Global phase diagram of two-dimensional dirty hyperbolic Dirac liquids

Abstract

Within the framework of the canonical nearest-neighbor tight-binding model for spinless fermions, a family of two-dimensional bipartite hyperbolic lattices hosts massless Diraclike excitations near half-filling with the iconic vanishing density of states (DOS) near zero energy. We show that a collection of such ballistic quasiparticles remains stable against sufficiently weak pointlike charge impurities, a feature captured by the vanishing average [$ρ_{a}(0)$] and typical [$ρ_{t}(0)$] DOS at zero energy, computed by employing the kernel polynomial method in sufficiently large $\{ 10, 3\}$ hyperbolic lattices (Schläfli symbol) with more than $10^8$ and $10^5$ sites, respectively, with open boundary conditions. However, at moderate disorder the system enters a metallic state via a continuous quantum phase transition where both $ρ_{a}(0)$ and $ρ_{t}(0)$ become finite. With increasing strength of disorder, ultimately an Anderson insulator sets in, where only $ρ_{t}(0) \to 0$. The resulting phase diagram for dirty Dirac fermions living on a hyperbolic space solely stems from the background negative spatial curvature, as confirmed from the vanishing $ρ_{t}(0)$ for arbitrarily weak disorder on honeycomb lattices, fostering relativistic fermions on a flatland, as the thermodynamic limit is approached.

Figures (3)

• Figure 1: (a) Average density of states (DOS) at zero energy $\rho_{\rm a}(0)$ as a function of disorder strength ($W$). Inset shows $\rho_{\rm a}(E)$ vs $E$ (energy). (b) Typical DOS at zero energy $\rho_{\rm t}(0)$, averaged over the system (see text), as a function of $W$. Inset shows $\rho_{\rm t}(E)$ vs $E$. Here, $N$ ($N_m$) denotes the number of sites (Chebyshev moments). (c) A schematic phase diagram of disordered hyperbolic Dirac semimetal (SM), where circles denote quantum critical points, separating the adjacent phases. All the results are for $\{10,3 \}$ lattices.
• Figure 2: (a) Variation of the average density of states (DOS) at zero energy $\rho_{\rm a}(0)$ in a $\{ 10,3\}$ system containing $N (>10^8)$ sites (see legend) with $N_m$ number of Chebyshev moments for disorder strengths ranging from $0.0$ to $1.0$ in steps of $0.1$ in the direction of the arrow. (b) Same as (a), but for the typical DOS at zero energy $\rho_{\rm t}(0)$, containing additional results for $W=2.0$. (c) Same as (b), but for disorder strengths ranging from $3.0$ to $10.0$ in steps of $1.0$ in the direction of the arrow. (d) Typical DOS at zero energy $\rho^n_{\rm t}(0)$, computed only on the decagon belonging to the $n$th generation of a $\{10,3 \}$ lattice with a total of $n_{\rm tot}=6$ generations hosting $N$ number of sites (see legend), where $1 \leq n \leq n_{\rm tot}$. (e) Variation of $\rho_{\rm t}(0)$ with $W$ on honeycomb or $\{ 6,3 \}$ lattices with increasing system size or number of sites $N$ therein (see legend) for a fixed $N_m=2^{15}$.
• Figure 3: Computation of density of states (DOS) exponents from the scaling of the (a) average ($\alpha_a$) and (b) typical ($\alpha_t$) DOS at finite energies ($E$) and order-parameter exponents from the scaling of the (c) average ($\beta_{a}$) and (d) typical ($\beta_{t}$) DOS at $E=0$ near the semimetal-to-metal critical point at $W=W_{c,1}$, where $W$ is disorder strength and $\delta=(W-W_{c,1})/W_{c,1}$. (e) Order-parameter exponent ($\beta_{A}$) near the Anderson critical point at $W=W_{c,2}$ from the scaling of typical DOS at zero energy with $\delta_{A}=(W_{c,2}-W)/W_{c,2}$. For (a) and (c) [(b), (d), and (e)], analyses were performed on $\{ 10, 3\}$ lattices with $n_{\rm tot}=9$ [$n_{\rm tot}=6$], containing more than $10^8$ ($10^5$) sites. The scaling forms and values of the exponents are quoted in the legends.