PT-Symmetry in $2\times 2$ Matrix Polynomials Formed by Pauli Matrices

Abstract

$2\times2$ matrix polynomials of the form $P_{n}(z)= \Sigma^{n}_{j=0}\,\sigma_{j}\,z^{j}$, for the cases $n=1,2,3$ are constructed, and the nature of PT-symmetry is examined across different points $z=(x,y)$ in the complex plane. The PT-symmetric properties of $P_{n}(z)$ can be characterized by two functions, denoted by $s(x,y)$ and $h(x,y)$. If the trace of the matrix polynomial is real, then the points at which it can exhibit PT-symmetry are defined by the family of curves $s(x,y)=0$. Additionally, at points where the function $h(x,y)\geq 0$, the matrix polynomial exhibits unbroken PT-symmetry; otherwise, it exhibits broken PT-symmetry. The intersection points of the curves $s(x,y)=0$ and $h(x,y)=k$, for a given $k\in \mathbb{R}$, are shown to lie on an ellipse, hyperbola, two lines passing through the origin, or a straight line, depending on the nature of PT-symmetry of the matrix polynomial. The PT-symmetric behaviour of $P_{n}(z)$ at the zeros of the matrix polynomial is also studied.