Entangled Pauli principles: The DNA of quantum Hall fluids

Abstract

A formalism is developed for the rigorous study of solvable fractional quantum Hall parent Hamiltonians with Landau level mixing. The idea of organization through "generalized Pauli principles" is expanded to allow for root level entanglement, giving rise to "entangled Pauli principles". Through the latter, aspects of the effective field theory description become ingrained in exact microscopic solutions for a great wealth of phases for which no similar single Landau level description is known. We discuss in detail braiding statistic, edge theory, and rigorous zero-mode counting for the Jain-221 state as derived from a microscopic Hamiltonian. The relevant root-level entanglement is found to feature an AKLT-type MPS structure associated with an emergent SU(2)-symmetry.

Fractional quantum Hall effect at ν=2+4/9

“…The Jain states, described by the wave function Ψ Jain ν=n/(2pn±1) = P LLL Φ ±n Φ 2p 1 , can be re-interpreted as (2p + 1)-parton states where 2p partons form a ν = 1 IQHE state and one parton forms a ν = ±n IQHE state. Numerous parton states, beyond the Laughlin and Jain states, have been proposed as promising candidates to describe FQHE states occurring in the SΛL [28], second LL (SLL) [30][31][32][33][34], wide quantum wells [35], and in the LLs of graphene [24,[36][37][38][39][40]. These works suggest that it is plausible that viable candidate parton states can be constructed to capture all the observed FQHE states [24,32].…”

Composite fermions in Fock space: Operator algebra, recursion relations, and order parameters

“…Using the methods of Ref. 27, which we later characterized as making use of an "entangled Pauli principle"(EPP) [28], we can also establish that these are the densest possible (highest filling factor) zero modes (see Ref. 28 for details).…”

“…31 on the basis of variational wave functions, and which moreover can be seen to be a property of the zero mode spaces associated to all composite fermion states, given appropriate parent Hamiltonians [35]. (This is quite a robust property of n > 1 special Hamiltonians, and generalizes even to more complicated "parton" states [28].) It is thus natural to define O k,r =c * k,r S k,k M N −r−δ+Lmax(N +1,n) R N,n,k .…”

“…The utility of this mixed first/second-quantized formalism was further demonstrated in Ref. 28, where the notion of an "entangled Pauli principle" was introduced, and much of the low-energy physics of the non-Abelian, mixed-Landaulevel Jain 221 state [5,16,29] was linked to properties of its microscopic two-body parent Hamiltonian.…”

“…In this case, some of us have recently shown [24][25][26] (for the simplest, Laughlin state parent Hamiltonians) that the rigorous characterization of the zero mode space may be done in a purely second-quantized framework that does not reference the analytic polynomial wave functions of the traditional approach at all. These developments were very useful for the study of the case n > 1, which we argued [27,28] is absolutely necessary to consider if we wish to establish a zero mode paradigm for a more representative set of FQH states. Indeed, already for the phases described by Jain states, which are elemental to both theory and experiment, we argued that a zero mode paradigm requires n > 1, with the n = 2 case extensively discussed in Ref.…”