This work displays the min-max and max-min principles involved in solving different 2+1 Dirac fermion problems. The first problem concerns a charged fermionic oscillator in an external homogeneous magnetic field, for which a quantum phase transition was predicted [Bermudez et al., Phys. Rev. A2008, 77, 063815/1-13]. The ‘zero-mass’ fermion near Dirac point in graphene monolayer constitutes the second example. Here, the min-max and max-min theorems arise from lattice symmetry, being otherwise independent of the trial function. The min-max and max-min principles hold for the trial solution of the fermion with ‘nearly zero’ mass in diatomic materials with graphene-type lattice.

这项工作显示了解决不同的2 + 1 Dirac费米子问题涉及的最小-最大和最大-最小原理。第一个问题涉及外部均质磁场中的带电费米子振荡器，对此已预测出量子相变[Bermudez et al。，Phys。版本A 2008，77，063815 / 1-13]。石墨烯单层中狄拉克点附近的“零质量”费米子构成了第二个示例。此处，最小-最大和最大-最小定理由晶格对称性产生，否则与试验函数无关。最小-最大和最大-最小原理适用于具有石墨烯型晶格的双原子材料中质量几乎为零的费米子的试验解决方案。