Emergent commensurability from Hilbert space truncation in fractional quantum Hall fluids

Abstract

We show that model states of fractional quantum Hall fluids at all experimentally detected plateaus can be uniquely determined by imposing translational invariance with a particular scheme of Hilbert space truncation. The truncation is based on classical local exclusion conditions, motivated by constraints on physical measurements. The scheme allows us to identify filling factors, topological shifts and clustering of topological quantum fluids universally without resorting to microscopic Hamiltonians. This prompts us to propose the notion of emergent commensurability as a fundamental property for many known FQH states, which allows us to predict families of new FQH state that can be realised in principle. We also discuss the implications of certain missing states proposed from other phenomenological approaches, and suggest that the physics of FQH effect could fundamentally arise from the algebraic structure of the Hilbert space in a single Landau level.A large number of fractional quantum Hall (FQH) states with distinct topological orders have been observed experimentally and proposed theoretically, ever since the surprising discovery of the quantised Hall conductivity at 1/3 filling factor [1, 2]. The physics of the FQH effect is mainly derived from the formation of an incompressible quantum fluid with a charge excitation gap, which could be realised at specific rational filling factors when a two-dimensional electron gas system is subject to a perpendicular magnetic field. We now understand that both Abelian and non-Abelian FQH states are likely observed in the experiments [1, 3]. In addition to the single component FQH states (e.g. the Read-Rezayi series [4,5]), there can also be multi-component or hierarchical states from the coexistence of more than one type of quantum fluids in a strongly correlated manner [6,7].There has been much development in the microscopic theories of the FQH effect since the first proposition of the Laughlin wavefunctions[2] and later on, the model Hamiltonians [8]. One major approach is the phenomenological formation of "composite fermions" (CF) with flux attachment [7], and the parton construction inspired from it [9,10]. It leads to the systematic construction of microscopic wavefunctions for almost all observed and proposed FQH states [11]. Another major approach is to exploit the rich algebraic structures of many-body wavefunctions in a single Landau level (LL) on genus 0 geometries (e.g. sphere or disk), leading to very efficient constructions of microscopic model wavefunctions with the Jack polynomial formalism [1,12,14,15]. The method is particularly useful for the Read-Rezayi (RR) series including the coveted non-Abelian states, revealing the particle clustering properties in an intuitive manner. The Jack polynomial formalism and the related techniques are also closely linked to the wavefunction constructions from parafermion correlators in conformal field theory (CFT) [4,5], and in contrast to the CF approach, in many cases model projection Hamiltonians can be found [16], o...

Microscopic theory for nematic fractional quantum Hall effect

“…). These did not seem to be recognized before in the literature, but are easy to see from the recently developed LEC formalism [40] (i.e., the L = 2 state satisfies the LEC condition {2, 1, 2} ∨ {6, 2, 6}, while the L > 2 state satisfies the LEC condition {2, 1, 2} ∨ {5, 2, 5}), using the root configurations in Eq. (28) and the associated squeezed basis.…”

Gaffnian and Haffnian: Physical relevance of nonunitary conformal field theory for the incompressible fractional quantum Hall effect

“…The easiest way to see that the M = −2 state is a zero-energy state of the Haffnian model Hamiltonian is that it satisfies the local exclusion condition (LEC) [48,49] of {4, 2, 4} at the center of the disk (corresponding to the north pole of the sphere). Similarly, we know the M < −2 states are the zero-energy states of the Gaffnian model Hamiltonian because they satisfy the LEC of {2, 1, 2} ∨ {5, 2, 5}.…”

Fractional Quantum Hall Effect from Frustration-Free Hamiltonians

“…The construction of the local projections is reminiscent of the classical local exclusion conditions proposed in Ref. [27], but the lattice of such projections is a welldefined quantum Hamiltonian in the continuum that can, in principle, be realized experimentally. There are interesting analogies between this new form of FQH Hamiltonians and the spin Hamiltonian for the AKLT model.…”

“…The projection Hamiltonians.-It has been numerically established [27] that the model ground states of many FQH phases can be uniquely determined by two constraints: translational or rotational invariance and the classical reduced density matrix constraint. The latter is denoted with a triplet of non-negative integers ĉ ¼ fn; n e ; n h g. Physically, it dictates for any small droplet containing n fluxes (thus, with an area of 2nπl 2 B , l B being the magnetic length), a measurement in this droplet can never detect more than n e number of electrons or n h number of holes [unoccupied orbitals in a single Landau Level (LL)].…”

“…This is because a coherent state can be a droplet containing any number of fluxes, with different arrangements of electrons in the droplet, in analogy to the local exclusion conditions (LEC) constraints introduced in Ref. [27].…”