Abstract:
An interlayer phase coherence develops spontaneously in the bilayer quantum Hall system at the filling factor $\nu =1$. On the other hand, the spin and pseudospin degrees of freedom are entangled coherently in the canted antiferromagnetic phase of the bilayer quantum Hall system at the filling factor $\nu =2$. There emerges a complex Nambu-Goldstone mode with a linear dispersion in the zero tunneling-interaction limit for both cases. Then its phase field provokes a Josephson supercurrent in each layer, which is dissipationless as in a superconductor. We study what kind of phase coherence the Nambu-Goldstone mode develops in association with the Josephson supercurrent and its effect on the Hall resistance in the bilayer quantum Hall system at $\nu=1,2$, by employing the Grassmannian formalism.

Abstract:
We study the interlayer coherent incompressible phase in Trilayer Quantum Hall systems (TLQH) at total filling factor $ \nu_{T}=1 $ from three approaches: Mutual Composite Fermion (MCF), Composite Boson (CB) and wavefunction approach. Just like in Bilayer Quantum Hall system, CB approach is superior than MCF approach in studying TLQH with broken symmetry. The Hall and Hall drag resistivities are found to be quantized at $ h/e^{2} $. Two neutral gapless modes with linear dispersion relations are identified and the ratio of the two velocities is close to $ \sqrt{3} $. The novel excitation spectra are classified into two classes: Charge neutral bosonic 2-body bound states and Charge $ \pm 1 $ fermionic 3-body bound states. In general, there are two 2-body Kosterlize-Thouless (KT) transition temperatures and one 3-body KT transition. The Charge $ \pm 1 $ 3-body fermionic bound states may be the main dissipation source of transport measurements. The broken symmetry in terms of $ SU(3) $ algebra is studied. The structure of excitons and their flowing patterns are given. The coupling between the two Goldstone modes may lead to the broadening in the zero-bias peak in the interlayer correlated tunnelings of the TLQH. Several interesting features unique to TLQH are outlined. Limitations of the CB approach are also pointed out.

Abstract:
Based on the known physics of the excitonic superfluid or 111 state of the quantum Hall bilayer, we create a simple trial wavefunction ansatz for constructing a low-energy branch of (Goldstone) excitations by taking the overall ground state and boosting one layer with respect to the other. This ansatz works extremely well for any interlayer spacing. For small , this is simply the physics of the Goldstone mode, whereas for large , this is a reflection of composite fermion physics. We find hints that certain aspects of composite fermion physics persist to low whereas certain aspects of Goldstone mode physics persist to high . Using these results, we show nonmonotonic behavior of the Goldstone mode velocity as a function of . The quantum Hall bilayer is a remarkably rich system [1, 2]. At small enough spacing between the layers, , the system is known to be an excitonic superfluid [3] known sometimes as the 111 phase [4]. At larger layer spacing, a phase transition or crossover is observed experimentally [5–11] leading to a compressible phase which is well described by two weakly coupled composite fermion Fermi liquids. The nature of this crossover, as well as whether there are intervening phases between small and large , has been a matter of some debate in the community [12–20]. There are some results, however, that are extremely well established theoretically. In the limit where becomes very small, it is known that the Halperin 111 trial wavefunction becomes exact [4]. In a more BCS-like language, this wavefunction can be expressed as [3] where and indicate the layer index (we assume the real spin is frozen throughout this paper) and constitutes the orbital index within the lowest Landau level (chosen to be the -directed momentum in Landau gauge, e.g.). (Strictly speaking this second quantized form of the wavefunction must be projected to fixed number of particles within each layer to generate a Halperin 111 wavefunction. However, in the thermodynamic limit these two descriptions are essentially equivalent.) The BCS-like form of (1) allows one to consider long wavelength Goldstone excitations of the form [3, 21] These modes are expected to form a linearly dispersing low energy branch with energy proportional to for small . Physically, this Goldstone mode corresponds to superflow—one layer being boosted with respect to the other. Both a linearly dispersing mode [22] and excitonic superflow [7, 9, 23–25] were observed experimentally in this system. Some properties of the Goldstone mode were discussed in a numerical study on the torus [17]. Away from the

Abstract:
We study the ground states and low-energy excitations of a generic Dirac material with spin-orbit coupling and a buckling structure in the presence of a perpendicular magnetic field. The ground states can be classified into three types under different conditions: SU(2), easy-plane, and Ising quantum Hall ferromagnets. For the SU(2) and the easy-plane quantum Hall ferromagnets there are goldstone modes in the collective excitations, while all the modes are gapped in an Ising-type ground state. We compare the Ising quantum Hall ferromagnet with that of bilayer graphene and present the domain wall solution at finite temperatures. We then specify the phase transitions and transport gaps in silicene in Landau levels 0 and 1. The phase diagram strongly depends on the magnetic field and the dielectric constant. We note that there exists triple points in the phase diagrams in Landau level N = 1 that could be observed in experiments.

