Emergent Multi-Flavor ${\mathrm{QED}}_3$ at the Plateau Transition between Fractional Chern Insulators: Applications to Graphene Heterostructures

Recent experiments in graphene heterostructures have observed Chern insulators—integer and fractional quantum Hall states made possible by a periodic substrate potential. Here, we study theoretically that the competition between different Chern insulators, which can be tuned by the amplitude of the periodic potential, leads to a new family of quantum critical points described by ${\mathrm{QED}}_3$-Chern-Simons theory. At these critical points, $N_f$ flavors of Dirac fermions interact through an emergent U(1) gauge theory at Chern-Simons level $K$, and remarkably, the entire family (with any $N_f$ or $K$) can be realized at special values of the external magnetic field. Transitions between particle-hole conjugate Jain states realize “pure” ${\mathrm{QED}}_3$, in which multiple flavors of Dirac fermions interact with a Maxwell U(1) gauge field. The multiflavor nature of the critical point leads to an emergent $\mathrm{SU}(N_f)$ symmetry. Specifically, at the transition from a $\nu =1/3$ to $2/3$ quantum Hall state, the emergent SU(3) symmetry predicts an octet of charge density waves with enhanced susceptibilities, which is verified by DMRG numerical simulations on microscopic models applicable to graphene heterostructures. We propose experiments on Chern insulators that could resolve open questions in the study of ($2+1$)-dimensional conformal field theories and test recent duality inspired conjectures.

Popular Summary

At the boundary between two different quantum phases, a system can fluctuate severely. This is called quantum criticality. One of the most interesting possibilities at quantum criticality is the existence of massless fractionalized particles interacting strongly via gauge bosons, much like quarks interact strongly via gluons inside a proton. However, after a few decades of study, the precise properties of these theories remain unknown because of a lack of controlled theoretical techniques or experimental systems to study them. To fill this gap, we propose that a certain family of critical theories, know as QED3-Chern-Simons theory, can be studied in the phase transitions of fractional Chern insulators (FCIs).

Remarkably, we find that experiments with graphene could realize the entire family of the QED3-Chern-Simons theory (containing an infinite number of different critical theories). These experiments rely on techniques, namely, creating and manipulating an artificial lattice in graphene, that have recently been achieved in labs. By observing the behaviors of electrons, researchers should be able to measure properties of the critical theories such as critical exponents and emergent symmetry. Our theoretical and numerical analyses show that FCI transitions have emergent symmetry and exist in simple theoretical models that mimic experiments.

We expect that our proposed experiments on FCIs should help resolve many open questions concerning criticality theories.

Figure 1

Red and blue lines represent gapped trajectories labeled by $(C,s)$ and $(C^{\prime },s^{\prime })$, respectively, which intersect at $(n_{*},{\varphi }_{*})$. For a generic value of the tuning parameter $m$ (superlattice potential strength), one type of gap will “win” over the other at the crossing. At the critical value $m=m_c$, the gap closes and their relative strength is exchanged; this is the manifestation of a Chern-number changing transition.

Figure 2

Landau levels in the presence of a superlattice potential Eq. (8). We consider $\varphi =2/3$, $V_1=V_2=1$, $V_3=V_4=1.4$. The superlattice is projected into the $N=1$ and $N=2$ Landau levels with spacing $\mathrm{\Delta }{E}_{12}=1$. Each Landau level splits into two subbands, so that, at $n=2/3$, the two lowest subbands are filled. (a) Chern number and gap versus potential strength $U_0$ at $n=2/3$. (b) The band structure at the transition ($U_0\approx 0.79$) shows three emergent Dirac cones which mediate the transition between the $C=1$ and $C=-2$ insulators.

Figure 3

The lowest Landau level (LLL) under the square lattice potential [Eq. (8)] with the magnetic flux density $\varphi =2$ in one unit cell. We take $V_1=V_2=1$, (a) $V_3=V_4=0$, (b) $V_3=V_4=0.5$, (c) $V_3=V_4=1.4$. The upper panel shows the potential pattern, and the lower panel shows the band structure. The LLL splits into two subbands; the lowest band has Chern number $C=1$ when $V_3=V_4>0$. Tuning the potential pattern gives a nearly flat band as shown in (c).

