Quantum Holography in a Graphene Flake with an Irregular Boundary

Electrons in clean macroscopic samples of graphene exhibit an astonishing variety of quantum phases when strong perpendicular magnetic field is applied. These include integer and fractional quantum Hall states as well as symmetry broken phases and quantum Hall ferromagnetism. Here we show that mesoscopic graphene flakes in the regime of strong disorder and magnetic field can exhibit another remarkable quantum phase described by holographic duality to an extremal black hole in two-dimensional anti–de Sitter space. This phase of matter can be characterized as a maximally chaotic non-Fermi liquid since it is described by a complex fermion version of the Sachdev-Ye-Kitaev model known to possess these remarkable properties.

Figure 1

Schematic depiction of the proposed device. Irregular shaped graphene flake in applied magnetic field $B$ forms the ($0+1$)-dimensional many-body system equivalent to a black hole in ($1+1$) anti–de Sitter space. Inset: lattice structure of graphene with $A$ and $B$ sublattices marked and nearest neighbor vectors denoted by ${\mathit{\delta }}_j$.

Figure 2

Electronic properties of an irregular graphene flake in the absence of interactions. (a) Single-particle energy levels ${\epsilon }_j$ of the Hamiltonian $H_0$ as a function of the magnetic flux $\mathrm{\Phi }=SB$ through the flake. The flake used for this calculation, depicted in the inset, consists of 1952 carbon atoms with equal number of $A$ and $B$ sites. The energy spectrum, calculated here in the Landau gauge $\mathit{A}=Bx\widehat{\mathit{y}}$ and with open boundary conditions, shows the same generic features irrespective of the detailed flake geometry. (b) Typical wave function amplitudes of the eigenstates ${\mathrm{\Phi }}_j(\mathit{r})$ belonging to ${\mathrm{LL}}_0$ at $\mathrm{\Phi }=40{\mathrm{\Phi }}_0$ and the edge modes. The numerals above each panel denote the energy ${\epsilon }_j$ of the state in eV, scale bar shows the magnetic length $l_B=\sqrt{\hslash c/eB}$.

Figure 3

Statistical properties of the coupling constants and the thermal entropy. (a) Histogram of $|J_{ij;kl}|$ as calculated from Eq. (4) with $V_1=1$ for the graphene flake depicted in Fig. 2 and $N=16$, compared to the Gaussian distribution (orange line) with the same variance $0.000805V_1$. Inset shows the histogram of real and imaginary components of $J_{ij;kl}$. The mirror symmetry about the horizontal follows from the Hermiticity property $J_{ij;kl}=J_{kl;ij}^{*}$. (b) Entropy $S(T)$ of the SY Hamiltonian (1) calculated with $J\mathrm{s}$ shown in panel (a).

Figure 4

Many-body level statistics for the interacting electrons in ${\mathrm{LL}}_0$ of the graphene flake. Blue bars show the calculated distributions for the graphene flake. Orange, green and red curves indicate the expected distributions given by Eq. (6) for GOE, GUE, and GSE, respectively. To obtain smooth distributions, results for $N=14,15,(16)$ have been averaged over 8 (4) distinct flake geometry realizations while $N=17$, 18 reflect a single realization.