Abstract:
Weakly disordered bilayer quantum Hall systems at filling factor $\nu=1$ show spontaneous interlayer phase coherence if the layers are sufficiently close together. We study the collective modes in the system, the current-voltage characteristics and their evolution with an in-plane magnetic field in the phase-coherent regime.

Abstract:
Based on the construction of generalized Halperin wave functions, we predict the possible existence of a large class of broken spin symmetry states in bilayer quantum Hall structures, generalizing the recently suggested canted antiferromgnetic phase to many fractional fillings. We develop the appropriate Chern-Simons theory, and establish explicitly that the low-lying neutral excitation is a Goldstone mode and that the charged excitations are bimerons with continuously tunable (through the canted antiferromagnetic order parameter) electric charge on the individual merons.

Abstract:
At the filling factor $\nu=2$ the bilayer quantum Hall system has three phases, the ferromagnetic phase (spin phase), the spin singlet phase (ppin phase) and the canted antiferromagnetic phase. We analyze soft waves and quasiparticle excitations in the spin and ppin phases. It is shown that the dynamic field is the Grassmannian G$_{4,2}$ field carrying four complex degrees of freedom. In each phase there are four complex soft waves (pseudo-Goldstone modes) and one kind of skyrmion excitations (G$_{4,2}$ skyrmions) flipping either spins or pseudospins coherently. An intriguing property is that a quasiparticle is a G$_{4,2}$ skyrmion essentially consisting of two CP$^{3}$ skyrmions and thus possesses charge $2e$.

Abstract:
We study the long-wavelength collective modes in the magnetic-field-induced spin-density-wave (FISDW) phases experimentally observed in organic conductors of the Bechgaard salts family, focusing on phases that exhibit a sign reversal of the quantum Hall effect (Ribault anomaly). We have recently proposed that two SDW's coexist in the Ribault phase, as a result of Umklapp processes. When the latter are strong enough, the two SDW's become circularly polarized (helicoidal SDW's). In this paper, we study the collective modes which result from the presence of two SDW's. We find two Goldstone modes, an out-of-phase sliding mode and an in-phase spin-wave mode, and two gapped modes. The sliding Goldstone mode carries only a fraction of the total optical spectral weight, which is determined by the ratio of the amplitude of the two SDW's. In the helicoidal phase, all the spectral weight is pushed up above the SDW gap. We also point out similarities with phase modes in two-band or bilayer superconductors. We expect our conclusions to hold for generic two-SDW systems.

Abstract:
The electromagnetic characteristics of bilayer quantum Hall systems in the presence of interlayer coherence and tunneling are studied by means of a pseudospin-texture effective theory and an algebraic framework of the single-mode approximation, with emphasis on clarifying the nature of the low-lying neutral collective mode responsible for interlayer tunneling phenomena. A long-wavelength effective theory, consisting of the collective mode as well as the cyclotron modes, is constructed. It is seen explicitly from the electromagnetic response that gauge invariance is kept exact, this implying, in particular, the absence of the Meissner effect in bilayer systems. Special emphasis is placed on exploring the advantage of looking into quantum Hall systems through their response; in particular, subtleties inherent to the standard Chern-Simons theories are critically examined.

Abstract:
We study properties of some known trial wavefunctions in bilayer quantum Hall systems at the total filling factor $ \nu_{T}=1 $. In particular, we find that the properties of a meron wavefunction and a natural "quasi-hole" wave function are dramatically different due to the broken symmetry and the associated Goldstone mode in the bulk. Although the (smallest) meron has localized charge $ 1/2 $ and logarithmically divergent energy, the charge of the quasi-hole excitation extends over the whole system and its energy diverges linearly with the area of the system. This indicates that the natural quasi-hole wavefunction is not a good trial wavefunction for excitations. It also shows that the energy of the naive candidate for a pair of meron wavefunction written down previously increases quadratically instead of logarithmically as their separation increases. Our results indicate that qualitatively good trial wave functions for the ground state and the excitations of the interlayer coherent bilayer quantum Hall system at finite $ d $ are still not available and searching for them remains an important open problem.