Higher-Spin Theory of the Magnetorotons

Abstract

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

Positions of the magnetoroton minima in the fractional quantum Hall effect

“…The positions of the magnetoroton minima agree well (within 15%) with the recent prediction of these magnetoroton minima positions by Golkar et al 30 . We also note that the composite fermion Chern-Simons theory 28 predicts the positions of the roton minima that are in good agreement with the microscopic CF theory.…”

“…Recently, Golkar et al 30 treated the neutral excitations as quantized shape deformations of the composite fermion Fermi surface at 1/2. From this approach, they predict that the magnetoroton minima at ν = s/(2s + 1) are located at:…”

The hierarchy of excitation lifetimes in two-dimensional Fermi gases

“…This approach interprets the low-energy excitations of the Fermi liquid as fluctuations of the shape of the Fermi surface, treating these fluctuations as bosonic fields. Recently, these ideas were successfully applied to the problem of the Fractional Quantum Hall effect 4,5 and to Fermi liquids with spin-orbit coupling 6 . Here we show that similar ideas help to gain new insight into the long-standing problem of quasiparticle lifetimes and angular relaxation in 2D Fermi liquids.…”

Acoustic wave absorption as a probe of dynamical geometrical response of fractional quantum Hall liquids

“…What is more interesting, however, is the longwavelength mode with k → 0, which is the graviton mode that is of primary interest of the present paper [32]. It is known [12,29,34] …”

“…This demonstrates the observability of this internal geometry. It has also been argued [1,[10][11][12] that this internal metric may be viewed as a dynamical degree of freedom, whose long-wave length dynamics corresponds to the collective excitations of the system that can be viewed as "gravitons" [13]. In a parallel stream of works, much effort has been devoted to studying FQH liquids in a curved background space [14][15][16][17][18][19][20][21][22][23], following earlier seminal work by Wen and Zee [24].…”