**2. Gauge fields, fluctuations and chaos in nanoscale graphene structures**

Dirac fermions confined in graphene quantum dots [30] have provided yet another surprising situation, in which a piece of handbook knowledge needed a careful revision [31].

Quantum chaotic behavior appears generically for systems, whose classical dynamics are chaotic, and manifest itself via the fact that energy levels show statistical fluctuations following those of Gaussian ensembles of random matrices [32]. In particular, if such a system posses the time‐reversal symmetry (TRS), its spectral statistics follows the Gaussian orthogonal ensemble (GOE). A system with TRS and half‐integer spin has the symplectic symmetry and, in turn, shows spectral fluctuations of the Gaussian symplectic ensemble (GSE). If TRS is broken, as in the presence of nontrivial gauge fields, and the system has no other antiunitary symmetry [33], spectral statistics follow the Gaussian unitary ensemble (GUE). For a particular case of massless spin‐1/2 particles, it was pointed out by Berry and Mondragon [34], that the confinement may break TRS in a persistent manner (i.e., even in the absence of gauge fields), leading to the spectral fluctuations of GUE.

When applying the above symmetry classification to graphene nanosystems [24, 25], one needs, however, to take into account that Dirac fermions in graphene appear in the two valleys, *K* and *K* ' , coupled by TRS. (In particular, real magnetic field breaks TRS and has the same sign in the two valleys, whereas the strain‐induced gauge field preserves TRS and has opposite signs in the two valleys.) If the valley pseudospin is conserved, a special (symplectic) time‐ reversal symmetry (STRS) becomes relevant, playing a role of an effective TRS in a single valley [24]. Both real magnetic and strain‐induced gauge fields may break STRS leading to the spectral fluctuations of GUE [35]. As demonstrated numerically in Ref. [25], such fluctuations also appear for particular closed nanosystems in graphene in the presence of random scalar potentials slowly varying on the scale of atomic separation. Such nanosystems include equilateral triangles with zigzag or Klein edges, i.e., with terminal atoms belonging to one sublattice. Generic graphene nanoflakes with irregular edges show spectral fluctuations of GOE [24], as strong intervalley scattering restores TRS in the absence of gauge fields (see **Figure 1**). In contrast, the boundary effects are suppressed in open graphene systems, for which signatures of the symplectic symmetry class were reported [36].

**Figure 1.** Transitions between symmetry classes and random matrix ensembles relevant for *closed* nanosystems in gra‐ phene characterized by the disorder strength, the intervalley scattering rate, and (optionally) placed in the weak mag‐ netic field *B*. (Reprinted with permission from Ref. [25].)

It is worth mention here, that triangular graphene flakes, similar to studied theoretically in Ref. [25], have been recently fabricated [37, 38]. However, due to the hybridization with metallic substrates, quantum‐dot energy levels in such systems are significantly broaden, making it rather difficult to determine the symmetry class via spectral statistics.
