Complex Dynamics in Circular and Deformed Bilayer Graphene-Inspired Billiards with Anisotropy and Strain

Abstract

While billiard systems of various shapes have been used as paradigmatic model systems in the fields of nonlinear dynamics and quantum chaos, few studies have investigated anisotropic billiards. Motivated by the tremendous advances in using and controlling electronic and optical mesoscopic systems with bilayer graphene (BLG), representing an easily accessible anisotropic material for electrons when trigonal warping is present, we investigate billiards of various anisotropies and geometries using a trajectory-tracing approach founded on the concept of ray–wave correspondence. We find that the presence of anisotropy can change the billiards’ dynamics dramatically from its isotropic counterpart. It may induce chaotic and mixed dynamics in otherwise integrable systems and may stabilize originally unstable trajectories. We characterize the dynamics of anisotropic billiards in real and phase space using Lyapunov exponents and the Poincaré surface of section as phase space representation.

1. Introduction

Billiards of all kinds have proven to be versatile model systems for mathematics and physics. Besides a variation in the shape, or geometry, of hard-wall billiards typically studied in the field of (classical) nonlinear dynamics, one can use physical inspiration and change their properties, such as their boundary conditions or equations of motion, to apply them to studies of electrons or light [1]. This has been carried out in the context of quantum chaos [2], for example, for electronic billiards realized in the form of quantum dots or light confined in dielectric cavities of certain shapes by total internal reflection. An important topic has been the investigation of wave billiards and their classical counterparts, where wave–ray or quantum–classical correspondence is expected and has been confirmed in many examples, e.g., [3,4].

Here, we will focus on anisotropic billiards for electrons, inspired by technological progress in the fabrication of bilayer graphene (BLG) samples [5,6,7], and employ their unique properties. Therefore, we use a classical approach based on wave–particle correspondence, which is justified by the ballistic behavior of electrons in bilayer graphene on the microscale [8,9,10,11]. Epitaxial graphene even finds applications in metrology, e.g., as a reference for Kelvin probe force microscopy [7]. Here, we focus on Bernal-stacked BLG (BBLG), which has very recently become a topic of in-depth experimental and theoretical investigations: a plethora of ordered electronic phases have been obtained in external fields [12], which comprise spin and valley magnetism, correlated insulators, correlated metals, charge and spin-density waves and superconductivity [13,14,15].

In the absence of external fields, neutral BBLG is semiconducting as a result of the interlayer interaction between the two graphene sheets; applying a transverse electric field lifts the equivalence of the two layers (and may induce a so-called gap), and at low temperatures, the resistance varies non-monotonously with the gap closing and reopening with increasing field strength [16,17,18]. A DFT-based model Hamiltonian recently attributed this finding to a charge-ordered state, which originates from the interplay of interlayer van der Waals interactions and ripple formation [19].

Adding extrinsic charge carriers shifts the Fermi level into the frontier states at the K and $K^{\prime }$ points (see below), where the threefold symmetry of the graphene lattice and the resulting trigonal warping are reflected in the trigonal symmetry of the resulting Fermi lines [12,20]. With increasing carrier density, the Fermi line evolves from a small, all convex, onigiri-type (inspired by the Japanese rice snack of similar geometry) triangle with rounded corners to a larger triangle with concave edges. Adding a transverse electric field enhances the concave inward bending of the edges, which is predominantly observed for holes [12]. Similar transitions from convex to concave triangular shapes have also been obtained theoretically with model Hamiltonians, which include coupled-charge and spin-density waves [15]. For weak Hund’s coupling, in addition to long-range Coulomb with Ising-type spin–orbit coupling, predominantly concave trigonally symmetric shapes dominate [20].

