## Abstract

Spectral statistics of weakly disordered triangular graphene flakes with zigzag edges are revisited. Earlier, we have found numerically that such systems may show spectral fluctuations of Gaussian unitary ensemble (GUE), signaling the time‐reversal symmetry (TRS) breaking at zero magnetic field, accompanied by approximate twofold valley degeneracy of each energy level. Atomic‐scale disorder induces the scattering of charge carriers between the valleys and restores the spectral fluctuations of Gaussian orthogonal ensemble (GOE). A simplified description of such a nonstandard GUE‒GOE transition, employing the mixed ensemble of 4 × 4 real symmetric matrices was also proposed. Here, we complement our previous study by analyzing numerically the spectral fluctuations of large matrices belonging the same mixed ensemble. Resulting scaling laws relate the ensemble parameter to physical size and the number of atomic‐scale defects in graphene flake. A phase diagram, indicating the regions in which the signatures of GUE may by observable in the size‐doping parameter plane, is presented.

### Keywords

- graphene
- quantum chaos
- random matrix
- time-reversal symmetry
- gaussian ensemble

## 1. Introduction

The notion of emergent phenomena was coined out by Anderson in his milestone science paper of 1979 [1]. In brief, emergence occurs when a complex system shows qualitatively different properties then its building blocks. Numerous examples of emergent systems studied in contemporary condensed matter physics, including high‐temperature superconductors and heavy‐fermion compounds [2], are regarded as systems with spontaneous symmetry breaking [3]. A link between emergence and spontaneous symmetry breaking, however, does not seem to have a permanent character. In a wide class of electronic systems, such as semiconducting heterostructures containing a two‐dimensional electron gas (2DEG), physical properties of itinerant electrons are substantially different than properties of free electrons (or electrons in atoms composing the system), and are also highly‐tunable upon variation of external electromagnetic fields [4]. To give some illustration of this tunability, we only mention that electrons in GaAs heterostructures can be usually described by a standard Schrödinger equation of quantum mechanics with the effective mass

It is rather rarely noticed that graphene, a two‐dimensional form of carbon just one atom tick [7], also belongs to the second class of emergent systems (i.e., without an apparent spontaneous symmetry breaking) described briefly above. In a monolayer graphene, effective Hamiltonian for low‐energy excitations has a Dirac‒Weyl form, namely

where

Strictly speaking,

*are slightly different. It is also worth to mention so‐called artificial graphenes, in which waves (of different kinds) obey their effective Dirac equations [11‒13].*other Dirac systems

A peculiar nature of Dirac fermions in graphene originates from the chiral structure of the Hamiltonian

Although the interest in graphene and other Dirac systems primarily focus on their potential applications [28, 29], quite often linked to the nonstandard quantum description [8], we believe that the fundamental perspective sketched in the above also deserves some attention. In the remaining part of this article, we first overview basic experimental, theoretical and numerical findings concerning signatures of quantum chaos in graphene and its nanostructures (Section 2). Next, we present our new numerical results concerning the additive random matrix model originally proposed in Ref. [25] to describe a nonstandard GUE‒GOE transition, accompanied by lifting out the valley degeneracy (Section 3). The consequences of these findings for prospective experiments on graphene nanoflakes, together with the phase diagram depicting the relevant matrix ensembles in the system size‐doping plane, are described in Section 4. The concluding remarks are given in Section 5.

## 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,

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.

## 3. Transition GUE‒GOE for real symmetric matrices

### 3.1. Additive random‐matrix models: brief overview

Additive random‐matrix models are capable of reproducing the evolutions of spectral statistics in many cases when a complex system undergoes transition to quantum chaos or transition between symmetry classes [32, 39]. The discussion usually focus on the auxiliary random Hamiltonian of the form

where

To describe transition to quantum chaos rather then transition between symmetry classes in a chaotic system, one can choose

For instance, if elements of

where

The above follows from the normalization condition

The limiting forms of the spacing distribution given by Eqs. (3) and (4) are

coinciding with well‐known Wigner surmises for GOE and GUE, respectively [32]. For

Relatively recently, spectra of models employing self‐dual random matrices have attracted some attention [41]. In such models, the matrix

where random matrix

In the remaining part of section, we focus on the transition between self‐dual GUE to GOE, show that the corresponding Hamiltonian

