Depiction of Hamiltonian PT-symmetry
Authors: Zhang Hui-Jie , He Kan ,   School of Mathematics, Taiyuan University of Technology, Taiyuan 030024, China
Published: December 2023

The theory of PT-symmetry describes the non-hermitian Hamiltonian with real energy levels, which means that the Hamiltonian <i>H</i> is invariant neither under parity operator <i>P</i>, nor under time reversal operator <i>T</i>, <i>PTH</i> = <i>H</i>. Whether the Hamiltonian is real and symmetric is not a necessary condition for ensuring the fundamental axioms of quantum mechanics: real energy levels and unitary time evolution. The theory of PT-symmetry plays a significant role in studying quantum physics and quantum information science, Researchers have paid much attention to how to describe PT-symmetry of Hamiltonian. In the paper, we define operator <i>F</i> according to the PT-symmetry theory and the normalized eigenfunction of Hamiltonian. Then we first describe the PT-symmetry of Hamiltonian in dimensionless cases after finding the features of commutator and anti-commutator of operator <i>CPT</i> and operator <i>F</i>. Furthermore, we find that this method can also quantify the PT-symmetry of Hamiltonian in dimensionless case. <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> represents the part of PT-symmetry broken, and <i>J</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> represents the part of PT-symmetry. If <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> = 0, Hamiltonian <i>H</i> is globally PT-symmetric. Once <i>I</i>(<i>CPT</i>, <i>F</i>) = ||[<i>CPT</i>, <i>F</i>]||<i>CPT</i> ≠ 0, Hamiltonian <i>H</i> is PT-symmetrically broken. In addition, we propose another method to describe PT-symmetry of Hamiltonian based on real and imaginary parts of eigenvalues of Hamiltonian, to judge whether the Hamiltonian is PT symmetric. Re<i>F</i> = 1/4||(<i>CPTF</i>+<i>F</i>)||CPT represents the sum of squares of real part of the eigenvalue <i>E<sub>n</sub></i> of Hamiltonian <i>H</i>, Im<i>F</i> = 1/4||(<i>CPTF</i>–<i>F</i>)||CPT is the sum of imaginary part of the eigenvalue <i>E<sub>n</sub></i> of a Hamiltonian <i>H</i>. If Im<i>F</i> = 0, Hamiltonian <i>H</i> is globally PT-symmetric. Once Im<i>F</i> ≠ 0, Hamiltonian <i>H</i> is PT-symmetrically broken. Re<i>F</i> = 0 implies that Hamiltonian <i>H</i> is PT-asymmetric, but it is a sufficient condition, not necessary condition. The later is easier to realize in the experiment, but the studying conditions are tighter, and it further requires that <i>CPT</i> <inline-formula><tex-math id="Z-20240108115351">\begin{document}$\phi_n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115351.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115351.png"/></alternatives></inline-formula>(<i>x</i>) = <inline-formula><tex-math id="Z-20240108115401">\begin{document}$\phi_n $\end{document}</tex-math><alternatives><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115401.jpg"/><graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="4-20230458_Z-20240108115401.png"/></alternatives></inline-formula>(<i>x</i>). If we only pay attention to whether PT-symmetry is broken, it is simpler to use the latter method. The former method is perhaps better to quantify the PT-symmetrically broken part and the part of local PT-symmetry.