Jain体系态的拓扑纠缠与聚簇
Topological Entanglement and Clustering of Jain Hierarchy States
摘要：
我们得到的聚簇性质和部分结构的Jain状态的零填充$\frac{k}{2k+1}$：它们是一个直接的产品的范德蒙行列式（必须存在的任何费米子状态）和玻色子多项式填充$\frac{k}{k+1}$时消失的$k+1$粒子簇在一起。我们发现，所有的Jain状态满足“压缩规则”（他们是“压缩多项式”），这大大减少了必要的希尔伯特空间的维数来生成它们。压缩规则也证明了这些态满足的成簇条件。计算了Jain $ν={2/5}$态的拓扑纠缠谱，并与库仑基态和非幺正Gaffnian态进行了比较.这三种状态都有非常相似的“低能”结构。然而，Jain态纠缠“边缘”态计数与库仑计数以及两个解耦的U（1）自由玻色子相匹配，而Gaffnian边缘计数错过了库仑谱的一些“边缘”态。谱分解以及边缘结构证明了Jain态与$ν={2/5}$的库仑哈密顿量的基态是普适等价的。这些证据比通常的重叠研究强得多，后者无法有意义地区分耆那教和加夫尼教国家。我们计算的纠缠间隙，并提出证据表明，它仍然是恒定的热力学极限。我们还分析了纠缠间隙和重叠的依赖性，因为我们通过相变驱动复合费米子系统。
英文摘要：
We obtain the clustering properties and part of the structure of zeroes of the Jain states at filling $\frac{k}{2k+1}$: they are a direct product of a Vandermonde determinant (which has to exist for any fermionic state) and a bosonic polynomial at filling $\frac{k}{k+1}$ which vanishes when $k+1$ particles cluster together. We show that all Jain states satisfy a "squeezing rule" (they are "squeezed polynomials") which severely reduces the dimension of the Hilbert space necessary to generate them. The squeezing rule also proves the clustering conditions that these states satisfy. We compute the topological entanglement spectrum of the Jain $ν={2/5}$ state and compare it to both the Coulomb ground-state and the non-unitary Gaffnian state. All three states have very similar "low energy" structure. However, the Jain state entanglement "edge" state counting matches both the Coulomb counting as well as two decoupled U(1) free bosons, whereas the Gaffnian edge counting misses some of the "edge" states of the Coulomb spectrum. The spectral decomposition as well as the edge structure is evidence that the Jain state is universally equivalent to the ground state of the Coulomb Hamiltonian at $ν={2/5}$. The evidence is much stronger than usual overlap studies which cannot meaningfully differentiate between the Jain and Gaffnian states. We compute the entanglement gap and present evidence that it remains constant in the thermodynamic limit. We also analyze the dependence of the entanglement gap and overlap as we drive the composite fermion system through a phase transition.
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