Widely Tunable Quantum Phase Transition from Moore-Read to Composite Fermi Liquid in Bilayer Graphene

We develop a proposal to realize a widely tunable and clean quantum phase transition in bilayer graphene between two paradigmatic fractionalized phases of matter: the Moore-Read fractional quantum Hall state and the composite Fermi liquid metal. This transition can be realized at total fillings $\nu =\pm 3+1/2$ and the critical point can be controllably accessed by tuning either the interlayer electric bias or the perpendicular magnetic field values over a wide range of parameters. We study the transition numerically within a model that contains all leading single particle corrections to the band structure of bilayer graphene and includes the fluctuations between the $n=0$ and $n=1$ cyclotron orbitals of its zeroth Landau level to delineate the most favorable region of parameters to experimentally access this unconventional critical point. We also find evidence for a new anisotropic gapless phase stabilized near the level crossing of $n=0/1$ orbits.

Figure 1

(a) Phase diagram of two-orbital model. ${\mathrm{\Delta }}_{10}$ is the orbital splitting and $\gamma$ parametrizes form factors controlled by magnetic field. We identify the MR, two types of CFL states, and an intermediate AGP. The shaded region is the expected range of parameters accessed in BLG by tuning $B[T]$ and the interlayer electric bias $u$. (b) The phase diagram of the SU(2) two-valley model [see Eq. (1)] as a function of $B[T]$ or $\gamma$ expected to be realized at $u=0$. There are three phases: the valley polarized MR, and the valley polarized, and unpolarized CFL states.

Figure 2

(a) The single particle splittings of BLG as a function of interlayer bias $u$. (b) The schematic depiction of the zero Landau level manifold of BLG $|\tau n\sigma ⟩$ on $(A,B^{\prime },A^{\prime },B)$ sites. Here, $\tau =\{K,K^{\prime }\}\equiv \{+,-\}$ denotes valleys, $n$ and $\sigma$ are the LL and spin index, respectively. (c) The relationship between parameter $\gamma$ and the magnetic field $B$. Figures (a) and (b) are from Ref. [15], the data of (c) is from Ref. [4].

Figure 3

(a) The SU(2) Casimir operator $S^2$ as a function of $\gamma$. (b) and (c) The energy spectra as a function of $\gamma$ for the valley-unpolarized sector (b) and valley-polarized sector (c). (d) The cluster of momentum determined by the trial wave function for polarized CFL (upper) and unpolarized CFL (lower).

Figure 4

The energy spectra of a two-orbital model as a function of $\gamma$ at different single particle splitting ${\mathrm{\Delta }}_{10}$ between ${\psi }_0$ and ${\psi }_1$ orbits: (a) ${\mathrm{\Delta }}_{10}=0.08$, (b) ${\mathrm{\Delta }}_{10}=0.2$.