Mixed-spin incompressible states in the fractional quantum Hall effect

Quench Dynamics of Collective Modes in Fractional Quantum Hall Bilayers

We introduce different types of quenches to probe the nonequilibrium dynamics and multiple collective modes of bilayer fractional quantum Hall states. We show that applying an electric field in one layer induces oscillations of a spin-1 degree of freedom, whose frequency matches the long-wavelength limit of the dipole mode. On the other hand, oscillations of the long-wavelength limit of the quadrupole mode, i.e., the spin-2 graviton, as well as the combination of two spin-1 states, can be activated by a sudden change of band mass anisotropy. We construct an effective field theory to describe the quench dynamics of these collective modes. In particular, we derive the dynamics for both the spin-2 and the spin-1 states and demonstrate their excellent agreement with numerics.

Light-Induced Fractional Quantum Hall Phases in Graphene

We show how to realize two-component fractional quantum Hall phases in monolayer graphene by optically driving the system. A laser is tuned into resonance between two Landau levels, giving rise to an effective tunneling between these two synthetic layers. Remarkably, because of this coupling, the interlayer interaction at nonzero relative angular momentum can become dominant, resembling a hollow-core pseudopotential. In the weak tunneling regime, this interaction favors the formation of singlet states, as we explicitly show by numerical diagonalization, at fillings ν=1/2 and ν=2/3. We discuss possible candidate phases, including the Haldane-Rezayi phase, the interlayer Pfaffian phase, and a Fibonacci phase. This demonstrates that our method may pave the way towards the realization of non-Abelian phases, as well as the control of topological phase transitions, in graphene quantum Hall systems using optical fields and integrated photonic structures.

Landau-Level Mixing and Particle-Hole Symmetry Breaking for Spin Transitions in the Fractional Quantum Hall Effect

The spin transitions in the fractional quantum Hall effect provide a direct measure of the tiny energy differences between differently spin-polarized states and thereby serve as an extremely sensitive test of the quantitative accuracy of the theory of the fractional quantum Hall effect, and, in particular, of the role of Landau-level mixing in lifting the particle-hole symmetry. We report on an accurate quantitative study of this physics, evaluating the effect of Landau-level mixing in a nonperturbative manner using a fixed-phase diffusion Monte Carlo method. We find excellent agreement between our calculated critical Zeeman energies and the experimentally measured values. In particular, we find, as also do experiments, that the critical Zeeman energies for fractional quantum Hall states at filling factors ν=2-n/(2n±1) are significantly higher than those for ν=n/(2n±1), a quantitative signature of the lifting of particle-hole symmetry due to Landau-level mixing.

Luttinger Theorem for the Strongly Correlated Fermi Liquid of Composite Fermions

While an ordinary Fermi sea is perturbatively robust to interactions, the paradigmatic composite-fermion (CF) Fermi sea arises as a nonperturbative consequence of emergent gauge fields in a system where there was no Fermi sea to begin with. A mean-field picture suggests two Fermi seas, of composite fermions made from electrons or holes in the lowest Landau level, which occupy different areas away from half filling and thus appear to represent distinct states. Using the microscopic theory of composite fermions, which satisfies particle-hole symmetry in the lowest Landau level to an excellent approximation, we show that the Fermi wave vectors at filling factors ν and 1-ν are equal when expressed in units of the inverse magnetic length, and are generally consistent with the experimental findings of Kamburov et al. [Phys. Rev. Lett. 113, 196801 (2014)]. Our calculations suggest that the area of the CF Fermi sea may slightly violate the Luttinger area rule.

SU(3) and SU(4) Singlet Quantum Hall States at ν=2/3

We report on an exact diagonalization study of fractional quantum Hall states at a filling factor of ν=2/3 in a system with a fourfold degenerate n=0 Landau level and SU(4) symmetric Coulomb interactions. Our investigation reveals previously unidentified SU(3) and SU(4) singlet ground states which appear at a flux quantum shift 2 when a spherical geometry is employed and lie outside the established composite-fermion or multicomponent Halperin state patterns. We evaluate the two-particle correlation functions of these states and discuss quantum phase transitions in graphene between singlet states with a different number of components as the magnetic field strength is increased.

Broken SU(4) symmetry and the fractional quantum Hall effect in graphene

We describe a variational theory for incompressible ground states and charge gaps in the N=0 Landau level of graphene that accounts for the fourfold Landau level degeneracy and the short-range interactions that break SU(4) spin-valley invariance. Our approach explains the experimental finding that gaps at odd numerators are weak for 1<|ν|<2 and strong for 0<|ν|<1. We find that in the SU(4) invariant case the incompressible ground state at |ν|=1/3 is a three-component incompressible state, not the Laughlin state, and discuss the competition between these two states in the presence of SU(4) spin-valley symmetry-breaking terms.

Competing crystal phases in the lowest Landau level

We show that the solid phase between the 1/5 and 2/9 fractional quantum Hall states arises from an extremely delicate interplay between type-1 and type-2 composite fermion crystals, clearly demonstrating its nontrivial, strongly correlated character. We also compute the phase diagram of various crystals occurring over a wide range of filling factors and demonstrate that the elastic constants exhibit nonmonotonic behavior as a function of the filling factor, possibly leading to distinctive experimental signatures that can help mark the phase boundaries separating different kinds of crystals.

