Fermionic fractional quantum Hall states: A modern approach to systems with bulk-edge correspondence

TL;DR

The paper develops a systematic method to construct cylinder partition functions for fermionic fractional quantum Hall states from bulk gapless excitations, unifying edge-mode data with bulk topological order through a fermionic CFT framework. By introducing a $Z_2$ simple current and fermionic parity, it derives twisted/untwisted sectors and analyzes modular properties, including a fermionic T duality that links edge and bulk descriptions. A modular-invariant sector is obtained via an imaginary gapping that yields a product of NS fermionic CFT content with Laughlin/CFT sectors, providing RG-oriented insight into edge protection. The Moore–Read Pfaffian state is treated as a concrete example, where the Pfaffian fusion structure and $Z_2$ duality illuminate edge-mode organization and potential dualities. The work connects BCFT/Open-Closed duality, Schottky doubling, and bulk RG flow to offer a broad, RG-informed platform for understanding topologically ordered phases across geometries and dimensions.

Abstract

In contemporary physics, especially in condensed matter physics, fermionic topological order and its protected edge modes are one of the most important objects. In this work, we propose a systematic construction of the cylinder partition corresponding to the fermionic fractional quantum Hall effect (FQHE) and a general mechanism for obtaining the candidates of the protected edge modes. In our construction, when the underlying conformal field theory has the $Z_{2}$ duality defects corresponding to the fermionic $Z_{2}$ electric particle, we show that the FQH partition function has a fermionic T duality. This duality is analogous to (hopefully the same as) the dualities in the dual resonance models, typically known as supersymmetry, and gives a renormalization group (RG) theoretic understanding of the topological phases. We also introduce a modern understanding of bulk topological degeneracies and topological entanglement entropy. This understanding is based on the traditional tunnel problem and the recent conjecture of correspondence between the bulk renormalization group flow and the boundary conformal field theory. Our formalism gives an intuitive and general understanding of the modern physics of the topologically ordered systems in the traditional language of RG and fermionization.

Figures (9)

• Figure 1: Bulk and boundary RG picture of the bulk-edge correspondence. As we discuss in the subsequent sections, we expect the edge modes in $CFT_{D}/BTQFT_{D+1}$ as a (emergent) renormalizable theory.
• Figure 2: Connectivity diagram of bosonic and fermionic TO to FQHE. The bosonic TO ($M$) and fermionic TO ($FM$) are related by the Jordan-Wigner transformation. It should be worth noting that this transformation acts nonlocally on the Hilbert space. By this transformation, the bosonic system, such as a spin system, is transformed into the fermionic particle system. By attaching $U(1)$ flux corresponding to this particle system, one can obtain FQHE. In this sense, FQHE should be interpreted as a kind of gauge theory.
• Figure 3: Duality interpretation of the partition functions. One can see the mapping of Hilbert space by adding the quasiparticle corresponding to the duality of the bosonic CFT.
• Figure 4: Mapping of tunnel problem. Based on the cut and gluing operation, one can interpret the ground state of FQHE as the ground states of interacting edges, corresponding to the full CFT with relevant perturbations.
• Figure 5: QFT/BCFT correspondence. By using the correspondence proposed in Cardy:2015xaa, one can map the low energy physics of interacting edges to the physics of (smeared) BCFT, typically Cardy states.