Hofstadter's Cocoon
Hofstadter's Cocoon
摘要：
霍夫施塔特表明，晶格上的电子能级作为磁场的函数，形成了一个美丽的结构，现在被称为“霍夫施塔特的蝴蝶”。我们研究了Hofstadter模型的非厄米延拓;随着非厄米参数$g$的增加超过一个临界值序列，特征值连续地在一个“双叉分叉”序列中变得复杂，其中对实特征值退化，然后成为复共轭对。相关的波函数进行自发对称性破缺过渡，我们阐明。在跃迁之外，本征值的实部对磁场的曲线类似于霍夫施塔特蝴蝶;虚部对磁场的曲线形成了一个复杂的结构，我们称之为霍夫施塔特茧。描述了茧的对称性。波多野和纳尔逊研究了安德森局部化模型的非厄米延拓，它与这里研究的模型有密切的相似之处。我们的工作的关系，波多野和纳尔逊和PT量子力学研究的PT跃迁进行了讨论。
英文摘要：
Hofstadter showed that the energy levels of electrons on a lattice plotted as a function of magnetic field form an beautiful structure now referred to as "Hofstadter's butterfly". We study a non-Hermitian continuation of Hofstadter's model; as the non-Hermiticity parameter $g$ increases past a sequence of critical values the eigenvalues successively go complex in a sequence of "double-pitchfork bifurcations" wherein pairs of real eigenvalues degenerate and then become complex conjugate pairs. The associated wavefunctions undergo a spontaneous symmetry breaking transition that we elucidate. Beyond the transition a plot of the real parts of the eigenvalues against magnetic field resembles the Hofstadter butterfly; a plot of the imaginary parts plotted against magnetic fields forms an intricate structure that we call the Hofstadter cocoon. The symmetries of the cocoon are described. Hatano and Nelson have studied a non-Hermitian continuation of the Anderson model of localization that has close parallels to the model studied here. The relationship of our work to that of Hatano and Nelson and to PT transitions studied in PT quantum mechanics is discussed.
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