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Exact Operator Fractionalization and Edge Physics in the Lowest-Landau-Level

Arkadiusz Bochniak, Zohar Nussinov, Alexander Seidel, Gerardo Ortíz

Published Oct 12, 2021• 0 citations
OpenAlex

Abstract

Advancing a microscopic field-theoretic framework that rigorously unveils the underlying topological hallmarks of fractional quantum Hall fluids is a prerequisite for making progress in the classification of strongly-coupled topological matter. Here we expand on such a program and, in particular, address the problems of Laughlin's quasiparticle(hole) operator fractionalization and condensation of non-local operators characterizing topological order in the lowest-Landau-level (LLL). We express the sequence of (both bosonic and fermionic) Laughlin second-quantized states in a manner that highlights the lack of local condensation. In addition, we present a rigorous constructive subspace bosonization dictionary for the bulk fluid and establish the universal long-distance behavior of edge excitations by formulating a conjecture complementary to Wen's.

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Advancing a microscopic field-theoretic framework that rigorously unveils the underlying topological hallmarks of fractional quantum Hall fluids is a prerequisite for making progress in the classification of strongly-coupled topological matter. Here we expand on such a program and, in particular, address the problems of Laughlin's quasiparticle(hole) operator fractionalization and condensation of non-local operators characterizing topological order in the lowest-Landau-level (LLL). We express the sequence of (both bosonic and fermionic) Laughlin second-quantized states in a manner that highlights the lack of local condensation. In addition, we present a rigorous constructive subspace bosonization dictionary for the bulk fluid and establish the universal long-distance behavior of edge excitations by formulating a conjecture complementary to Wen's.
Abstract Advancing a microscopic framework that rigorously unveils the underlying topological hallmarks of fractional quantum Hall (FQH) fluids is a prerequisite for making progress in the classification of strongly-coupled topological matter. We present a second-quantization framework that reveals an exact fusion mechanism for particle fractionalization in FQH fluids, and uncovers the fundamental structure behind the condensation of non-local operators characterizing topological order in the lowest-Landau-level. We show the first exact analytic computation of the quasielectron Berry connections leading to its fractional charge and exchange statistics, and perform Monte Carlo simulations that numerically confirm the fusion mechanism for quasiparticles. We express the sequence of (bosonic and fermionic) Laughlin second-quantized states, highlighting the lack of local condensation, and present a rigorous constructive subspace bosonization dictionary for the bulk fluid. Finally, we establish universal long-distance behavior of edge excitations by formulating a conjecture based on the DNA, or root state, of the FQH fluid.
Motivated by the structure of the quasiparticle wavefunction in the composite fermion (CF) theory for fractional quantum Hall filling factor [Formula: see text] (m odd), I consider a suitable quasiparticle operator in differential form, as a modified form of Laughlin's quasiparticle operator, that reproduces quasiparticle wave function as predicted in the CF theory, without a priori assumption of the presence of CFs. I further consider the conjugate of this operator as quasihole operator for obtaining a novel quasihole wave function for 1/m state. Each of these wave functions is interpreted as expelling an electron into a different Hilbert subspace from the original Hilbert space of the Laughlin condensate while still maintaining its correlation (although changed) with the electrons in the condensate such that the expelled electron behaves as a CF with respect to the electrons in the condensate. With this interpretation, I show that the ground state wavefunctions for general states at filling fractions [Formula: see text], respectively, can be constructed as coherent superposition of n coupled Laughlin condensates and their 'conjugates', formed at different Hilbert subspaces. The corresponding wave functions, specially surprising for [Formula: see text] sequence of states, are identical with those proposed in the theory of noninteracting CFs. The states which were considered as fractional quantum Hall effect of interacting CFs, can also be treated in the same footing as for the prominent sequences of states describing as the coupled condensates among which one is a non-Laughlin condensate in a different Hilbert subspace. Further, I predict that the half filling of the lowest Landau level is a quantum critical point for phase transition between two topologically distinct phases each corresponding to a family of states: one consists of large number of coupled Laughlin condensates of filling factor 1/3 and the other corresponds to large number of coupled conjugate Laughlin condensates of filling factor 1, which may be distinguished, respectively, by the absence and presence of upstream edge modes.
I propose quasiparticle and quasihole operators which operating on the Laughlin wave functions describing the Laughlin condensates (LCs) at the filling factors ν=1/m\nu =1/mν=1/m in a specific Hilbert-subspace, {\em generate} composite fermions (CFs) by expelling electrons from the condensate to a different Hilbert-subspace. The condensation of these expelled electrons into ν=1/m\nu =1/mν=1/m together with the original LC, form a new condensate at ν=2/[2(m−1)+1]\nu = 2/[2(m- 1) + 1]ν=2/[2(m−1)+1]. In general, hierarchically constructed states are coupled LCs formed at different Hilbert-subspaces and the corresponding wave functions are {\em identical} with those proposed in the CF theory. This theory further predicts that the half and the quarter filled lowest Landau level are quantum critical points for topological phase transitions.
Starting from Halperin multilayer systems we develop a hierarchical scheme, dubbed symmetrized multicluster construction, that generates bosonic and fermionic single-layer quantum Hall states (or vacua) of arbitrary filling factor. Our scheme allows for the insertion of quasiparticle excitations with either Abelian or non-Abelian statistics and quantum numbers that depend on the nature of the original vacuum. Most importantly, it reveals a fusion mechanism for quasielectrons and magnetoexcitons that generalizes ideas about particle fractionalization introduced in A. Bochniak, Z. Nussinov, A. Seidel, and G. Ortiz, Commun. Phys. 5, 171 (2022) for the case of Laughlin fluids. In addition, in the second quantization representation, we uncover the inherent topological quantum order (or the off-diagonal long-range order) characterizing these vacua. In particular, we illustrate the methodology by constructing generalized composite (generalized Read) operators for the non-Abelian Pfaffian and Hafnian quantum fluid states.
Establishing the fusion rules of anyonic quasiparticles in fractional quantum Hall fluids is essential for understanding their underlying topological order. Building on the conjecture that key topological properties are encoded in the ''DNA'' of candidate many-body wave functions - that is, the pattern of dominant orbital occupations restricted to a finite number of lowest Landau levels - we propose a combinatorial framework that derives these fusion rules directly from microscopic data. By extending Schrieffer's counting argument and introducing classes of topological excitations, our framework provides a unified route to the fusion rules for both Abelian and non-Abelian excitations. This approach elucidates the emergence of topological features from first principles in both fermionic and bosonic systems. In particular, we unveil the previously unknown fusion rules of the Hafnian fluid.
We define a measurable spin for the edge of a lowest Landau level and incompressible fractional quantum Hall state in the presence of an Abelian or non-Abelian bulk quasiparticle. We show that this quantity takes a fractional value inherited from the fractional spin of the bulk quasiparticle. We present a geometric picture that does not rely on global symmetries of the wave function, but is able to treat quasiparticles and edges with different shapes. We study finite-size many-body wave functions on the cylinder with circular quasiparticles and straight edges. Our results are supported by matrix-product-state calculations for the Laughlin and the k = 3 Read-Rezayi states.