The manifold all-electrical control mechanism for BLG via gate voltages and their importance for potential applications based on electronic transport motivate the studies presented here. The anisotropy we first focus on (Section 2) is the preference for certain propagation directions in space in the form of trigonal warping at the K-points. As outlined above, in BBLG, this can be obtained through proximity with a spin–orbit-coupled two-dimensional (2D) layer [20], by strong doping with charge carriers [13], or by applying external fields [14,15]. According to the theorem of Emmy Noether, angular momentum then cannot be conserved anymore—and that is the moment when symmetry considerations are in order. Indeed, the dynamics of the anisotropic billiard have to differ from those of its isotropic counterpart. In fact, chaotic dynamics have been observed in circular BLG billiards [21], as well as in BLG billiards of noncircular shapes [22].

Note that there are two types of K-points (so-called valleys) with mirrored Fermi lines in BLG. Since we do not consider intervalley scattering and want to keep the system as simple as possible, we focus on one K-point throughout. This scenario corresponds to isospin-conserving electron dynamics within one valley, which may be obtained in Bernal-stacked BLG at low carrier densities, i.e., in the weakly coupling regime and for external fields, which increase the degeneracy of the multitude of valleys. We will discuss the consequences of the electron’s pseudospin on the billiard’s dynamics in detail below, but first, we point out that it breaks the mirror symmetry of the Poincaré surface of section (PSOS) for a trigonally warped Fermi line, as shown in the right panel of Figure 1, while a deformed billiard shape with isotropic dispersion always possesses this symmetry, cf. the left panel of Figure 1.

Mechanical strain is another source of anisotropy and is induced here by an applied external force [23], and we will consider this in the second part of this paper in Section 3. We will focus on closed (hard-wall) billiards in this systematic ray-tracing study. We close with a summary and outlook and argue that, based on ray–wave correspondence, our results are relevant for realistic (quantum) systems.

2. Circular and Deformed Billiards in Anisotropic Materials

We investigate the billiard’s dynamics in real and phase space using a ray-tracing algorithm. Therefore, we need a link between the billiard’s dynamics in anisotropic momentum space and real space. This link is provided by the group velocity (or the Poynting vector), which is always normal to the Fermi line (or to the index ellipsoid for optical systems). Thus, a certain wave vector, $\overrightarrow{k}$, leads to a group velocity, which is, in anisotropic media, usually not parallel to the wave vector. The angles of incidence, $\alpha$, and reflection, $\beta$, at the billiard boundary are derived from the conservation of the momentum component, $k_{\Vert }$, parallel to the boundary interface and are given as the angle to the corresponding group velocity, cf. Figure 2. For an isotropic and thus circular dispersion relation (Fermi line), we immediately recover the well-known reflection law $\alpha =\beta$. This relation even holds for situations where the Fermi line is symmetric with regard to the $k_{\Vert }$ axis. However, for generically anisotropic media, such a situation is usually not present, and consequently, the angle of incidence is not equal to the angle of reflection. Consequently, symmetry breaking in the momentum space destroys the so-called whispering-gallery orbits characteristic of circular cavities and replaces them with complex dynamics with chaotic features, even in disk resonators. This is illustrated in the right panel of Figure 1. Although the cavity is circular, the anisotropic dispersion relation (onigiri-type; see below in Equation (2)) restructures the phase space by bringing up new stable (triangular) orbits associated with the shape of the dispersion relation. This implies that the PSOS is not mirror-symmetric with respect to the $k_{\Vert }=0$ axis because the stability of these triangular orbits changes (from stable to unstable and vice versa) when their sense of rotation is reversed. This is, however, not the case when the dispersion relation is isotropic and the cavity geometry has the (same) onigiri-type shape (Equation (1)), as illustrated in the left panel of Figure 1.

