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

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.

Figure 1

Effective $S=1$ spin chains for the $\nu =p/(2p+1)$ FQH states. There are $p$ spins in each unit cell.

Figure 2

Energy gaps of the original FQH systems (a), and those of the corresponding effective spin chains (b). $L$ is the system size which describes the number of electrons in FQH systems ($L/\nu =15\sim 27,10\sim 25,14\sim 28$ for $\nu =1/3,2/5,3/7$) and the number of spins in effective spin chains ($L=8\sim 12,6\sim 14,6\sim 15$).

Figure 3

Correlation functions of the effective spin chains with (a) and without (b) static terms obtained by iTEBD, while (a${}^{\prime }$) and (b${}^{\prime }$) are their logarithms. $S^z\left(n\right)$ is the total spin of the $n$th unit cell $S^z\left(n\right)={\sum }_{j=np+1}^{(n+1)p}{\widehat{S}}_{j,n}^z$. $r$ denotes the distance between two total spins where the lengths of unit cells are set to be unity.