Interaction-driven quantum phase transitions between topological and crystalline orders of electrons

Abstract

Topological and crystalline orders of electrons both benefit from enhanced Coulomb interactions in partially filled Landau levels. In bilayer graphene (BLG), the competition between fractional quantum Hall liquids and electronic crystals can be tuned electrostatically. Applying a displacement field leads to Landau-level crossings, where the interaction potential is strongly modified due to changes in the orbital wave functions. Here, we leverage this control to investigate phase transitions between topological and crystalline orders at constant filling factors in the lowest Landau level of BLG. Using transport measurements in high-quality hBN-encapsulated devices, we study transitions as a function of displacement field near crossings of $N=0$ and $N=1$ orbitals. The enhanced Landau-level mixing near the crossing stabilizes electronic crystals at all fractional fillings, including a resistive state at $ν= \frac{1}{3}$ and a reentrant integer quantum Hall state at $ν= \frac{7}{3}$. On the $N=0$ side, the activation energies of the crystal and fractional quantum Hall liquid vanish smoothly and symmetrically at the transition, while the $N=1$ transitions out of the crystal appear discontinuous. Additionally, we observe quantized plateaus forming near the crystal transition at half filling of the $N=0$ levels, suggesting a paired composite fermion state stabilized by Landau level mixing.

Figures (7)

• Figure 1: FQH--WC and FQH--RIQH competition at Landau-level crossings. (a) Evolution of symmetry-broken LLs with displacement field. (b) Illustration of crossing $N=0$ and $N=1$ Landau levels as a function of $D$. (c) Transverse and longitudinal conductances, $G_{xy}$ and $G_{xx}$, as a function of $D$, show the transition from the $\nu=\frac{1}{3}$ plateau to an insulating state and back. (d) $G_{xy}$ and $G_{xx}$ as a function of $D$ showing the transition from the $\nu=\frac{7}{3}$ plateau to the RIQH effect and back. (e) Numerically obtained energies in units of Coulomb energy of competing Laughlin FQH and WC states as a function of $D$ relative to the crossing point $\delta D$. Here, $\varepsilon$ is the permittivity and $\ell_B$ is the magnetic length. For strong positive or negative $\delta D$, the FQH is energetically favorable in pure $N=0$ and $N=1$ levels, while the WC is favored near the crossing.
• Figure 2: FQH--WC--FQH phase transitions. (a) $R_{xx}$ as a function of $\nu$ and $D$ showing the disappearing and reemerging FQH states. Black dashed lines indicate data shown in panels b and c. The white dashed lines indicate the expected crossing of partially filled $N=0$ and $N=1$ levels. Colored labels mark quantum numbers of Landau levels. Crossings at lower $|D|$ were omitted for clarity. (b) and (c) $R_{xx}$ as a function of $\nu$ for different temperatures at $D=-500~\mathrm{mV/nm}$ and $D=-550~\mathrm{mV/nm}$, respectively. (d) Arrhenius plot of $R_{xx}$ highlighting the opposite temperature dependence of $R_{xx}$ above and below $D_{c,1}$. (e) Activation energies of FQH and WC states at $\nu=\frac{1}{3}$ showing the FQH--WC--FQH transition. Closed circles were obtained by fitting to $R_{xx} \propto \exp\left(\frac{E_\mathrm{A}}{2k_\mathrm{B}T}\right)$, open circles with $R_{xx} \propto \exp\left(-\frac{\Delta}{2k_\mathrm{B}T}\right)$.
• Figure 3: FQH--RIQH--FQH phase transitions at $2<\nu<3$. (a) $R_{xx}$ as a function of $\nu$ and $D$ showing the disappearing and reemerging FQH effect. Colored labels mark quantum numbers of Landau levels. Crossings at lower $|D|$ were omitted for clarity. (b) and (c) $R_{xx}$ as a function of $\nu$ for different temperatures at $D=-420~\mathrm{mV/nm}$ and $D=-600~\mathrm{mV/nm}$, respectively. (d) Arrhenius plot of $R_{xx}$ showing similar temperature dependence of $R_{xx}$ above and below $D_{c,1}$. (e) Activation gaps of FQH and RIQH states across the two transitions. All gaps were obtained by fitting $R_{xx}\propto\exp\left(-\frac{\Delta}{2k_\mathrm{B}T}\right)$.
• Figure 4: Half-filled states at Landau-level crossings. (a) $R_{xx}$ as a function of $\nu$ and $D$ centered around $\nu=\frac{1}{2}$. The $R_{xx}$ minimum at $\nu=\frac{1}{2}$ is marked by a black arrow. (b) $R_{xx}$ and $R_{xy}$ as a function of $\nu$ at $D=-660~\mathrm{mV/nm}$, clearly showing a minimum and a plateau, respectively. The dashed line marks the expected $R_{xy}$ value for $\nu=\frac{1}{2}$. (c) $R_{xx}$ as a function of $\nu$ and $D$ centered around $\nu=\frac{5}{2}$. The $R_{xx}$ minimum at $\nu=\frac{5}{2}$ is marked by a black arrow. (d) $R_{xx}$ and $R_{xy}$ as a function of $\nu$ at $D=-680~\mathrm{mV/nm}$, clearly showing a minimum and a plateau, respectively. The dashed line marks the expected $R_{xy}$ value for $\nu=\frac{5}{2}$.
• Figure 5: The energies of the eight Landau levels of bilayer graphene are plotted as a function of $D$. The level crossings of the $N=0$ and $N=1$ Landau levels with the same spin and valley occur at $|D|\approx 343 mV/nm$. The partially occupied Landau level in the range $0<\nu<1$ is marked in gray for reference.