2.1. Lyapunov Exponents as Characteristics of the Billiard’s Dynamics

In this section, we investigate the nonlinear dynamics of billiards of various shapes while simultaneously changing their dispersion relation. Motivated by the trigonal warping of the Fermi line in BLG, we focus on this type of geometry both in real and momentum space and refer to it as the onigiri shape. The geometry of the cavity in real space is defined in polar coordinates as follows:

$$R\left(\varphi \right)=R_0(1+{ϵ}_{ng}cos\left(n(\varphi -{\theta }_0)\right)$$

(1)

where $R_0$ is the mean radius of the cavity and ${ϵ}_{ng}$ is a deformation parameter. For $n=0$, a circle is obtained, and for $n=3$, a $C_3$-symmetric onigiri geometry is obtained.

The geometry of the Fermi line in momentum space is defined analogously to that of the cavity:

$$R_f\left(\varphi \right)=k_0(1+{ϵ}_{nf}cos\left(n\varphi \right))$$

(2)

The rotation angle, ${\theta }_0$, indicates the tilt between the cavity geometry axis and the underlying BLG lattice or the Fermi line, which becomes important if both the cavity and the Fermi line are noncircular. Here, ${\theta }_0$ is defined such that the geometry is rotated counter-clockwise by this angle with respect to the (fixed) BLG lattice or Fermi line.

In a previous study [22] addressing realistic BLG systems, it was found that a highly structured PSOS—concerning the presence of islands and their sizes—was achieved when the shapes of the Fermi line and the billiard cavity possessed a very similar geometry. Then, the PSOS of the billiards showed a high portion of regular trajectory dynamics confined to correspondingly large islands. Moreover, these emerging island chains protected whispering-gallery (WG) trajectories by hindering (“blocking”) their transition into the chaotic sea of the PSOS. It is desirable to quantify these qualitative statements to allow for a profound comparison and optimization of the different geometries. To this end, we introduce generalized Lyapunov exponents in order to characterize the chaotic behavior of the billiard’s overall dynamics.

The Lyapunov exponent is used to make statements about the stability of a given trajectory. In order to determine the Lyapunov exponent of the reference trajectory, a trajectory infinitesimally adjacent to it is considered. Let the perturbation $ϵ\left(t\right)$ be the distance between the two trajectories at any point in time. For $t\to \infty$ and $ϵ\left(t_0\right)\to 0$, the Lyapunov exponent, $\lambda$, is defined as the mean exponential divergence or convergence of the trajectories [24,25]:

$$ϵ\left(t\right)\approx e^{\lambda t}·ϵ\left(t_0\right),$$

(3)

$$\lambda ={\displaystyle \frac{1}{t}}·ln{\displaystyle \frac{ϵ\left(t\right)}{ϵ\left(t_0\right)}}.$$

(4)

The perturbation $ϵ\left(t\right)$ is limited from above since it cannot be larger than half the circumference of the cavity. Thus, it is necessary to renormalize the perturbation to $ϵ\left(t_0\right)$ when it becomes larger than an upper limit, ${ϵ}_{\max }$, and to calculate the Lyapunov exponent after each renormalization. The average of those Lyapunov exponents results in the maximal Lyapunov exponent of the reference trajectory. This corresponds to the so-called Wolff method. If the perturbation $ϵ\left(t\right)$ stays smaller than ${ϵ}_{\max }$, the maximal Lyapunov exponent of the trajectory will be set to 0 [24]. This is justified since the model is a non-dissipative system, and thus, the Lyapunov exponent cannot be negative.

A positive Lyapunov exponent characterizes trajectories that are exponentially drifting apart. The reference trajectory is then chaotic, i.e., unstable. A stable trajectory, on the other hand, has a Lyapunov exponent that is equal to zero. The choice of reference trajectory is therefore not arbitrary in a mixed-phase space. In order to obtain a characteristic value for the entire system, the mean value of the Lyapunov exponents of the trajectories must be calculated (cf. Figure 3). Thus, a lower average Lyapunov exponent indicates a higher proportion of stable islands in the PSOS.

We characterize the system as a whole by starting 2500 fixed initial conditions (both stable and chaotic), followed by 1000 reflections for each. Stable trajectories yield Lyapunov exponents equal to zero, as explained above. Due to the ergodicity of chaotic trajectories, they all yield similar Lyapunov exponents.

