Localization of Interacting Dirac Fermions

Using exact quantum Monte Carlo calculations, we examine the interplay between localization of electronic states driven by many-body correlations and that by randomness in a two-dimensional system featuring linearly vanishing density of states at the Fermi level. A novel disorder-induced nonmagnetic insulating phase is found to emerge from the zero-temperature quantum critical point separating a semimetal and a Mott insulator. Within this phase, a phase transition from a gapless Anderson-like insulator to a gapped Mott-like insulator is identified. Implications of the phase diagram are also discussed.

Figure 1

Phase diagram of the disordered Hubbard model on the honeycomb lattice at half-filling. $\mathrm{\Delta }$ labels the disorder strength and $U$ represents the local Coulomb repulsion. Phase boundary lines are guides to the eye. The metallic phase boundary is determined by the temperature dependence of the conductivity ${\sigma }_{\mathrm{dc}}$ and the region of long range AFM order by finite size scaling of the AFM structure factor. The (black) triangle point is obtained using the Drude weight data presented in the Supplemental Material [29]. Lines are guides to the eyes. Although these transitions coincide in the clean limit, for nonzero $\mathrm{\Delta }$ an intermediate, magnetically disordered, insulator phase intervenes. This phase itself contains a transition from Anderson-like to Mott-like insulators. The inset shows the geometry of the $L=6$ lattice where sublattices are labeled by blue and red colors.

Figure 2

(a) dc conductivity ${\sigma }_{\mathrm{dc}}$ versus temperature $T$ in the clean limit $\mathrm{\Delta }=0$ computed at various coupling strengths for the $L=12$ honeycomb lattice. (b) Scaling behavior of the normalized AFM spin structure factor $S_{\mathrm{AFM}}/N_c$ at corresponding $U$ values. Solid and dashed lines represent third-order polynomial fits to the data.

Figure 3

Temperature dependence of the dc conductivity ${\sigma }_{\mathrm{dc}}$ measured on the $L=12$ lattice with disorder at (a) $U=1.0$, (b) $U=2.0$, (c) $U=3.0$, and (d) $U=4.0$. In each figure, lines are guides to the eyes. Metallic and insulating behaviors are indicated by solid and dashed lines, respectively. In panels (a)–(c), the low-$T$ behavior of ${\sigma }_{\mathrm{dc}}$ clearly indicates a disorder-driven metal-insulator transition.

Figure 4

Charge compressibility $\kappa$ versus chemical potential $\mu$ computed for the linear size $L=12$ disordered lattice at inverse temperature $\beta =10$. To distinguish between gapped and gapless phases, we have adopted a finite threshold $\kappa \lesssim 0.04$ deduced from the procedure described in the Supplemental Material [29].

Figure 5

Finite-size scaling studies of the AFM spin structure factor. Statistical errors of DQMC results are smaller than the symbol size. Lines represent cubic polynomial (in $1/L$) fits to the data. A finite $y$-axis intercept in the $L\to \infty$ limit indicates the existence of long-range magnetic order. Here again, we have used solid and dashed lines to label magnetically ordered and disordered phases.