE: Panels (a)–(c) depict the behavior of moment $M_1\equiv \langle g\rangle$ with respect to ${\Lambda }_g$ for AE dots connected to single-channel, symmetric leads for different matrix sizes $N$: (a) Poisson $\to$ GOE, (b) Poisson $\to$ GUE, and (c) GOE $\to$ GUE. The ${\Lambda }_g$ used in the (a) and (b) is given by Eq. (38) with ${\lambda }_a=1$ and by Eq. (41) in (c) with $\gamma =1$. Note $\langle g\left(\infty \right)\rangle =0.25$ in case (a) and $0.31$ in cases (b) and (c) here, which is in agreement with Eq. (29).

Figure 2

Size dependence of ${\Lambda }_g$ for BE: Panels (a)–(c) depict the behavior of $M_1\equiv \langle g\rangle$ with respect to ${\Lambda }_g$ for the BE dots connected to single-channel, symmetric leads for different matrix sizes $N$: (a) Poisson $\to$ GOE, (b) Poisson $\to$ GOE, and (c) GOE $\to$ GOE. The ${\Lambda }_g$ used in (a) and (b) is given by Eq. (39) with ${\lambda }_b\approx 15$ and by Eq. (42) in (c).

Figure 3

Dots with time-reversal symmetry: Poisson $\to$ GOE transition. Panels (a)–(c) show the $P(lng)$ behavior for the $n$-channels leads: (a) $n=1$, (b) $n=2$, and (c) $n=3$. Each part depicts the behavior of three AEs [described by case (i) with ensemble complexity parameter $Y$ given by Eq. (32)] and their BE analogs [case (iv) with $Y$ given by Eq. (36) and $c_1\equiv c$]. For different disorders of the AE case (i), the numerics gives following values: (1) $W=7.0,F\left(0\right)\approx 0.128,I_2^{\mathrm{typ}}=0.011,N{\Lambda }_g=1.489$; (2) $W=5.0,F\left(0\right)\approx 0.15,I_2^{\mathrm{typ}}=0.0056,N{\Lambda }_g=4.018$; (3) $W=1.1,F\left(0\right)\approx 0.27,I_2^{\mathrm{typ}}=0.0026,N{\Lambda }_g=28.04$, where $F\left(E\right)=(N\Delta \left(E\right){)}^{-1}$ and ${\Lambda }_g$ is obtained from Eq. (38) along with ${\lambda }_a=1$, $F\left(E\right)=(N\Delta \left(E\right){)}^{-1}$, and $I_2^{\mathrm{typ}}={\xi }^d$, which gives ${\Lambda }_{g,a}=F^2/\left(N{I}_2^{\mathrm{typ}}\right)$. Using Eq. (39) with $\frac{{\lambda }_a}{{\lambda }_b}\approx 15$, it is easy to check that the BE analogs shown in the figure for each disorder satisfy the theoretical condition given by Eq. (43). Panels (d)–(f) depict the behavior of moments $M_k\left({\Lambda }_g\right)$, $k=1-3$; ${\mathrm{AE}}_n$ and ${\mathrm{BE}}_n$ refer to the AE and BE dots connected to the leads with $n$ channels.

Figure 4

Dots without time-reversal symmetry: Poisson $\to$ GUE transition. Here again panels (a)–(c) show the $P(lng)$ behavior for the $n$ channels leads: (a) $n=1$, (b) $n=2$, and (c) $n=3$. Each part depicts AE case (ii) corresponding to three different values of the diagonal disorder $w$ (with $\varphi \ne 0$) and the BEs represented by the case (v) (for three different $c=c_1=c_2$), respectively. The numerically obtained values for each AE case with $N=1200$ are as follows: (1) $W=1.0,F\left(0\right)\approx 0.28,I_2^{\mathrm{typ}}=0.0018,N{\Lambda }_g=43.56$; (2) $W=3.0,F\left(0\right)\approx 0.19,I_2^{\mathrm{typ}}=0.0026,N{\Lambda }_g=13.89$; (3) $W=5.0,F\left(0\right)\approx 0.15,I_2^{\mathrm{typ}}=0.0048,N{\Lambda }_g=4.69$, where $F\left(E\right)=(N\Delta \left(E\right){)}^{-1}$ and ${\Lambda }_g$ is obtained from Eq. (38) along with ${\lambda }_a=1$, $F\left(E\right)=(N\Delta \left(E\right){)}^{-1}$ and $I_2^{\mathrm{typ}}={\xi }^d$, which gives ${\Lambda }_{g,a}=F^2/\left(N{I}_2^{\mathrm{typ}}\right)$. Again, using Eq. (39) with $\frac{{\lambda }_a}{{\lambda }_b}=5$, one can verify that the BE analogs for each disorder case satisfy the theoretical condition given by Eq. (43). Panels (d)–(f) depict the behavior of moments $M_k\left({\Lambda }_g\right)$, $k=1-3$; ${\mathrm{AE}}_n$ and ${\mathrm{BE}}_n$ refer to the AE and BE dots connected to the leads with $n$ channels.