Phase diagram of the two-component fractional quantum Hall effect

We calculate the phase diagram of the two component fractional quantum Hall effect as a function of the spin or valley Zeeman energy and the filling factor, which reveals new phase transitions and phase boundaries spanning many fractional plateaus. This phase diagram is relevant to the fractional quantum Hall effect in graphene and in GaAs and AlAs quantum wells, when either the spin or valley degree of freedom is active.

Integer quantum Hall effect for bosons

A simple physical realization of an integer quantum Hall state of interacting two dimensional bosons is provided. This is an example of a symmetry-protected topological (SPT) phase which is a generalization of the concept of topological insulators to systems of interacting bosons or fermions. Universal physical properties of the boson integer quantum Hall state are described and shown to correspond with those expected from general classifications of SPT phases.

Multi-component fractional quantum Hall states in graphene: SU(4) versus SU(2)

Because of the spin and Dirac-valley degrees of freedom, graphene allows the observation of one-, two- or four-component fractional quantum Hall effects in different parameter regions. We address the stability of various states in the SU(2) and SU(4) limits. In the SU(4) limit, we predict that new low-energy Goldstone modes determine the stability of the fractional quantum Hall states at 2/5, 3/7, etc; SU(4) skyrmions are not found to be relevant for the low-energy physics. These results are discussed in light of experiments.

Bipartite composite fermion States

We study a class of ansatz wave functions in which composite fermions form two correlated "partitions." These "bipartite" composite fermion states are demonstrated to be very accurate for electrons in a strong magnetic field interacting via a short-range 3-body interaction potential over a broad range of filling factors. Furthermore, this approach gives accurate approximations for the exact Coulomb ground state at 2+3/5 and 2+4/7 and is thus a promising candidate for the observed fractional quantum Hall states at the hole conjugate fractions at 2+2/5 and 2+3/7.

Atypical fractional quantum Hall effect in graphene at filling factor 1/3

We study, with the help of exact-diagonalization calculations, a four-component trial wave function that may be relevant for the recently observed graphene fractional quantum Hall state at a filling factor ν(G) = 1/3. Although it is adiabatically connected to a 1/3 Laughlin state in the upper spin branch, with SU(2) valley-isospin ferromagnetic ordering and a completely filled lower spin branch, it reveals physical properties beyond such a state that is the natural ground state for a large Zeeman effect. Most saliently, it possesses at experimentally relevant values of the Zeeman gap low-energy spin-flip excitations that may be unveiled in inelastic light-scattering experiments.

Understanding the 5/2 fractional quantum Hall effect without the Pfaffian wave function

It is demonstrated that an understanding of the 5/2 fractional quantum Hall effect can be achieved within the composite fermion theory without appealing to the Pfaffian wave function. The residual interaction between composite fermions plays a crucial role in establishing incompressibility at this filling factor. This approach has the advantage of being amenable to systematic perturbative improvements, and produces ground as well as excited states. It, however, does not relate to non-Abelian statistics in any obvious manner.

Direct measurement of the g factor of composite fermions

The activation gap Delta of the fractional quantum Hall states at constant fillings nu=2/3 and 2/5 has been measured as a function of the perpendicular magnetic field B. A linear dependence of Delta on B is observed while approaching the spin-polarization transition. This feature allows a direct measurement of the g factor of composite fermions which appears to be heavily renormalized by interactions and strongly sensitive to the electronic filling factor.

New phase transition between partially and fully polarized quantum Hall states with charge and spin gaps at nu = 2/3

The average electron spin polarization Rho of a two-dimensional electron gas confined in GaAs/GaAlAs multiple quantum wells was measured by NMR near the fractional quantum Hall state with filling factor nu = 2/3. Above this filling factor (2/3< or = nu < 0.85), a strong depolarization is observed corresponding to two spin flips per additional flux quantum. The most remarkable behavior of the polarization is observed at nu = 2/3, where a quantum phase transition from a partially polarized (Rho approximately 3/4) to a fully polarized (Rho = 1) state can be driven by increasing the ratio between the Zeeman and the Coulomb energy above a critical value eta(c) = Delta(Z)/Delta(C) = 0.0185.

Composite fermion hofstadter problem: partially polarized density wave states in the nu = 2/5 fractional quantum hall effect

It is well known that the nu = 2/5 state is unpolarized at zero Zeeman energy, while it is fully polarized at large Zeeman energies. A novel state with a charge/spin density wave order for composite fermions is proposed to exist at intermediate values of the Zeeman coupling for nu = 2/5. This state has half the maximum possible polarization, and can be extended to other incompressible fractions. A Hartree-Fock calculation based on the new approach for all fractional quantum Hall states developed by R. Shankar and the author is used to demonstrate the stability of this state to single-particle excitations and to compute gaps. A very recent experiment shows direct evidence for this state.

Composite Fermions in the Quantum Hall Regime

Recent progress in the understanding of the two-dimensional electron system under the influence of a strong magnetic field is reviewed. This system is characterized by the existence of a particle called the composite fermion, which manifests itself in several dramatic experimental observations.