Universality class

According to random matrix theory, ordinary disordered quantum wires are classified into the standard universality classes, orthogonal, unitary and symplectic. The universality classes describe transport properties, which are independent of the microscopic details of disordered wires. These classes can be specified by time-reversal and spin–rotation symmetry. The orthogonal class consists of systems having both time-reversal and spin–rotation symmetries, while the unitary class is characterized by the absence of TRS. The systems having TRS without spin–rotation symmetry belong to the symplectic class. These universality classes have been believed to inevitably cause Anderson localization although typical behaviors are different from class to class.

In the graphene system, the presence or absence of inter-valley scattering affects the

TRS of the system. If inter-valley scattering is absent, i.e. u ⁰ _X (r) = 0, the Hamiltonian imp becomes invariant under the transformation of ^S = −i(σ ^y ⊗τ⁰) C, where C is the complex-conjugate operator. This operation corresponds to the special time-reversal operation for pseudospins within each valley, and supports that the system has symplectic symmetry. However, in the presence of inter-valley scattering due to SRI, the invariance under S is broken. In this case, the TRS across two valleys described by the operator T = (σ ^z ⊗τ ^x) C becomes relevant, which indicates orthogonal universality class. Thus as noted in [72], graphene with LRI belongs to symplectic symmetry, but that with SRI belongs to orthogonal symmetry (see figure 9).

Graphene

Orthogonal Symplectic

d

a _T

Nano−graphene WL φ

Zigzag

Orthogonal Unitary

d

a _T

Armchair

Orthogonal Orthogonal

d

a _T

Figure 9. Summary concerning the universality crossover. On increasing the range of the impurity potential, graphene is known to be orthogonal for SRIs and symplectic for LRIs. However, zigzag nanoribbons are unitary class for SRIs. Armchair ribbons are classified into orthogonal class for the whole impurity range. L _φ is the phase coherence length. W is the width of graphene ribbons.

However, in the zigzag nanoribbons, the boundary conditions, which treat the two sublattices asymmetrically leading to edge states give rise to a single special mode in each valley. Considering now one of the two valleys separately, say the one around k = k ₊, we see that the pseudo TRS is violated in the sense that we find one more left-moving than rightmoving mode. Thus, as long as disorder promotes only intra-valley scattering, the system has no TRS. On the other hand, if disorder yields inter-valley scattering, the pseudo TRS disappears but the ordinary TRS is relevant making a complete set of pairs of time-reversed modes across the two valleys. Thus we expect to see qualitative differences in the properties if the range of the impurity potentials is changed.

The presence of one PCC has been recently found in disordered metallic carbon nanotubes with LRI [61]. The PCC in this system originates from the skew-symmetry of the reflection matrix, ^t r = − r [61], which is special to the symplectic symmetry with an odd number of channels. The electronic transport properties of such systems have been studied on the basis of the random matrix theory [62, 63]. On the other hand, zigzag ribbons without inter-valley scattering are not in the symplectic class, since they break TRS in a special way. The decisive feature for a PCC is the presence of one excess mode in each valley as discussed in the previous section.

In view of this classification we find that the universality class of the disordered zigzag nanoribbon with LRI potential (no inter-valley scattering) is the unitary class (no TRS). On the other hand, for SRI potentials with inter-valley scattering the disordered ribbon belongs to the orthogonal class (with overall TRS). Consequently, we can observe a crossover between two universality classes when we change the impurity range continuously.

However, in the disordered armchair nanoribbons, the special TRS within each valley is broken even in the case of LRI. This is because u ⁰ _X (r) 6= 0 as we have seen in section 2.3. Thus, irrespective of the range of impurities, the armchair nanoribbons are classified into orthogonal universality class. Since the disordered zigzag nanoribbons are classified into unitary class for LRI but orthogonal class for SRI [23], it should be noted that the universality crossover in a

Figure 10. The magnetic field dependence of the averaged conductance for disordered zigzag nanoribbons with (a) LRIs and (b) SRIs. Similarly, the case for armchair nanoribbons with (c) LRIs and (d) SRIs. φ is the magnetic flux through a hexagonal ring, which is measured in units of ch / e.

nanographene system can occur not only due to the range of impurities but also due to the edge boundary conditions (see figure 9).

The application of a magnetic field enforces the above arguments. In figure 10, the magnetic field dependence of the averaged conductance for disordered zigzag nanoribbons with (a) LRIs and (b) SRIs is presented. Similarly, the case for armchair nanoribbons with (c) LRIs and (d) SRIs is presented. Here we have included the magnetic field perpendicular to the graphite plane, which is incorporated via the Peierls phase: γ _{i , j} → γ _{i , j} exp[i2π _ch ^{e R} _i^j d l · A ], where A is the vector potential. Since the time-reversal symmetry within the valleys for zigzag ribbons is already broken, the averaged conductance h g i for LRIs (absence of inter-valley scattering) is quite insensitive to the application of magnetic field as can be seen in figure 10(a). This is consistent with the behavior in the unitary class. For higher energies, weak magnetic field dependence appears due to inter-valley scattering. For all the other cases a weak magnetic field improves the conductance, i.e. weak localization behavior, which is typical for the orthogonal class.