Figure 4

The DMRG simulation result of the electrons on the infinite cylinder with circumference size $L_y=31.9l_B$. We consider the LLL under a square lattice potential [Eq. (8)] with $V_1=V_2=1$, $V_3=V_4=1.4$, magnetic flux density (per unit cell) $\varphi =2$, electron density $n=2/3$. We change the strength of lattice potential $U_0$ with a step size $\mathrm{\Delta }U=0.001$, with the strength of the short-ranged Coulomb interaction being constant, $E_C=1$. The blue line represents the change of Hall conductivity, which is measured at bond dimension $\chi =2000$. It shows a sharp direct transition between the ${\sigma }_{xy}=1/3$ and ${\sigma }_{xy}=2/3$ state at $U_0\sim 0.06$. Red lines represent the change of correlation length (in the unit of lattice constant $a$ in the $y$-direction) with different bond dimensions $\chi$. At the phase transition point ($U_0\sim 0.06$), the correlation seems to diverge with the bond dimensions. The truncation error is ${\epsilon }_{\mathrm{trun}}<{10}^{-6}$ at $U_0=0$, ${\epsilon }_{\mathrm{trun}}\sim 6\times {10}^{-5}$ at the critical point, and ${\epsilon }_{\mathrm{trun}}\sim 4\times {10}^{-5}$ at the ${\sigma }_{xy}=2/3$ phase.

Figure 5

Correlation length spectra of operators carrying zero electric charge versus crystal momenta $\overrightarrow{k}$ in three different regimes: (i) ${\sigma }_{xy}=1/3$ FQH phase; (ii) critical point; (iii) ${\sigma }_{xy}=2/3$ FCI phase. The spectrum is obtained from simulations with DMRG bond dimension $\chi =6000$. The color bar represents the inverse of correlation length $1/\xi$. Note that we have a finite circumference size $L_y=9$ in the units of the lattice constant, so the $k_y$ take discretized values $k_y=2\pi n/L_y$, $n=0,1,\dots L_y-1$. At the critical point, the correlation lengths at the eight momenta $(k_x,k_y)=(2\pi n_x/3,2\pi n_y/3)\ne 0$ are the largest and identical. These modes correspond to the eight SU(3) adjoint mass terms. Numerically, correlation lengths at other $k$-points have at least twice smaller correlation lengths in this data. The spectra at the ${\sigma }_{xy}=1/3$ and $2/3$ phases have no additional symmetries other than two reflections ($k_x$ and $k_y$) and approximate $C_4$ rotation inherent to the simulation. At ${\sigma }_{xy}=2/3$, one may notice that there are also eight points with almost identical correlation lengths near the aforementioned eight special momenta. However, (i) they deviate from the exact momenta of SU(3) adjoint mass terms, and (ii) their correlation lengths are not well separated from correlation lengths of other $k$-points, e.g., $(k_x,k_y)\sim (\pm 2\pi /3,\pm 4\pi /9)$.

Figure 6

DMRG simulation result on infinite cylinder geometry with circumference size $L_y=24.6l_B$. $\varphi =3/2$, $n=1/2$, $V_1=V_2=1$, $V_3=V_4=1.6$, while changing the potential strength $U_0$. The blue line represents the change of Hall conductivity, which is measured at bond dimension $\chi =2000$. Red lines represent the change of correlation length with bond dimensions $\chi =6000$, 7000.

Figure 7

The response of the ${\sigma }_{xy}=1/3$ to ${\sigma }_{xy}=2/3$ transition under the adiabatic flux insertion from 0 to $6\pi$. Through this procedure, we can also calculate the Hall conductance by measuring the charge transported through the flux insertion, where we changed the flux by $2\pi /10$ in each step. Here, the potential strength $U_0=0.6$ and the bond dimension $\chi =2000$. Both the energy and correlation length change as a function of $\mathrm{\Phi }$ with a periodicity of $6\pi$, three flux quanta. Peak structures in both plots reveal the existence of the emergent Dirac fermions in the low-energy effective theory.