The Jordan-Wigner transformation is frequently utilised to rewrite quantum spin chains in terms of fermionic operators. When the resulting Hamiltonian is bilinear in these fermions, i.e. the fermions are free, the exact spectrum follows from the eigenvalues of a matrix whose size grows only linearly with the volume of the system. However, several Hamiltonians that do not admit a Jordan-Wigner transformation to fermion bilinears still have the same type of free-fermion spectra. The spectra of such "free fermions in disguise" models can be found exactly by an intricate but explicit construction of the raising and lowering operators. We generalise the methods further to find a family of such spin chains. We compute the exact spectrum, and generalise an elegant graph-theory construction. We also explain how this family admits an N=2 lattice supersymmetry.

中文翻译：

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超越乔丹和维格纳的自由费米子

乔丹-维格纳变换经常被用来根据费米子算子重写量子自旋链。当所得的哈密顿量在这些费米子中是双线性的，即费米子是自由的时，精确的谱来自于矩阵的特征值，该矩阵的大小仅随系统的体积线性增长。然而，一些不承认乔丹-维格纳变换为费米子双线性的哈密顿量仍然具有相同类型的自由费米子谱。这种“伪装的自由费米子”模型的谱可以通过复杂但明确的升高和降低算子的构造来准确地找到。我们进一步推广这些方法来找到这样的自旋链家族。我们计算精确的频谱，并概括出一个优雅的图论结构。我们还解释了这个族如何承认 N=2 晶格超对称性。