The advantage of this simple and numerically efficient method is that it works for all mixed-phase spaces, and the same algorithm can be used for each cavity geometry. The average Lyapunov exponent of short chaotic trajectories (which we consider here) is equal to the Lyapunov exponent of a long (ergodic) chaotic trajectory, which can be considered as being assembled from the short ones as long as the total length yields an ergodic trajectory. This is evident in our algorithm, as shown by several PSOS (for example, Figure 4b,c,e,f), which demonstrate a homogeneous coverage of phase space already for only 100 reflections.

2.2. Deviation from Circular Geometry in Real and Momentum Space

We start our analysis by using the average Lyapunov exponent to investigate whether breaking the circular symmetry in momentum space or in real space has a greater influence on the cavity dynamics. To this end, the deformation parameter, ${ϵ}_3$, of the Fermi line (${ϵ}_{3f}$) or the cavity (${ϵ}_{3g}$) was varied from 0 to 0.12, while the cavity or the Fermi line remained circular. For each cavity–Fermi line pair, the mean Lyapunov exponent, $\overline{\lambda }$, and its error were calculated, and these are plotted as a function of the deformation parameter, ${ϵ}_3$, in Figure 3.

It can be seen in Figure 3 that a larger deformation parameter, both in real and momentum space, leads to a larger $\overline{\lambda }$. This is due to the fact that chaotic behavior increasingly dominates. Furthermore, it can be seen that for higher deformation parameters, a deformed Fermi line results in a larger Lyapunov exponent than a deformed geometry. Consequently, the deformation of the Fermi line has a greater influence on the dynamics of the electrons. One possible reason for this is that when a trajectory moves along the wall of the cavity, the deviation from the circular curvature is very small for each reflection. The curvature of the Fermi line, on the other hand, plays a role in every reflection, as the normal vector indicates the direction of the electrons.

Figure 4a–c show the Poincaré sections of a circular cavity with an onigiri-shaped Fermi line with different deformation parameters, ${ϵ}_{3f}$. The PSOS for the reversed case of an onigiri-shaped cavity with a circular Fermi line are shown in Figure 4d–f. Here, an interpretation in terms of the Kolmogorov–Arnold–Moser (KAM) theorem [26] is suitable.

In the case of small deformation parameters, the PSOS is mostly filled with wavy lines, which can be seen in Figure 4a,d. These represent deformed KAM surfaces, while the share of chaotic parts increases with the deformation parameter. Furthermore, in both cases, one can see six big islands, which belong to two stable triangular orbits. In the case of a deformed cavity, these two orbits are stabilized by the three flat sides of the onigiri geometry. On the other hand, a deformed Fermi line results in an anisotropic velocity distribution, with, in the case of the onigiri geometry, three preferred propagation directions. These three preferred directions create two stable triangular orbits, leading to the six big islands observed in Figure 4a–c. However, the upper and lower rows of these islands are not symmetric to the $k_{\Vert }=0$ axis (in contrast to Figure 4d–f) but shifted by $\Delta s={\displaystyle \frac{1}{6}}$. In fact, there are also triangular orbits in between the islands, but these are unstable since their group velocities correspond to the curved sides of the Fermi line.

In both cases, with a deformed Fermi line or cavity, the islands shrink when increasing the deformation parameter, ${ϵ}_3$. In the next section, we will discuss how the interplay between cavity and Fermi line deformation can increase or decrease the islands’ sizes and whether there is an optimal setting for the largest possible islands.

2.3. Influence of the Interplay Between the Deformation of the Cavity and Fermi Line

In this section, we deform both the Fermi line and the cavity. Then, the orientation angle, ${\theta }_0$, between the Fermi line and the cavity becomes crucial. As long as at least one is circularly symmetric, varying the orientation, ${\theta }_0$, only results in a shift in phase space. However, the tilt angle, ${\theta }_0$, now affects the entire dynamics of the system.

