Spin-chain description of fractional quantum Hall states in the Jain series

TL;DR

This paper establishes a connection between fractional quantum Hall states in the Jain series and quantum spin chains, demonstrating how FQH states can be mapped to effective spin chains and analyzing their energy gaps and correlations.

Contribution

It introduces a novel mapping of FQH states at specific filling factors to effective quantum spin chains, providing insights into their energy gaps and correlation functions.

Findings

Mass gaps decrease as p increases, similar to S=p integer Heisenberg chains.

Effective spin chains accurately reproduce FQH state properties.

The results connect FQH hierarchy with the Haldane conjecture.

Abstract

We discuss the relationship between fractional quantum Hall (FQH) states at filling factor $\nu=p/(2p+1)$ and quantum spin chains. This series corresponds to the Jain series $\nu=p/(2mp+1)$ with $m=1$ where the composite fermion picture is realized. We show that the FQH states with toroidal boundary conditions beyond the thin-torus limit can be mapped to effective quantum spin S=1 chains with $p$ spins in each unit cell. We calculate energy gaps and the correlation functions for both the FQH systems and the corresponding effective spin chains, using exact diagonalization and the infinite time-evolving block decimation (iTEBD) algorithm. We confirm that the mass gaps of these effective spin chains are decreased as $p$ is increased which is similar to $S=p$ integer Heisenberg chains. These results shed new light on a link between the hierarchy of FQH states and the Haldane conjecture for quantum spin chains.