Exactly Solvable Fermion Chain Describing a $\nu =1/3$ Fractional Quantum Hall State

We introduce an exactly solvable fermion chain that describes a $\nu =1/3$ fractional quantum Hall (FQH) state beyond the thin-torus limit. The ground state of our model is shown to be unique for each center-of-mass sector, and it has a matrix product representation that enables us to exactly calculate order parameters, correlation functions, and entanglement spectra. The ground state of our model shows striking similarities with the BCS wave functions and quantum spin-1 chains. Using the variational method with matrix product ansatz, we analytically calculate excitation gaps and vanishing of the compressibility expected in the FQH state. We also show that the above results can be related to a $\nu =1/2$ bosonic FQH state.

Figure 1

(a) Density functions $⟨{\widehat{n}}_i⟩$ in three-sites unit cell, and (b) String order parameter $O_{\mathrm{string}}^z$ and dual string order parameter ${\overline{O}}_{\mathrm{string}}^z$ as functions of $t$. $O_{\mathrm{string}}^z$ (${\overline{O}}_{\mathrm{string}}^z$) dominant in the large (small) $t$ regime plays a role to characterize the “superconducting” (“normal”) component in an analogy of the BCS theory.

Figure 2

Entanglement spectra $\{{\xi }_i\}$, and entanglement entropy $S_A$, as functions of $t$ for (a) infinite-size system and (b) finite-size ($N=2L=32$) system.

Figure 3

The lowest excitation gaps of the model (3) at $\alpha =1$, $t=1/3$. The neutral gaps for $\Delta K=1$, 2 are obtained by exact diagonalization (ED) for $N_s=27$ (dots) and the variational calculation with the MP ansatz for $N_s\to \infty$ (lines). $E(N_s-1)$ for $N_s\to \infty$ is also obtained by the MP ansatz. Here, ${\beta }^2=0.77$ corresponds to a FQH system on a torus with circumference $L_1=5.99$.