In Figure 5, the Lyapunov exponents for equally shaped cavities and Fermi lines, i.e., ${\epsilon }_{3f}={\epsilon }_{3g}={\epsilon }_3$, are shown depending on the orientation angle, ${\theta }_0$ (we only present the interval ${\theta }_0\in [0,{120}^{\circ }]$, corresponding to the C₃ symmetry of the Fermi line and the cavity). In all cases, and most obviously for ${\epsilon }_3=0.08$, there are minima at ${\theta }_0={30}^{\circ }$ and ${\theta }_0={90}^{\circ }$. These minima are due to the interplay between the stabilization properties in the momentum and real space. Figure 6 explains this interplay. Figure 6a shows a triangular orbit, where the preferred directions (corresponding to the flat sides of the Fermi line; see Figure 6e) point to the regions of the cavity with high curvature, which have a destabilizing effect. Thus, the islands in the PSOS become a little bit smaller. However, the unstable directions, cf. Figure 6f, point to the stabilizing flat sides of the cavity, so they become stable and form additional islands in the PSOS; see Figure 6b. If both stabilizing effects coincide, huge islands arise, cf. Figure 6d, which reduce the Lyapunov exponent. In addition, the arising chain of six islands in the upper half establishes a barrier in phase space that stops WG-like trajectories from entering the chaotic sea, further reducing the Lyapunov exponent.

In Figure 7a, we show the Lyapunov exponents for different orientation angles, ${\theta }_0$, of a system with an onigiri-shaped Fermi line and cavity over the Fermi line deformation parameter, ${\epsilon }_{3f}$, while the deformation parameter of the cavity, ${\epsilon }_{3g}$, is kept constant. The Lyapunov exponent reaches a clear minimum if the orientation angle is ${\theta }_0={30}^{\circ }$ and the deformation parameter of the cavity is equal to that of the Fermi line. Figure 7c shows how optimal settings for ${\epsilon }_3$ and ${\theta }_0$ can be used to maximally structure the phase space via the presence of large stable islands. However, further increasing the deformation parameter leads to higher Lyapunov exponents, and the dependency trend is similar to that shown in Figure 3; i.e., symmetry breaking in real space barely has any influence on the Lyapunov exponent once the anisotropy is strong enough.

In Figure 8a, the cavity deformation parameter, ${\epsilon }_{3g}$, is varied while the Fermi line deformation, ${\epsilon }_{3g}=0.06$, remains constant. Again, the average Lyapunov exponent is minimal when the orientation angle is ${\theta }_0={30}^{\circ }$ and both deformation parameters are equal.

In order to further examine the influence of the deformation on the Lyapunov exponent, we chose a fixed orientation angle, ${\theta }_0={30}^{\circ }$, and varied ${\epsilon }_{3g}$ for several constant values of ${\epsilon }_{3f}$, and vice versa, as shown in Figure 9. Starting with ${\epsilon }_{3g}=0$ (${\epsilon }_{3f}=0$), a larger ${\epsilon }_{3f}$ (${\epsilon }_{3g}$) leads to higher Lyapunov exponents since stronger deformation of the Fermi line (cavity) results in a larger number of chaotic trajectories if the cavity (Fermi line) is not deformed. However, the Lyapunov exponent decreases when increasing the other deformation parameter until both are equal. We observe clear minima at ${\epsilon }_{3f}={\epsilon }_{3g}$ if ${\epsilon }_{3f}<0.07$ or ${\epsilon }_{3g}<0.07$. Increasing the deformation parameter, ${\epsilon }_{3g}$ (${\epsilon }_{3f}$), further yields higher Lyapunov exponents, but the differences between the several ${\epsilon }_{3f}$ values (${\epsilon }_{3g}$) are less striking. We conclude that the deformation of the Fermi line (cavity) does not play a role if the deformation of the cavity (Fermi line) is much stronger, a behavior similar to the one described in Figure 3.

