Thierry Jolicoeur

We study the fractional quantum Hall effect in the central Landau level of AB stacked (Bernal) bilayer graphene. By tuning the external applied magnetic field and the electric bias between the two layers one can access a regime where there is a degeneracy between Landau levels with orbital characters corresponding to $N=0$ and $N=1$ Galilean- Landau levels. While the Laughlin state is generically the ground state for filling $\nu =1/3$ we find that it can be destroyed and replaced by a Wigner crystal at the same filling factor by tuning the bias and applied field. This competition does not take place at $\nu =2/3$ where the incompressible ground state remains stable. The possibility of electrically inducing the Wigner crystal state opens a new range of studies of this state of matter.

We study spin and valley ordering in the quantum Hall fractions in monolayer graphene at Landau level filling factors ${\nu }_G=-2+n/3(n=2,4,5)$. We use exact diagonalizations on the spherical as well as toroidal geometry by taking into account the effect of realistic anisotropies that break the spin/valley symmetry of the pure Coulomb interaction. We also use a variational method based on eigenstates of the fully $\text{SU}\left(4\right)$ symmetric limit. For all the fractions we study, there are two-component states for which the competing phases are generalizations of those occurring at neutrality ${\nu }_G=0$. They are ferromagnetic, antiferromagnetic, charge-density wave, and Kékulé phases, depending on the values of Ising or $XY$ anisotropies in valley space. The varying spin-valley content of the states leads to ground-state quantum numbers that are different from the ${\nu }_G=0$ case. For filling factor ${\nu }_G=-2+5/3$, there is a parent state in the $\text{SU}\left(4\right)$ limit that has a flavor content $(1,1/3,1/3,0)$ where the two components that are one-third filled form a two-component singlet. The addition of anisotropies leads to the formation of new states that have no counterpart at ${\nu }_G=0$. While some of them are predicted by the variational approach, we find notably that negative Ising-like valley anisotropy leads to the formation of a state that is a singlet in both spin and valley space and lies beyond the reach of the variational method. Also fully spin polarized two-component states at $\nu =-2+4/3$ and $-2+5/3$ display an emergent $\text{SU}\left(2\right)$ valley symmetry because they do not feel point-contact anisotropies. We discuss implications for current experiments concerning possible spin transitions.

A recent work [Balram, Jain, and Barkeshli, Phys. Rev. Res. 2, 013349 (2020)] has suggested that an unconventional state describing ${\mathbb{Z}}_n$ superconductivity of composite bosons, which supports excitations with charge $1/\left(3n\right)$ of the electron charge, is energetically better than the Laughlin wave function at $\nu =7/3$ in GaAs systems. All experiments to date, however, are consistent with the latter. To address this discrepancy, we study the effect of finite width on the ground state and predict a phase transition from an unconventional ${\mathbb{Z}}_n$ state at small widths to the Laughlin state for widths exceeding $\sim 1.5$ magnetic lengths. We also determine the parameter region where an unconventional state is stabilized in the one-third filled zeroth Landau level in bilayer graphene. The roles of Landau level mixing and spin are also considered.

We investigate fractional quantum Hall states for model interactions restricted to a repulsive hard core. When the hard core excludes relative angular momentum $m=1$ between spinless electrons the ground state at Landau level filling factor $\nu =\frac{1}{3}$ is known to be exactly given by the Laughlin wave function. When we exclude relative angular momentum 3 only, Wójs, Quinn, and Yi have suggested the appearance of a liquid state with non-Laughlin correlations. We study this special hard-core interaction at filling factor $\frac{1}{3}$ on the sphere, torus, and cylinder geometry. An analysis of the charged and neutral gaps on the sphere geometry points to a gapless state. On the torus geometry the projected static structure factor has a two-peak feature pointing to one-dimensional density ordering. To clarify the nature of the ground state we perform extended density matrix renormalization group studies on the cylinder geometry for up to 30 particles. The pair correlation function allows us to conclude that the ground state is a two-particle bubble phase.

Monolayer graphene under a strong perpendicular magnetic field exhibits quantum Hall ferromagnetism with spontaneously broken spin and valley symmetry [K. Nomura and A. H. MacDonald, Phys. Rev. Lett. 96, 256602 (2006)]. The approximate SU(4) spin/valley symmetry is broken by small lattice-scale effects in the central Landau level corresponding to filling factors $\nu =0,\pm 1$. Notably, the ground state at $\nu =0$ is believed to be a canted antiferromagnetic (AF) or a ferromagnetic (F) state depending on the component of the magnetic field parallel to the layer and the strength of small anisotropies. We study the skyrmions for the filling factors $\nu =\pm 1,0$ by using exact diagonalizations on the spherical geometry. If we neglect anisotropies, we confirm the validity of the standard skyrmion picture generalized to four degrees of freedom. For filling factor $\nu =-1$, the hole skyrmion is an infinite-size valley skyrmion with full spin polarization because it does not feel the anisotropies. The electron skyrmion is also always of infinite size. In the F phase it is always fully polarized, while in the AF phase it undergoes continuous magnetization under increasing Zeeman energy. In the case of $\nu =0$, the skyrmion is always maximally localized in space both in F and AF phases. In the F phase it is fully polarized, while in the AF phase it has also progressive magnetization with Zeeman energy. The magnetization process is unrelated to the spatial profile of the skyrmions, contrary to the SU(2) case. In all cases, the skyrmion physics is dominated by the competition between anisotropies and Zeeman effect but not directly by the Coulomb interactions, breaking universal scaling with the ratio Zeeman to Coulomb energy.

We reexamine the nature of the ground states of bilayer graphene at odd integer filling factors within a simplified model of nearly degenerate $n=0$ and $n=1$ Landau levels. Previous Hartree-Fock studies have found that ferroelectric states with orbital coherence can be stabilized by tuning the orbital splitting between these levels. These studies indicated that, in addition to a uniform ferroelectric state, a helical ferroelectric phase with spontaneously broken translational symmetry is possible. By performing exact diagonalization on the torus, we argue that the system does not have a uniform coherent state but instead transitions directly from the uniform incoherent state into the ferroelectric helical phase. We argue that there is a realistic prospect to stabilize the helical ferroelectric state in bilayer graphene by tuning the interlayer electric field in a model that includes all single-particle corrections to its zero energy eightfold multiplet of Landau levels.