### 3.2. Self‐dual GUE to GOE via 4 × 4 real‐symmetric matrices

We focus now on the situation, when the matrix

For

where

The matrix on the right‐hand side of Eq. (10) is self‐dual, and can be further transformed as

where

Exchanging the second with the third row and column in the rightmost matrix in Eq. (11) we arrive to

where the blocks

Spectral statistics of the Hamiltonian

The nearest‐neighbor spacings distribution can be approximated by

with

and

Eqs. (15) and (16) represent simplified versions of the corresponding formulas given in Ref. [43]. A comparison with the numerical will be given later in this section.

### 3.3. Self‐dual GUE to GOE via 2 N × 2 N real‐symmetric matrices

We consider now the case of large random matrices (

Ensembles of large pseudo‐random Hamiltonians

We find that nearest‐neighbor level spacings of large matrix

with the empirical relations of Eqs. (15) and (16) [see blue solid lines in Figure 3] now replaced by

and

The above formulae are marked in Figure 3 with red dashed lines. We also find that the scaling law

## 4. Consequences for graphene nanoflakes

### 4.1. Level‐spacing distributions revisited

In this section, the empirical distribution

At zero magnetic field, the tight‐binding Hamiltonian for weakly‐disordered graphene can be written as

where

A model of substrate‐induced disorder, constituted by Eqs. (20) and (21), was widely used to reproduce numerically several transport properties of disordered graphene samples [45‒48]. Here, we revisit the spectra of closed graphene flakes considered in Ref. [25], within a simplified empirical model

A compact measure of the disorder strength is given by the dimensionless correlator

where the system area

For

The numerical results are presented in Figure 5, where we have fixed the remaining disorder parameters at

The disorder parameters are actually same as in Figures 8 and 9 of Ref. [25], where we have mistakenly omitted the factor

where the total number of terminal sites

### 4.2. Phase diagram for triangular flakes with zigzag edges

Eq. (25) is now employed to estimate the maximal system size

On the other hand, system size and the number of energy levels taken into account must be large enough to distinguish between spectral fluctuations of GUE and spectral fluctuations of other ensembles.

Density of states (per one direction of spin) for bulk graphene reads

The number of energy levels

Physically, occupying _{2}‐based substrates [49].

Level‐spacing distributions

where Eq. (33) refers to the empirical distribution

We use the property of

where

where the factor

For

Limiting values of

## 5. Concluding remarks

We have revisited level‐spacing statistics of triangular graphene nanoflakes with zigzag edges, subjected to weak substrate‐induced disorder. Our previous study of the system is complemented by comparing the spectral fluctuations with these of large random matrices belonging to a mixed ensemble interpolating between GUE with self‐dual symmetry and generic GOE. The results show that for a fixed value of maximal Fermi energy

In conclusion, we expect that triangular graphene flakes with * perfect* zigzag edges may show signatures of TRS breaking starting from physical sizes exceeding

## Acknowledgments

The author thanks to Huang Liang for the correspondence. The work was supported by the National Science Centre of Poland (NCN) via Grant no. 2014/14/E/ST3/00256. Computations were partly performed using the PL‐Grid infrastructure.

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## Notes

- Strictly speaking, ℋeff Eq. (1) applies to quasiparticles near the K valley in the dispersion relation. To obtain the effective Hamiltonian for other valley (K') it is sufficient to substitute σy→−σy.
- To describe transition to quantum chaos rather then transition between symmetry classes in a chaotic system, one can choose H0 to be a diagonal random matrix, elements of which follow a Gaussian distribution with zero mean and the variance 〈(H0)ij2〉=δij.
- The disorder parameters are actually same as in Figures 8 and 9 of Ref. [25], where we have mistakenly omitted the factor π in the numerical evaluation of K0.
- We use the property of m-th cumulant of the distribution P(S)=12[aP1(aS)+bP2(bS)], which is equal to 〈Sm〉P=1/2(a(−m)〈Sm〉(P1)+b(−m)〈Sm〉(P2)).. For P1=PGOE and P2=PGOE−GUE, see Eqs. (6) and (3), necessary integrals for m=2, 3, 4 can be calculated analytically.