3. Strain as a Source of Irregular Anisotropy

In this section, we investigate how a distortion of the underlying (graphene) lattice induced by an uniaxial strain force will affect mesoscopic electron dynamics (again, we focus on one valley). We illustrate the behavior on BLG with an initially trigonally warped Fermi line that is further perturbed by an external strain [27,28,29]; however, our results can be generalized to other strained systems. After discussing the overall influence of strain on the phase-space structure, we illustrate exemplarily how increasing strain alters the details of the island structure.

3.1. Modeling of Uniaxially Strained Graphene-Type Media

In order to model the mechanical deformation of a hexagonal lattice within Hooke’s regime, we consider one carbon atom trigonally connected to three neighbors via the bonding vectors ${\overrightarrow{\tau }}_1$, ${\overrightarrow{\tau }}_2$, and ${\overrightarrow{\tau }}_3$, cf. Figure 10. It is then always possible to assume that an external, uniaxial force, ${\overrightarrow{F}}_{\sigma }$, acts on one of the outer atoms, compensated by other forces, ${\overrightarrow{F}}_{\sigma }/2$, acting on the two remaining outer atoms. Furthermore, we assume a harmonic potential, V, for both the angular deformations, $d\phi$, and the binding distance deformations, $dr$, $\overrightarrow{\tau }:=r_0·{[\cos \left({\phi }_0\right),\sin \left({\phi }_0\right)]}^{\mathrm{T}}$ (cf. Figure 10).

$$V_r\left(dr\right)={\displaystyle \frac{D_r}{2}}{(r_0+dr)}^2,V_{\phi }\left(d\phi \right)={\displaystyle \frac{D_{\phi }}{2}}{({\phi }_0+d\phi )}^2.$$

(5)

Here, $D_r$ and $D_{\phi }$ are the corresponding lattice- and material-dependent spring constants expressing the proportionality between the applied force and the resulting deformation, $d{\left(F_{\sigma }\right)}_i=-D_{i}d{q}_i$, with $q_i\in \{dr,d\phi \}$ in Hooke’s regime.

In order to obtain the resulting distorted lattice corresponding to a given force, ${\overrightarrow{F}}_{\sigma }$, we increase the force successively by introducing a switch-on factor, S, starting from $S=0$. In each step, the force was increased by ${\overrightarrow{F}}_{\sigma }/S$ and the resulting lattice distortion was computed assuming a linear response until the final value ${\overrightarrow{F}}_{\sigma }/S={\overrightarrow{F}}_{\sigma }$ was reached to ensure quasistatic conditions in each step. Numerically, we used 1000 steps for S since the distorted lattice vectors, ${\overrightarrow{\tau }}_1^{\prime }$, ${\overrightarrow{\tau }}_2^{\prime }$ and ${\overrightarrow{\tau }}_3^{\prime }$, did not change any further, even for more values of S. Figure 10b shows the decomposition of the bonding vector’s distortion due to an infinitesimal increase in $F_{\sigma }$ in its radial and angular components.

Having computed the distorted lattice resulting from the mechanical strain, we now have to quantify its effect on the Fermi line. This is achieved by inserting the distorted lattice vectors, ${\overrightarrow{\tau }}_1$, ${\overrightarrow{\tau }}_2$ and ${\overrightarrow{\tau }}_3$, into the dispersion relation of the underlying lattice (which can result, e.g., from an analytical tight-binding calculation as in single-layer graphene or can be obtained numerically). Here, we use a generic trigonally warped Fermi line inspired by BLG as an ad hoc model system with an isotropic undistorted initial state. We parameterize the undistorted Fermi vector, ${\overrightarrow{k}}_{\mathrm{Fl}}$, as a function of the polar angle, $\lambda$, in momentum space in the parametric component form:

$${\overrightarrow{k}}_{\mathrm{Fl}}\left(\lambda \right):={\left[a_1\cos \left(\lambda \right)-a_2\cos \left(2\lambda \right)/\sqrt{3},a_1\sin \left(\lambda \right)+a_2\sin \left(2\lambda \right)/\sqrt{3}\right]}^{\mathrm{T}},$$

(6)

with $a_1$ being the radius of the underlying isotropic Fermi line. The parameter $a_2$ induces the trigonally warped shape, with $a_2$ marking the maximum deviation from the isotropic Fermi line circle reached at angles of $\lambda =\pm \pi /3,\pi$.

