Search for exact local Hamiltonians for general fractional quantum Hall states
Search for exact local Hamiltonians for general fractional quantum Hall states
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搜索一般分数量子霍尔态的精确局部哈密顿量
作者：
S. Jaya;M. Fremling;G. Jeon;J. Jain
我们报告了我们系统地寻找局域相互作用的尝试，对于这些局域相互作用，最低朗道能级的投影复合费米子波函数是唯一的零能基态。为此，我们详细地研究了Laughlin态以外最简单的非平凡系统，即填充的玻色子
U=Frc{2}{3}$并识别处于基本状态的粒子簇之间的局部约束。通过显式计算，我们证明了直到(包括)四个粒子相互作用的哈密顿量都不会产生与基态完全相同的哈密顿量，并推测即使当包含更多粒子的相互作用项时，这仍然是正确的。令人惊讶的是，我们可以识别出一种相互作用，它对四个相对角动量为6hbar$的粒子的特定纠缠配置施加了能量惩罚，从而产生了唯一的零能量解(正如我们已经证实的那样，多达12个粒子)。这种状态，被称为$lambda$-态，与投影的复合费米子态不同，但下列事实表明，两者可能在拓扑上是等价的：两个态有很高的重叠；它们具有相同的根分区；它们中性激发的量子数相同；准粒子激发的量子数也匹配。在准空穴方面，我们发现，即使最低能态的量子数与复合费米子理论的预测一致，这些态与其他态之间并没有明显可识别的间隙。这阻止了我们就$lambda$态和复合费米子态的拓扑等价性做出决定性的断言。我们的研究说明了如何从约束选定的许多粒子构型中识别新的候选态，并对其进行拓扑分类将是有趣的。
We report on our systematic attempts at finding local interactions for which the lowest-Landau-level projected composite-fermion wave functions are the unique zero energy ground states. For this purpose, we study in detail the simplest non-trivial system beyond the Laughlin states, namely bosons at filling $
u=frac{2}{3}$ and identify local constraints among clusters of particles in the ground state. By explicit calculation, we show that no Hamiltonian up to (and including) four particle interactions produces this state as the exact ground state, and speculate that this remains true even when interaction terms involving greater number of particles are included. Surprisingly, we can identify an interaction, which imposes an energetic penalty for a specific entangled configuration of four particles with relative angular momentum of $6hbar$, that produces a unique zero energy solution (as we have confirmed for up to 12 particles). This state, referred to as the $lambda$-state, is not identical to the projected composite-fermion state, but the following facts suggest that the two might be topologically equivalent: the two sates have a high overlap; they have the same root partition; the quantum numbers for their neutral excitations are identical; and the quantum numbers for the quasiparticle excitations also match. On the quasihole side, we find that even though the quantum numbers of the lowest energy states agree with the prediction from the composite-fermion theory, these states are not separated from the others by a clearly identifiable gap. This prevents us from making a conclusive claim regarding the topological equivalence of the $lambda$ state and the composite-fermion state. Our study illustrates how new candidate states can be identified from constraining selected many particle configurations and it would be interesting to pursue their topological classification.