Several recent works have proposed that electron-electron interactions in bilayer graphene can be tuned with the help of external parameters, making it possible to stabilize different fractional quantum Hall states. In these prior works, phase diagrams were calculated based on a single Landau level approximation. We go beyond this approximation and investigate the influence of polarization effects and virtual interband transitions on the stability of fractional quantum Hall states in bilayer graphene. We find that for realistic values of the dielectric constant, the phase diagram is strongly modified by these effects. We illustrate this by evaluating the region of stability of the Pfaffian state. Recent experimental observation of the fractional steps in the Hall conductivity of ultraclean suspended graphene 1,2 and in graphene on the hexagonal boron nitride (h-BN) substrate 3 has opened a new chapter in the physics of the fractional quantum Hall effect (FQHE) where new states of matter might be observed. For example, the peculiarities of the single-particle spectrum of graphene are predicted to result in novel strongly correlated Hall fractions such as the SU(4) generalization of the Halperin-Laughlin state. 4 It has also been proposed that graphene allows for unprecedented tunability of the electron-electron interaction potential within a partially filled Landau level, 5,6 allowing access to the experimental investigation of quantum phase transitions between different topological orders 7 and potentially stabilizing exotic topological states such as non-Abelian quantum Hall fluids. 6 Of particular interest from this perspective is bilayer graphene (BLG) in which the matrix elements of the Coulomb interaction within one Landau level (also known as the Haldane pseudopotentials 8 ) may be tuned in situ by the application of an external electric field breaking the symmetry between the two graphene layers. 6 The tunable quantum Hall effect in BLG was investigated theoretically in two recent papers 6,7 where the stability conditions for several Abelian and non-Abelian FQHE states are found. Both groups use the same methodology based on the single Landau level approximation (SLLA). In this approximation the Hilbert space is constrained onto a given Landau level (LL) and the Hamiltonian takes the form of two-body interaction encoded in Haldane pseudopotentials. 8 The SLLA Hamiltonian is then exactly diagonalized for a small number of particles. The advantages of this approach are (a) its relative simplicity and (b) its effectiveness in GaAs structures 9 due to the large values of cyclotron gaps achieved in the experiment and a large value of the dielectric constant, which suppresses transitions between different LLs.10 However, as was discussed in Ref. 11, BLG is a narrow gap semiconductor in the strong coupling regime, where the effects of virtual interband transitions are essential. Moreover, as will be discussed in this work, the large cyclotron gap condition can be easily violated due to the peculiar properties of...

In the quantum Zeno effect, quantum measurements can block the coherent oscillation of a two level system by freezing its state to one of the measurement eigenstates. The effect is conventionally controlled by the measurement frequency. Here we study the development of the Zeno regime as a function of the measurement strength for a continuous partial measurement. We show that the onset of the Zeno regime is marked by a cascade of transitions in the system dynamics as the measurement strength is increased. Some of these transitions are only apparent in the collective behavior of individual quantum trajectories and are invisible to the average dynamics. They include the appearance of a region of dynamically inaccessible states and of singularities in the steady-state probability distribution of states. These predicted dynamical features, which can be readily observed in current experiments, show the coexistence of fundamentally unpredictable quantum jumps with those continuously monitored and reverted in recent experiments.

The state of a quantum system, adiabatically driven in a cycle, may acquire a measurable phase depending only on the closed trajectory in parameter space. Such geometric phases are ubiquitous, and also underline the physics of robust topological effects such as the quantized Hall conductance. Equivalently, a geometric phase may be induced through a cyclic sequence of quantum measurements. We show that the application of a sequence of weak measurements renders the closed trajectories, hence the geometric phase, stochastic. We study the concomitant probability distribution and show that, when tuning the measurement strength, the mapping between the measurement sequence and the geometric phase undergoes a topological transition. Our finding has the potential to impact the study of measurement-induced state distillation, trajectory manipulation, and active error correction-all crucial ingredients for quantum information processing.

In a recent experimental paper [Bid et al., Nature 466, 585 (2010)] a qualitative confirmation of the existence of upstream neutral modes at the ν = 2 3 quantum Hall edge was reported. Using the chiral Luttinger liquid theory of the quantum Hall edge we develop a quantitative model of the experiment of Bid et al. A good quantitative agreement of our theory with the experimental data reinforces the conclusion of the existence of the upstream neutral mode. Our model also enables us to extract important quantitative information about nonequilibrium processes in Ohmic and tunneling contacts from the experimental data. In particular, for ν = 2 3 , we find a power-law dependence of the neutral mode temperature on the charge current injected from the Ohmic contact.