Crossover of conductance and local density of states in a single-channel disordered quantum wire

The probability distribution of the mesoscopic local density of states (LDOS) for a single-channel disordered quantum wire with chiral symmetry is computed in two different geometries. An approximate ansatz is proposed to describe the crossover of the probability distributions for the conductance and LDOS between the chiral and standard symmetry classes of a single-channel disordered quantum wire. The accuracy of this ansatz is discussed by comparison with a large-deviation ansatz introduced by H. Schomerus and M. Titov in Phys. Rev. B 67, 100201 (2003).

Figure 1

In this paper, we will consider two different simply connected geometries, which are imposed by suitable boundary conditions at $y=-L∕2$ and $y=+L∕2$, respectively, for a strictly 1D disordered quantum wire. In geometry (a) and (b) the disordered quantum wire is closed to the right and open to the left where it is connected to an ideal lead. The particle in geometry (b) is subjected to absorption whereas it is not in geometry (a). In geometry (c) the disordered quantum wire is open to the left and to the right where it is connected to ideal leads.

Figure 2

(a) A thin slice of length $\delta L$ with $a⪡\delta L⪡\ell ⪡L$ is added to the left of the disordered region of length $L$. (b) Two disordered regions 1 and 2 with scattering matrices $S_1$ and $S_2$, respectively, in a quantum wire.

Figure 3

Parameter space for the Fokker-Planck equations (3.7) or (3.12). The point $\mathrm{Ch}1$ has the coordinates $\epsilon ={\zeta }_0=0$, $\zeta =-1$. It represents the chiral symmetry class. The point $\mathrm{KW}1$ has the coordinates $\epsilon =0$, ${\zeta }_0=-\zeta =1$. It represents the Kappus-Wegner anomaly. The two shaded planes represent the standard symmetry class. The regions $\zeta >0$ and $\zeta <0$ are equivalent in that they are related by a rotation by an angle $\pi$ around ${\sigma }_3$ of the Pauli matrices in the Hamiltonian (2.1a). Especially, this rotation relates $\mathrm{Ch}1$ and $\mathrm{KW}1$ to the points $\mathrm{Ch}2$ and $\mathrm{KW}2$, respectively.

Figure 4

The mean (a) and variance (b) of the conductance computed from the probability distribution (4.3) as a function of $t=L∕\ell$ for $\alpha =0,0.5,1$ and $\zeta =-1$.

Figure 5

Numerical evaluation of $1∕(1-\alpha )$ as a function of $\epsilon$ in the crossover regime (4.3). The integral $\alpha$ defined in Eq. (4.15b) is evaluated numerically for the stationary distribution of the phase (4.9). The line representing $-(1∕2)\log \mid \epsilon \mid$ is a guide to the eyes.

Figure 6

(a) Disordered quantum wire of length $L$ closed at the right end and connected to a perfect lead at the left end. Energy levels within the wire of length $L$ are broadened beyond the mean level spacing by the coupling to the perfect lead. (b) Disordered quantum wire of length $L$ closed at both ends. Energy levels within the wire are broadened beyond the mean level spacing by the absorption $\mathrm{Im}E={\mathcal{E}}^{″}>0$.

Figure 7

(a) Three traces for the probability distribution of the LDOS are plotted with $\epsilon =0,0.1,+\infty$ (from the bottom up at $\nu =0.01$) so as to interpolate between the chiral and standard symmetry classes (6.3) and (6.4), respectively. Each trace is obtained from numerical integration of Eq. (6.7) for the semiopen geometry of Fig. 6a with $L_{\mathrm{L}}∕\ell =2$. (b) Three traces for the probability distribution of the LDOS are plotted with $\epsilon =0,0.1,+\infty$ (from the bottom up at $\nu =0.1$) so as to interpolate between the chiral and standard symmetry classes (6.3) and (6.4), respectively. Each trace is obtained from numerical integration of Eq. (6.7) for the closed geometry of Fig. 6b with $\omega =1∕12$. We have verified that numerical integration of Eq. (6.7) agrees with a representation of Eq. (6.7) in terms of elementary functions when possible.