Open chaotic Dirac billiards: Weak (anti)localization, conductance fluctuations, and decoherence

In this paper, we investigate the transport properties of open chaotic Dirac billiards and their intrinsic (chiral universal) symmetry classes. The prominent examples of these systems are some categories of topological insulators and graphene structures. We extend the diagrammatic method of integration over the unitary group and obtain analytical results for the semiclassical limit and for the high quantum limit in the universal regime. We show the emergence of quantum fingerprints characteristic of the chiral symmetries, which are amplified in the presence of a single open channel in each electronic terminals. We compare the chaotic Dirac billiards with the “Schrödinger billiards” in a myriad of regimes, exhibiting the differences between the chiral universal classes and the Wigner-Dyson classes. Two numerical methods were used to confirm our analytical findings, yielding also the distribution of conductances. We also investigate analytically the effect of dephasing using the characteristic time scales of the chaotic billiards and we show the appearance of peculiar numbers of chaos.

Figure 1

(a) Basic objects of the diagrammatic rules. In decreasing order, a random element ($U_{ij}$) of the $U$ matrix, a projector on the space of channels of the terminals ($C_i$), and a Kronecker delta element, representing the contraction of indices (${\delta }_{ij}$). (b) Diagrammatic representation of Eq. (8). (c) The ensemble average of the conductance, Eq. (9). Notice that diagrams 1, 2, and 3 have weights $V_{11}$, and diagrams 4 to 9 have weights $V_2$. The weights were calculated in Ref. [37].

Figure 2

The average and variance of conductance are plotted in units of $4e^2/h$ and $8e^4/h^2$, respectively. (a),(b) Symmetric terminals, $N=N_1=N_2$, and (c),(d) asymmetric terminals with $N_1=2$. The dotted lines are obtained from QR (Gram-Schmidt) factorization and from Hamiltonian model (HM) numeric simulations. Solid lines are the exact results of Eqs. (12) and (17).

Figure 3

(Left) The figure indicates the comparison between the variance $\mathrm{var}\left[G\right]$ (in units of $8e^4/h^2$) of a chaotic Dirac billiard of chiral ensembles (CH) with the variance $\mathrm{var}{\left[G\right]}_{wd}$ (in unitos of $4e^4/h^2$) of Wigner-Dyson chaotic billiards (WD), both with symmetric terminals, $N=N_1=N_2$. The dotted lines are the plots of Eq. (17) (DM-CH) and the solid lines are the plots of Eq. (B3) (DM-WD). The symbols refer to QR (Gram-Schmidt) factorization (QR) and Hamiltonian model (HM) numeric simulations for chiral ensembles (CH) and Wigner-Dyson ensembles (WD). (Right) We compare the distribution of conductance $P\left(G\right)$ in the high quantum regime ($N_1=N_2=1$), in units of $4e^2/h$, between a chaotic Dirac billiard of chiral ensembles (right column) and, in units of $2e^2/h$, with the chaotic quantum billiard of Wigner-Dyson ensembles (Ref. [38]) (left column) with ${10}^6$ realizations of the random $S$ matrix.

Figure 4

(a) Week localization and (b) antilocalization corrections as a function of $N_{\varphi }$ for the high quantum regime, $N_1=N_2=1$. The symbols denote QR (Gram-Schmidt) factorization (QR) and Hamiltonian model (HM) numeric simulations for chiral ensembles with ${10}^6$ realizations. Solid lines are interpolation formulas.

Figure 5

(a) Variance of the conductance as a function of $N_{\varphi }$ in the high quantum regime, $N_1=N_2=1$. Top to bottom curves specify $\beta =1,2,4$ behavior, respectively. (b) The ratios between variance of conductances for $N_{\varphi }\gg 1$. They tend to $2.5$ due to chCOE/chCUE, and $0.44$ due to chCSE/chCUE, as obtained in Eq. (30). The symbols designate the numeric simulations of QR (Gram-Schmidt) factorization (QR) and Hamiltonian model (HM) chiral ensembles, both with ${10}^6$ realizations. Solid lines are interpolation formulas.

Figure 6

Conductance distributions for the chaotic Dirac billiards (right column) and for the chaotic quantum billiard (left column). The distribution $P\left(G\right)$ tends to the Gaussian as $N_{\varphi }$ is increased. The symbols are for numerical simulations of QR (Gram-Schmidt) factorization (QR) and Hamiltonian model (HM) for chiral ensembles and Wigner-Dyson ensembles (Ref. [31]), both with ${10}^6$ realizations.

Figure 7

The figures show examples of the diagrams resulting from Eq. (16), each one having weight $V_{1111}$, $V_{1,1,2}$, $V_{2,2}$, $V_{1,3}$, and $V_4$, for diagrams 1, 2, 3, 4, and 5, respectively. The weights were calculated in Ref. [37].