Figure 5

Dots without time-reversal symmetry: GOE $\to$ GUE transition. Here again the $P(lng)$ behavior is compared for the $n$-channels leads: (a) $n=1$, (b) $n=2$, and (c) $n=3$ but the AEs now correspond to case (iii) with $Y$ given by Eq. (33) (three different nonzero $\varphi$ values) and BEs represented by the case (vi) with $Y$ given by Eq. (37) (three different $c_2$ values), respectively. The numerics gives following values for each AE case (iii): (1) $\varphi =2.27\times {10}^{-3},F\left(0\right)\approx 0.27,I_2^{\mathrm{typ}}=2.4\times {10}^{-3},{\Lambda }_g=0.01$; (2) $\varphi =4.53\times {10}^{-3},F\left(0\right)\approx 0.27,I_2^{\mathrm{typ}}=2.1\times {10}^{-3},{\Lambda }_g=0.04$; (3) $\varphi =1.13\times {10}^{-2},F\left(0\right)\approx 0.27,I_2^{\mathrm{typ}}=1.9\times {10}^{-3},{\Lambda }_g=0.25$, where $F\left(E\right)=(N\Delta \left(E\right){)}^{-1}$ and ${\Lambda }_g$ is obtained from Eq. (41). Again, using Eq. (42), one can verify that the BE analogs of AE case (iii) for each $\varphi$ case, shown in (a), (b), and (c), satisfy the theoretical condition given by Eq. (43). Here again (d)–(f) depict the behavior of moments $M_k\left({\Lambda }_g\right)$, $k=1$–3 with respect to ${\Lambda }_g$; ${\mathrm{AE}}_n$ and ${\mathrm{BE}}_n$ refer to the AE and BE dots connected to the leads with $n$ channels.

Figure 6

Comparison with theory: Dots with time-reversal symmetry. (a) Although the theoretical solution of Eq. (20) is obtained only as a series for the Poisson initial condition, Eq. (24) [obtained for $P(g,0)=\delta \left(g\right)$ initial condition] is well applicable for the case (i) in the weak-disorder limit [due to almost independence of the solution of Eq. (20) from the initial state in the large-${\Lambda }_g$ limit]. Here the function $f_1$ is given by Eq. (45) with ${\tau }_0\approx 2$: $f_1\left(x\right)=0.7977\exp (0.5x-2e^x)$. (b) As disorder increases, Eq. (24) is applicable only in the lower part (below the peak) of the $P_g\left(g\right)$. The full distribution now seems to be a superposition of Eq. (45) with ${\tau }_0\approx 2.33$ and a Gaussian; here $f_2,f_3$ are fitted functions with $f_2\left(x\right)=1.33\exp (0.5x-6.1e^x)$, $f_3\left(x\right)=\frac{0.75}{\sqrt{4\pi }}exp[-(1/4){(x+2.3)}^2]$. (c) For very strong disorder, i.e., $w=19.0$, AE case (i) approaches a Gaussian distribution, referred to as $f_4$: $f_4\left(x\right)=\frac{1}{\sqrt{86\pi }}exp[-(1/86){(x+11)}^2]$. The approach to Gaussian behavior in the insulator regime is also suggested by previous studies [2]. (d) $\Lambda =2N{\Lambda }_g$ governed evolution of average peak height $\langle g\left(\Lambda \right)\rangle$ and peak-height variance ${\sigma }^2\left(\Lambda \right)$ for AE case (i) for a single-channel lead with $N=1200$ and their comparison with theory given by Eqs. (27) and (28): ${\langle g\left(\Lambda \right)\rangle }_{th}=0.24(1.0-exp(-0.25\Lambda ))$ and ${\sigma }_{th}\left(\Lambda \right)=0.09(1.35-exp(-0.7\Lambda ))$. Note that the large ${\Lambda }_g$ results for both $\langle g\left(\Lambda \right)\rangle$ and ${\sigma }^2\left(\Lambda \right)$ agree with theoretical results (see Ref. [19]) for a GOE dot with single-channel symmetric leads: $\langle g\left(\infty \right)\rangle \approx 0.25$ and ${\sigma }^2\left(\infty \right)\approx 0.125$. As expected for symmetric leads, Eq. (29) then gives almost the same results for $t_1$ and $t_2$: $t_1\approx t_2=0.25N$, which also agrees with Eq. (22).