量子霍尔效应的超普适性与$\vartheta$角的“大$N$图”
Super universality of the quantum Hall effect and the "large $N$ picture" of the $\vartheta$ angle
摘要：
结果表明，“无质量手征边缘激发”是一个完整的和普遍的方面的低能量动力学的$\vartheta$真空，历史上已经被忽视。在$SU(M+N)/S(U(M) \times U(N))$非线性西格玛模型中，我们引入了一个有效的“边缘激发”理论，从根本上解释了量子霍尔效应。在形成鲜明对比的共同信念在该领域中，我们的研究结果表明，这种宏观量子化现象，事实上，一个{ \em超通用}强耦合功能的$\vartheta$角与副本限制$M=N=0$只发挥了次要作用。为了证明超通用性，我们重新审视了$CP^{N-1}$模型的大型$N$扩展。我们首次得到了包括量子霍尔平台跃迁的量子临界性在内的量子霍尔效应的显式标度结果。从而得到描述大$N$理论的弱耦合“瞬子相”和强耦合“量子霍尔相”之间交叉的标度图。我们的结果与$\vartheta$角的“瞬子图”一致，但从根本上否定了基于历史“大$N$图”的所有想法、期望和假设。"
英文摘要：
It is shown that the "massless chiral edge excitations" are an integral and universal aspect of the low energy dynamics of the $\vartheta$ vacuum that has historically gone unnoticed. Within the $SU(M+N)/S(U(M) \times U(N))$ non-linear sigma model we introduce an effective theory of "edge excitations" that fundamentally explains the quantum Hall effect. In sharp contrast to the common beliefs in the field our results indicate that this macroscopic quantization phenomenon is, in fact, a {\em super universal} strong coupling feature of the $\vartheta$ angle with the replica limit $M=N=0$ only playing a role of secondary importance. To demonstrate super universality we revisit the large $N$ expansion of the $CP^{N-1}$ model. We obtain, for the first time, explicit scaling results for the quantum Hall effect including quantum criticality of the quantum Hall plateau transition. Consequently a scaling diagram is obtained describing the cross-over between the weak coupling "instanton phase" and the strong coupling "quantum Hall phase" of the large $N$ theory. Our results are in accordance with the "instanton picture" of the $\vartheta$ angle but fundamentally invalidate all the ideas, expectations and conjectures that are based on the historical "large $N$ picture."
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