The strain-induced deformation is assumed to be small (less than 10%) and can therefore be modeled using a linear stretching transformation, $\mathbf{T}$, of the form

$$T_{ij}={\delta }_{ij}+(\sigma -1)n_i{n}_j\mathrm{or}\mathbf{T}(\overrightarrow{n},\sigma )=\mathbf{1}+(\sigma -1)\overrightarrow{n}\otimes \overrightarrow{n}.$$

(7)

Here, $\sigma$ is the linear stretching factor in k-space, which is the reciprocal of the stretching factor, ${\sigma }_l$, of the lattice in position space, $\sigma =1/{\sigma }_l$. The lattice stretching factor, ${\sigma }_l$, is obtained directly from the distorted lattice vectors, ${\overrightarrow{\tau }}_1,{\overrightarrow{\tau }}_2,$ and ${\overrightarrow{\tau }}_3$, computed above. The (uniaxial) stretching direction in the real-space direction, ${\phi }_s$, is given by the normal vector $\overrightarrow{n}$, where $\overrightarrow{n}={[\cos \left({\phi }_{\mathrm{s}}\right),\sin \left({\phi }_{\mathrm{s}}\right)]}^{\mathrm{T}}$.

The strained Fermi line, ${\overrightarrow{k}}_{\mathrm{Fl}}^{\prime }$, is thus given by

$${\overrightarrow{k}}_{\mathrm{Fl}}^{\prime }\left(\lambda \right)=\mathbf{T}(\overrightarrow{n},s)·{\overrightarrow{k}}_{\mathrm{Fl}}\left(\lambda \right).$$

(8)

We can now model the complex dynamics of electrons in a strained system using the distorted Fermi line of Equation (8) and investigate it in real and phase space for various parameters, namely $a_1$, $a_2$, $\sigma$ and ${\phi }_s$, as in the previous section. The results are shown in Figure 11 and Figure 12.

The literature suggests that Equation (8) is physically consistent with the Fermi line shapes expected in experiments for strained BLG with the physical parameters strain strength, ${\overrightarrow{F}}_s$, the ratio $D_r$/$D_{\phi }$, and Fermi energy, $E_F$ [30].

To this end, we minimized the functional

$$\mathcal{F}[a_1,a_2,s,{\phi }_s]:={\int }_0^{2\pi }|E\left({\overrightarrow{k^{\prime }}}_{\mathrm{Fl}}\left(\lambda \right)\right)-E_{\mathrm{F}}|d\lambda ,$$

(9)

where $E\left({\overrightarrow{k^{\prime }}}_{\mathrm{Fl}}\left(\lambda \right)\right)$ is the dispersion relation for distorted BLG from tight-binding calculations (using the distorted bonding vectors ${\overrightarrow{\tau }}_i$) to find the best fit for the parameters $a_1,a_2,\sigma$ and ${\phi }_s$.

The maximum relative error,

$${\displaystyle \frac{\mathcal{F}}{E_{\mathrm{F}}}}:={}_{\Delta }\mathcal{F}_{\mathrm{rel}}^{\max }\approx 4.1·{10}^{-4},$$

(10)

occurs at high electron energies (strongly pronounced trigonal shape) and the highest considered stretching factor, $\sigma =0.94$, and when the stretching direction, $\overrightarrow{n}$, is not aligned with any symmetry axis of the Fermi line. Our parametrization applies to $\sigma >0.9$.

