I am studying the spectrum of a particular kind of block-tridiagonal Hermitian Toeplitz matrix made of three bands $\{B,A,C\}$
$$ T_n = \begin{pmatrix} A & C & 0 & \dots & 0\\ B & A & C & \vdots & \vdots\\ 0 & B & \ddots & \ddots & \vdots\\ \vdots & \ddots & \ddots & A & C\\ 0 & \dots & \dots & B & A \end{pmatrix} $$
where each block is constructed with gamma matrices $\Gamma_{ij} = \sigma_i \otimes \sigma_j$ (Kronecker products of Pauli matrices) in the following way :
$$ A=\Gamma_{22},B=\Gamma_{22}-i\Gamma_{03},C=\Gamma_{22}+i\Gamma_{03} $$
Such kind of matrix has determinant 1 and I notice its spectrum has interesting localization property. In particular, there are only 4 eigenvalues in the range $(-1,1)$ no matter what value $n$ is. As $n$ increase, there are more eigenvalues with absolute value bigger than $1$ and thus the four degenerate eigenvalues are forced to tend to zero as $n\rightarrow\infty$. Moreover, this pattern is observed if $\Gamma_{22}$ and $\Gamma_{03}$ are replaced by any pair of anticommuting real Hermitian gamma matrices.
Is there any reference for these interesting spectral properties?