论文标题
晶格上费米子的局部旋转描述
Local spin description of fermions on a lattice
论文作者
论文摘要
在这项工作中提出并研究了从费米子操作员到旋转矩阵的局部转换。为此，考虑了一个晶格上的费米子系统，并应用了该方案将费米子变量替换为旋转矩阵，而转换仅将分配给附近晶格站点的费米子/自旋操作员。在一个维度上，该提案与著名的Jordan-Wigner转换产生相同的结果，而不仅限于$ d = 1 $尺寸。为了获得旋转图片中的等效描述，需要对旋转空间施加约束。由于发现减少的旋转希尔伯特空间构成了整个过程的实质阶段，因此限制受到特别关注。在两个表示中都确定了完整的必要约束。为了解决解决约束的任务，构建了合适的基础。在旋转表示中引入基础以及在此基础上的约束和哈密顿量的构建表明，如何应用这项工作中提出的转换以在旋转图片中获得可观察结果。在基础上明确构造了约束，可以解决它们，一旦发现了旋转希尔伯特空间的基础向量，则在此基础上表达自旋汉密尔顿人并对角度化。这些约束是根据上述讨论的基础构建的，并使用Wolfram Mathematica程序分析了晶格尺寸$ 3 \ times3 $，$ 4 \ times3 $和$ 4 \ times4 $。确定它们的相互关系，并指定了减少的自旋希尔伯特空间。哈密顿量是在这种表示形式中构建的，并对角线化。可以证实，在自旋图片中获得的特征力与费米代表的分析公式一致。
A local transformation from fermionic operators to spin matrices is proposed and studied in this work. For this purpose, a system of fermions on a lattice is considered and one applies the scheme to replace the fermionic variables with spin matrices, while the transformation relates only those fermionic/spin operators which are assigned to nearby lattice sites. In one dimension, this proposal yields the same result as the well-known Jordan-Wigner transformation, while not being restricted to $d=1$ dimension. To obtain the equivalent description in the spin picture, one needs to impose constraints on the spin space. Since finding the reduced spin Hilbert space constitutes a substantial stage of the whole procedure, the constraints are paid particular attention. The full set of necessary constraints is determined in both representations. To approach the task to solve the constraints, a suitable basis is constructed. The introduction of the basis in the spin representation along with the construction of the constraints and the Hamiltonian in this basis show how the transformation proposed in this work can be applied to obtain observables in the spin picture. Explicit construction of the constraints in the basis allows one to solve them and, once the basis vectors of the reduced spin Hilbert space are found, the spin Hamiltonian is expressed in this basis and diagonalized. The constraints are constructed in the basis as discussed above and analyzed with the Wolfram Mathematica programs for lattice sizes $3\times3$, $4\times3$ and $4\times4$. Their mutual relations are determined and the reduced spin Hilbert space is specified. The Hamiltonian is constructed in this representation and diagonalized. It is verified that the eigenenergies obtained in the spin picture agree with the analytic formulas from the fermionic representation.