Since the size of the Fermi line is irrelevant for the billiard dynamics, we can discard one of the two parameters in the analytic expression given in Equation (6) and set $a_1:=1$ and $a_2:=\alpha$. The parameter $\alpha$ sets the shape of the Fermi line ($\alpha \to 0$: more circular; $\alpha \to 0.4$: more triangular).

3.2. Mesoscopic Electron Dynamics in Strained Graphene-Based Media

In general, a small uniaxial compression of the Fermi line, as mentioned above, will change the shape, size, and location of the six main stability islands in the PSOS. In addition, smaller islands may arise or disappear. Notice that the symmetry of the cavity geometry and/or the dispersion relation (Fermi line) is always reflected in the symmetry of the PSOS. We shall now discuss what this implies if an onigiri-type ($C_3$) Fermi line is subject to mechanical strain.

A compression of the Fermi line preserving one axial symmetry axis is shown in Figure 11 and Figure 12. The three-island chain is dissolved, and the islands are noticeably smaller. The corresponding triangular orbits are elongated, reflecting the superimposed $C_{2v}$ symmetry of the Fermi line. We illustrate the effect of increasing the uniaxial strain, $\sigma$, by investigating the change in the structure of one of the three $k_{\Vert }>0$ stable islands in the PSOS by generalizing ideas from the Poincaré–Birkhoff theorem; see Figure 13.

We start with a weak triangular (onigiri) distortion of the Fermi line without mechanical strain, as shown in Figure 13a. The island has a well-known structure formed by stable trajectories revolving around the central stable fixed point. (Notice that this fixed point can be considered to result from a Poincaré–Birkhoff-like mechanism, as destroying the circular symmetry of the Fermi line resulted in a chain of three stable (elliptic) fixed points (with stable islands) with three associated unstable (hyperbolic) fixed points in between.)

Switching on the mechanical strain, as shown in Figure 13b–d, results in the disintegration of some of the stable trajectories in a sequence of new elliptic and hyperbolic fixed points that appear near the boundaries of the islands, according to the Poincaré–Birkhoff theorem. The size of the islands shrinks as the amount of chaotic motion is increased.

4. Summary and Outlook

In this paper, we illustrated complex dynamics induced by phase-space anisotropies. Using bilayer graphene-inspired model systems, we investigated, in detail, the effects of deviations from an isotropic dispersion relation (circular Fermi line). We highlighted that symmetry breaking in momentum space resembles mechanisms observed for its real-space counterpart, where the KAM and Poincaré–Birkhoff theorems cover the transition from regular dynamics to mixed and chaotic dynamics. The advantage of operating in momentum space is, from an experimental point of view, that the dispersion relation can rather easily be manipulated fully electronically, for example, via gate voltages, which do not require a change in the cavity geometry. Superimposing mechanical strain on a triagonally warped (onigiri) Fermi line can be carried out to fine-tune the complex electron dynamics of the system.

We compared how deviations from circular symmetry in real and momentum space, respectively, affect the billiard’s dynamics. We used average Lyapunov exponents and found that momentum space deformation had a somewhat larger influence on the overall behavior; the breaking of integrability and the emergence of chaotic motion are comparable. We pointed out a difference between real- and momentum-space deformation that is relevant for charge carriers in bilayer graphene-type systems, where pseudospin or the valley index is relevant. While the Poincaré SOS respects symmetry with respect to $k_{\Vert }=0$ for real-space deformation (clockwise and counter-clockwise motions are equivalent), this is not the case for momentum-space deformation, where this phase-space symmetry does not hold.

While we focused on closed billiard cavities here for simplicity, there is no specific problem with describing open systems by including the generalized Fresnel-like reflection coefficients at each reflection point, as in Refs. [21,22,31]. Knowing the details of the dynamics of the electronic charge carriers is essential to understand their transport properties, which are crucial for future applications.