【作者】 曹阳；

【导师】 杨孝森；

【作者基本信息】 江苏大学 ， 电子科学与技术， 2021， 硕士

【摘要】 体边对应关系一直是我们研究拓扑物态、拓扑相的一个重要基石,一般来说,如果体态是拓扑非平庸的,那么在开边界条件下就会出现拓扑边界态,也就是说由定义在布里渊区上的布洛赫哈密顿量得到的拓扑不变量,可以准确预测开边界条件下拓扑边界态的位置。近年来,随着非厄米量子系统的不断研究,越来越多的人意识到传统的体边对应关系在非厄米体系中已经不再适用。非厄米趋肤效应的发现,以及开边界条件下的能谱和周期边界条件下的能谱存在巨大差异,这些现象无法用布洛赫能带理论很好的解释。进一步研究发现,在这些体边对应关系破坏的非厄米系统中,趋肤效应和不同边界条件下的能谱差异往往是同时发生的。为了恢复体边对应关系,清华大学汪忠教授将波数k由实数域推广到了复数域,即将布里渊区推广到了广义布里渊区。借助广义布里渊区,很好的解释了非厄米系统中所有本征态都局域在边界上,以及不同边界条件下能谱巨大差异性的原因。同时,由定义在广义布里渊区上的非布洛赫拓扑不变量也可以忠实的刻画与之相对应的拓扑边界态的存在,至此,建立了非布洛赫体边对应关系,无论是厄米系统还是非厄米系统,都可以很好的匹配。本文借助非布洛赫能带理论,首先从厄米和非厄米系统在本征值、本征矢量的差异性入手,介绍了奇异点,复能隙的概念。然后,以一维非厄米Su-Schrieffer-Heeger模型为例阐述了非厄米体系下,趋肤效应、不同边界条件下能带的差异性,以及广义布里渊区这三者之间的内在联系。同时,给出了两种计算广义布里渊区的方法,并以此定义了非厄米系统中对称性的分类方法,以及不同对称性对能量、广义布里渊区的影响。在一维非厄米超导系统中,研究了不同边界条件下的能谱以及波函数所展示出来的Z₂趋肤效应,同时揭示了一个客观规律:具备粒子-空穴对称性的非厄米系统,广义布里渊区将成对出现,且互为倒数,在发生相变时,一定伴随广义布里渊区的相交,且交点一定位于单位圆上。然后,我们详细研究了周期驱动下的一维非厄米系统,发现即使在能带拓扑平庸的情况下,仍然能存在拓扑边界态,给出了能带拓扑与能隙拓扑之间的联系,建立了非厄米周期驱动系统下的体边对应关系。最后,我们通过低能近似方法,研究了三维非厄米外尔半金属的不同拓扑相,发现在拓扑非平庸相存在手征的边界态,可以使得波包沿边界单向演化,同时通过非布洛赫陈数建立了高维非厄米系统的体边对应关系。更多还原

【Abstract】 The bulk-boundary-correspondence is an essential principle for our research on topological states and topological phases.Generally speaking,a nontrivial bulk topology usually implies the emergence of edge states under the open boundary conduction,that is to say,the topological invariant obtained by the Bloch Hamiltonian defined on Brillouin zone can accurately predict the topological nontrivial edge states under the open boundary condition.Recently,with the continuous research of non-Hermitian quantum systems,more and more people have realized that traditional bulk-boundary-correspondence is no longer applicable in nonHermitian systems.For example,the discovery of the non-Hermitian skin effect,and the difference between the energy spectrum under open boundary conditions and periodic boundary conditions.These phenomena cannot be well explained by Bloch band theory.Further studies revealed that all the eigenstates localized the boundary which called the skin effect and the difference of energy spectrum under different boundary conditions often occur simultaneously.In order to restore the bulk-boundary-correspondence,professor Wangzhong of Tsinghua University extended the wave number k from the real number domain to the complex number domain,which is to extend the Brillouin zone to the generalized Brillouin zone.With the help of the generalized Brillouin zone,it is well explained that all eigenstates in the non-Hermitian system are localized to the boundary,and the huge difference in energy spectrum under different boundary conditions.At the same time,the non-Bloch topological invariant defined on the generalized Brillouin zone can also faithfully describe the existence of the corresponding topological nontrivial edge state.So far,the non-Bloch bulk-boundary-correspondence has been established.No matter it is a Hermitian system or a nonHermitian system,which can be matched very well.Based on the Non-Bloch band theory,this paper analyzes the differences in eigenvalues,eigenvectors,exceptional points,and complex energy gaps between non-Hermitian and Hermitian systems.We explain the relationship in non-Hermitian system among the skin effect,differernce of energy spectrum under different boundary condictions and generalized Brillouin zones by taking the one-dimensional non-Hermitian Su-Schrieffer-Heeger model as an example.Then,we introduce two methods for calculating the generalized Brillouin zone,the classification of symmetry in non-Hermitian system,and the influence of different symmetry on energy and generalized Brillouin zone.In one-dimensional non-Hermitian superconductors,we investigate the energy spectrum under different boundary conditions and the Z skin effect.According to the partice-hole symmetry,we found there exist reciprocal particle and hole loops of generalized Brillouin zone.The critical point of quantum phase transition,where the energy gap closes,appears when the particle and hole loops cross at Brillouin zone.Also,we interplay the non-Hermitian and the periodic driving in a one-dimensional Su-Schrieffer-Heeger model,a novel phenomenon can emerge: the robust edge states can appear even when the Floquet bands are topological trivial with zero non-Bloch band invariant,which is defifined in terms of the non-Bloch effective Hamiltonian.We also show the relation between the non-Bloch winding numbers and the non-Bloch band invariant.Then the bulk-boundary-correspondence in non-Hermitian periodically driven system is established.Finally,we use low-energy continuum case as the tool to obtain the topological phase diagram of the non-Hermitian Weyl semimetal,which is also confirmed by the energy spectra from our numerical results.Moreover,these Fermi-arc edge modes can manifest as the unidirectional edge motion,and their signatures are consistent with the non-Bloch bulk-boundary correspondence,which defined by the non-Bloch chern number.更多还原