Fractional quantum Hall liquids exhibit a rich set of excitations, the lowest-energy of which are the magneto-rotons with dispersion minima at finite momentum. We propose a theory of the magnetorotons on the quantum Hall plateaux near half filling, namely, at filling fractions ν = N/(2N + 1) at large N . The theory involves an infinite number of bosonic fields arising from bosonizing the fluctuations of the shape of the composite Fermi surface. At zero momentum there are O(N ) neutral excitations, each carrying a well-defined spin that runs integer values 2, 3, . . .. The mixing of modes at nonzero momentum q leads to the characteristic bending down of the lowest excitation and the appearance of the magneto-roton minima. A purely algebraic argument show that the magnetoroton minima are located at q B = zi/(2N + 1), where B is the magnetic length and zi are the zeros of the Bessel function J1, independent of the microscopic details. We argue that these minima are universal features of any two-dimensional Fermi surface coupled to a gauge field in a small background magnetic field. Introduction.-Interacting electrons moving in two dimensions in a strong magnetic field can form nontrivial topological states: the fractional quantum Hall liquids [1, 2]. When the lowest Landau level is filled at certain rational filling fractions, including ν = N/(2N + 1) and ν = (N + 1)/(2N + 1) (the Jain's sequences), the quantum Hall liquid is gapped, and the lowest energy mode is a neutral mode. Girvin, MacDonald, and Platzman [3] proposed, based on a variational ansatz that the neutral excitation has a broad minimum at q B ∼ 1 at the Laughlin plateau ν = 1/3. Several years later, the existence of a neutral mode was confirmed experimentally [4]. Later experiments reveal surprising richness in the structure of the spectrum of neutral excitations. Unexpectedly, the ν = 1/3 state may have more than one branches of excitations [5]. Furthermore, higher in the Jain sequence, i.e., for ν = 2/5, 3/7, etc., the lowest excitation has been found to have a dispersion with more than one minima [6,7]. Various theoretical approaches have been brought to the problem of the magneto-roton [8][9][10][11][12]. Currently, the most common viewpoint is based on the composite fermion picture of the fractional quantum Hall effect, in which the neutral modes are bound states of a composite fermion and a composite hole.The notion of the composite fermion is tightly connected to the Halperin-Lee-Read (HLR) field theory [13], proposed as the low-energy description of the half-filled Landau level. Recently, an analysis of the particle-hole symmetry of the lowest Landau level has lead to a revision of the HLR proposal: the low-energy degrees of freedom is now a Dirac composite fermion coupled to a gauge field [14]. Magneto-rotons provide a rare window into the dynamics of a Fermi surface coupled to a gauge field, a long-standing problem of condensed matter

We study fractional quantum Hall states at filling fractions in the Jain sequences using the framework of composite Dirac fermions. Synthesizing previous work, we write down an effective field theory consistent with all symmetry requirements, including Galilean invariance and particlehole symmetry. Employing a Fermi liquid description, we demonstrate the appearance of the Girvin-Macdonlald-Platzman algebra and compute the dispersion relation of neutral excitations and various response functions. Our results satisfy requirements of particle-hole symmetry. We show that while the dispersion relation obtained from the HLR theory is particle-hole symmetric, correlation functions obtained from HLR are not. The results of the Dirac theory are shown to be consistent with the Haldane bound on the projected structure factor, while those of the HLR theory violate it. arXiv:1709.07885v1 [cond-mat.mes-hall]

We consider gapped fractional quantum Hall states on the lowest Landau level when the Coulomb energy is much smaller than the cyclotron energy. We introduce two spectral densities, ρ T (ω) andρ T (ω), which are proportional to the probabilities of absorption of circularly polarized gravitons by the quantum Hall system. We prove three sum rules relating these spectral densities with the shift S, the q 4 coefficient of the static structure factor S 4 , and the high-frequency shear modulus of the ground state µ ∞ , which is precisely defined. We confirm an inequality, first suggested by Haldane, that S 4 is bounded from below by |S − 1|/8. The Laughlin wavefunction saturates this bound, which we argue to imply that systems with ground state wavefunctions close to Laughlin's absorb gravitons of predominantly one circular polarization. We consider a nonlinear model where the sum rules are saturated by a single magneto-roton mode. In this model, the magneto-roton arises from the mixing between oscillations of an internal metric and the hydrodynamic motion. Implications for experiments are briefly discussed.

Employing the fracton-elastic duality, we develop a low-energy effective theory of a zero-temperature vortex crystal in a two-dimensional bosonic superfluid which naturally incorporates crystalline topological defects. We extract static interactions between these defects and investigate several continuous quantum transitions triggered by the Higgs condensation of vortex vacancies/interstitials and dislocations. We propose that the quantum melting of the vortex crystal towards the hexatic or smectic phase may occur via a pair of continuous transitions separated by an intermediate vortex supersolid phase.

We derive a number of exact relations between response functions of holomorphic, chiral fractional quantum Hall states and their particle-hole (PH) conjugates. These exact relations allow one to calculate the Hall conductivity, Hall viscosity, various Berry phases, and the static structure factor of PH-conjugate states from the corresponding properties of the original states. These relations establish a precise duality between chiral quantum Hall states and their PH-conjugates. The key ingredient in the proof of the relations is a generalization of Girvin's construction of PH-conjugate states to inhomogeneous magnetic field and curvature. Finally, we make several non-trivial checks of the relations, including for the Jain states and their PH-conjugates.Introduction. Particle-hole (PH) transformation for fractional quantum Hall (FQH) states was introduced by Girvin [1]. This transformation relates a FQH state at filling fraction ν to a FQH state at filling fraction 1−ν. In the absence of Landau level mixing the projected lowest Landau level (LLL) Hamiltonian is PH-symmetric and, therefore, two states related by a PH transformation have the same energy (up to a shift in the chemical potential). Despite the physical clarity of PH-symmetry, the PHtransformed wave functions look quite complicated and are difficult to work with. PH-transformed states contain a different number of particles, have different transport properties and different topological order. In this Letter we will explain that all of the information about PH-transformed state is encoded in the original state, so that both states are a different representation for essentially the same physics. For this reason we feel it is more appropriate to refer to the PH-transformation as a particle-hole duality (PHD).Recent years have also brought the rise of interest in the role of PHD in the problem of the half-filled Landau level. To resolve the issue of the apparent absence of the PH-invariance in the Halperin-Lee-Read [2] theory, Son has proposed a manifestly PH-invariant effective theory of composite fermions with π Berry phase around the composite Fermi surface [3]. This theory can successfully be used to describe Jain states at fillings close to ν = 1/2 and a PH-invariant (or self-dual) version of the Pfaffian state [3,4], which is a viable candidate for the observed ν = 5/2 plateau [5].PH-transformation, as defined by Girvin [1], works in flat space and homogeneous magnetic field. It was recently appreciated that placing a FQH state in inhomogeneous background magnetic field and curved geometry allows one to extract considerable information about the flat space properties of the state [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23][24][25]. For example, the projected static structure factor (SSF) [26] in leading and sub-leading order in momentum, and long-wave corrections to Hall conductivity and Hall viscosity can be