Classification of Topological Phase Transitions and van Hove Singularity Steering Mechanism in Graphene Superlattices

We study quantum phase transitions in graphene superlattices in external magnetic fields, where a framework is presented to classify multiflavor Dirac fermion critical points describing hopping-tuned topological phase transitions of integer and fractional Hofstadter–Chern insulators. We argue and provide numerical support for the existence of transitions that can be explained by a nontrivial interplay of Chern bands and van Hove singularities near charge neutrality. This work provides a route to critical phenomena beyond conventional quantum Hall plateau transitions.

Figure 1

Parameterization of the Hofstadter-Chern bands by the Thouless function. (a) Honeycomb superlattice with lattice constant $a$ in the nanometers and magnetic unit cell $q$ times extended along ${\mathit{a}}_2$. (b) Momentum dependence on the Thouless function $\xi$. (c) Spectrum as function of $\xi$ for $\varphi =(1/3){\varphi }_0$.

Figure 2

TPTs of the $\varphi =(1/7){\varphi }_0$ lattice ($\alpha$ denotes band index). (a) At $(t_1,t_2,t_3)=(1.24,1,1)$, 7 Dirac cones (only one shown) form at ${\mathit{k}}_{\min }^{(n)}=(-\pi /7+2\pi n/7,0)$, $n=0,\dots ,6$. (b) At $(t_1,t_2,t_3)=(2.73,1,1)$, 7 Dirac cones (only one shown) form at ${\mathit{k}}_{\max }^{(n)}=(2\pi n/7,0)$, $n=0,\dots ,6$.

Figure 3

Comparison between the density of states of system A and B. (a) DOS of the Dirac center band at ${\varphi }_A=(1/4){\varphi }_0$ and $(t_1,t_2,t_3)=(1,1,1)$. Inset: eight gapless DFs with locations given by Eq. (5). (b) DOS at ${\varphi }_B=(11/45){\varphi }_0$ and $(t_1,t_2,t_3)=(1,1,1)$ reflecting the reconstruction of the Dirac band in (a). (c) DOS of the Dirac center band at ${\varphi }_A=(1/4){\varphi }_0$ and $(t_1,t_2,t_3)=(1.441,1,1)$. Inset: eight gapped DFs with the gap-opening threshold $t_1=2^{1/4}\approx 1.19$. (d) DOS at ${\varphi }_B=(11/45){\varphi }_0$ and $(t_1,t_2,t_3)=(1.441,1,1)$ reflecting the reconstruction of the gapped Dirac band in (c). Inset shows emergent Dirac fermions at the critical point.

Figure 4

TPTs of system B (${\varphi }_B=11/45$) steered by the VHSs of system A (${\varphi }_A=1/4$). Eleven B bands form near charge neutrality by splitting of the Dirac band of A in response to a flux deviation $\delta \varphi =-1/180$, where nine of these bands are shown. All energies are rescaled by the average band separation $w$ of the B system with $t_1=1$. The Chern numbers of the bands are indicated by color coding. $E_{\mathrm{VHS},1}^A$ and $E_{\mathrm{VHS},2}^A$ are represented, respectively, by solid green and purple lines. The composite fermion (IHCI) TPTs at $n=47/90$ and $n=50/90$ ($n=49/90$) are marked by vertical red (black) arrows.