三维对称类AIII，BDI，C，D和CI中Anderson跃迁的普适类
Universality classes of the Anderson transition in three-dimensional symmetry classes AIII, BDI, C, D and CI
摘要：
我们澄清普遍的临界性质的离域-本地化过渡的三维（3D）酉和正交类粒子孔和/或手征对称性（类AIII，BDI，D，C和CI）。我们首先介绍了分别属于这五个非标准对称类的立方晶格上的紧束缚模型。与超导体的Bogoliubov-de Gennes哈密顿量不同，所有五种模型在干净极限的动量空间中都有有限面积的费米面。因此，Anderson跃迁的标度理论保证了在这些模型中存在有限无序强度的离域-局域化跃迁。基于这一期望，我们进行了广泛的传输矩阵计算的零能量本征态的准一维几何结构的无序紧束缚模型的李雅普诺夫指数。在Anderson转变点附近，关联长度以普适的临界指数$ν$发散.使用有限尺寸标度分析的本地化长度，我们确定的临界指数的过渡和标度函数的（归一化）本地化长度在三个非标准的酉对称类：$ν_{\text{AIII}}=1.06\pm0.02$，$ν_{\text{D}}=0.87\pm0.03$，和$ν_{\text{C}}=0.996\pm0.012$。我们对C类的结果与经典网络模型[M. Ortuño等人，Phys. Rev. Lett. 102，070603（2009）]。两个非标准正交类的临界指数估计为$ν_{\text{CI}}=1.17 \pm 0.02$和$ν_{\text{BDI}}=1.12 \pm 0.06$。CI类的结果与另一个格点模型的结果一致，而BDI类的指数与节Dirac环模型的结果不一致[X. L. Luo等人，Phys.Rev.B. 101，020202（R）（2020）]。
英文摘要：
We clarify universal critical properties of delocalization-localization transitions in three-dimensional (3D) unitary and orthogonal classes with particle-hole and/or chiral symmetries (classes AIII, BDI, D, C and CI). We first introduce tight-binding models on cubic lattice that belong to these five nonstandard symmetry classes respectively. Unlike the Bogoliubov-de Gennes Hamiltonian for superconductors, all the five models have finite areas of Fermi surfaces in the momentum space in the clean limit. Thereby, the scaling theory of the Anderson transition guarantees the presence of the delocalization-localization transitions at finite disorder strength in these models. Based on this expectation, we carry out extensive transfer matrix calculations of the Lyapunov exponents for zero-energy eigenstates of the disordered tight-binding models with quasi-one-dimensional geometry. Near the Anderson transition point, the correlation length diverges with a universal critical exponent $ν$. Using finite-size scaling analysis of the localization length, we determine the critical exponent of the transitions and scaling function of the (normalized) localization length in the three non-standard unitary symmetry classes: $ν_{\text{AIII}}=1.06\pm0.02$, $ν_{\text{D}}=0.87\pm0.03$, and $ν_{\text{C}}=0.996\pm0.012$. Our result of the class C is consistent with a previous study of classical network model [M. Ortuño et al, Phys. Rev. Lett. 102, 070603 (2009)]. The critical exponents of the two non-standard orthogonal classes are estimated as $ν_{\text{CI}}=1.17 \pm 0.02$ and $ν_{\text{BDI}}=1.12 \pm 0.06$. Our result of the class CI is consistent with another previous study of a lattice model, while the exponent of the class BDI is at variance with the previous evaluation of nodal Dirac ring model [X. L. Luo et al, Phys. Rev. B. 101, 020202(R) (2020)].
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