Geometric fluctuation of conformal Hilbert spaces and multiple graviton modes in fractional quantum Hall effect

Article https://doi.org/10.1038/s41467-023-38036-0 Geometric fluctuation of conformal Hilbert spaces and multiple graviton modes in fractional quantum Hall effect Received: 10 August 2022 Wang Yuzhu 1 & Yang Bo 1,2 Check for updates 1234567890():,; 1234567890():,; Accepted: 6 April 2023 Neutral excitations in fractional quantum Hall (FQH) fluids define the incompressibility of topological phases, a species of which can show graviton-like behaviors and are thus called the graviton modes (GMs). Here, we develop the microscopic theory for multiple GMs in FQH fluids and show explicitly that they are associated with the geometric fluctuation of well-defined conformal Hilbert spaces (CHSs), which are hierarchical subspaces within a single Landau level, each with emergent conformal symmetry and continuously parameterized by a unimodular metric. This leads to several statements about the number and the merging/splitting of GMs, which are verified numerically with both model and realistic interactions. We also discuss how the microscopic theory can serve as the basis for the additional Haldane modes in the effective field theory description and their experimental relevance to realistic electronelectron interactions. Our universe has two important fundamental constants: the speed of light c, which parametrizes the Lorentz invariance (from the theory of relativity), and Planck’s constant ℏ, which parametrizes the quantum fluctuation. It turns out that in a two-dimensional “universe" realized by the quantum Hall effect, we also have two analogous “fundamental constants”: The Fermi velocity of the chiral Luttinger liquid of the edge transport is analogous to c, while the magnetic length is analogous to ℏ. Furthermore, in such a microscopic universe, these two parameters can be tuned experimentally1. This leads to rich physics from the interplay of geometry, topology, and emergent symmetry due to strong interactions2, which can even induce the emergence of the quasiparticles analogous to those theoretically proposed at high energy but have yet been observed in nature. One intriguing example is the gravitons, which are the hypothetical spin-2 bosons from the quantization of gravitational field3,4. There also exist the analogous graviton modes (GMs) in a fractional quantum Hall (FQH) fluid5–7 is a two-dimensional quantum fluid of electrons subject to a strong magnetic field at low temperatures. These modes are the quadrupole gapped excitations of the quantum Hall effect that emerge from the geometric fluctuation of the topological ground state, encoding topological information about their respective FQH phases. Their dynamics leads to rich physics ranging from ground state incompressibility to the dynamical phase transitions of the low-lying excitations8,9. The effective field theory studying these modes has been proposed using the Newton-Cartan formalism, and various experimental proposals for the observation of these modes have been put forward10–19. The standard technique in probing neutral excitations is to use the inelastic photon scattering20–24. The coupling between the GMs and the acoustic waves can also be used to simulate the behavior of gravitons interacting with the gravitational waves25. Meanwhile, the microscopic theory of the GMs with model Hamiltonians has also been established to provide insights for the experiments, where model Hamiltonians for the GMs have been constructed for FQH fluids at different filling factors9,26. Recently numerical results have implied the signature of multiple GMs in FQH states, the microscopic understanding of which is under development27,28. Given that most of the research on GMs is based on effective field theories and numerical analysis with model wavefunctions28, a detailed microscopic theory is needed for a complete characterization of the emergence and interaction between different GMs. 1 School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 639798, Singapore. 2Institute of High Performance Computing, e-mail: A*STAR, Singapore 138632, Singapore. Nature Communications | (2023)14:2317 1 Article https://doi.org/10.1038/s41467-023-38036-0 In this work, we show analytically that multiple GMs are a generic feature of FQH fluids, from the splitting of the long-wavelength limit of the Girvin-MacDonald-Platzman (GMP) mode5 in different subspaces in a single Landau level (LL). Using the analytic tools we developed earlier9, we demonstrate that the number of observable GMs is dynamical in nature and is only meaningful when referring to specific interaction Hamiltonians. Each GM can be interpreted as the metric fluctuation of a conformal Hilbert space (or the null spaces of model Hamiltonians, as explained later) within a single LL. For short-range two-body interactions, we show all non-Laughlin FQH states around the filling factor ν = 1/(2n) with n > 1, including the interacting composite fermion (CF) states, have at least two GMs. In particular, the Jain states at ν = N=ð2nN ± 1Þ and the Pfaffian states at ν = 1/(2n) all have two GMs if n, N > 1. The Laughlin states (N = 1) and the Jain states with n = 1 all have a single GM. This agrees with the special cases studied numerically in both refs. 27,28 at ν = 2/7, 2/9, 1/4, at the same time providing an analytic explanation and geometric interpretation to their numerical observations. Furthermore, the microscopic theory can easily predict the chirality of the gravitons19 without numerical computations. This work is organized as follows: First we conceptually introduce the geometrical origin of the GMs and the hierarchical structure of the conformal Hilbert spaces (CHSs) as the null spaces of model Hamiltonians, the combination of which leads to the microscopic explanation of the emergence of multiple GMs, with the spectral function calculated from the single-mode approximation wave function to distinguish these GMs in the Hilbert space. Then we focus on the GMs in the CHSs defined by short-range two-body interactions and nonAbelian three-body interactions, where both analytical and numerical evidence shows the signature of multiple GMs; This adds yet another tool for the experimental probing of topological orders in low-temperature, two-dimensional electronic systems where the Coulomb interaction can be slightly tuned and how such orders are affected by the conformal symmetry that may or may not be fully realized in experiments; How our theory serves as the basis for the effective field theory is explained with more technical details, in the discussion section, where we show the necessity of additional Haldane modes depends on the proper identification of the base space of such theories. Table 1 | Definition of various symbols used in the text ^a R i Guiding center operator ^q ρ Guiding center density operator q δρ Regularized guiding center density operator ^ ia π Dynamical momentum operator  ab g Guiding center metric ~ ab g Cyclotron metric gab α (...truncated)