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The theory of PT-symmetry describes the non-hermitian Hamiltonian with real energy levels, which means that the Hamiltonian H is invariant neither under parity operator P, nor under time reversal operator T, PTH = H. 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 F 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 CPT and operator F. Furthermore, we find that this method can also quantify the PT-symmetry of Hamiltonian in dimensionless case. I(CPT, F) = ||[CPT, F]||CPT represents the part of PT-symmetry broken, and J(CPT, F) = ||[CPT, F]||CPT represents the part of PT-symmetry. If I(CPT, F) = ||[CPT, F]||CPT = 0, Hamiltonian H is globally PT-symmetric. Once I(CPT, F) = ||[CPT, F]||CPT ≠ 0, Hamiltonian H 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. ReF = 1/4||(CPTF+F)||CPT represents the sum of squares of real part of the eigenvalue En of Hamiltonian H, ImF = 1/4||(CPTF–F)||CPT is the sum of imaginary part of the eigenvalue En of a Hamiltonian H. If ImF = 0, Hamiltonian H is globally PT-symmetric. Once ImF ≠ 0, Hamiltonian H is PT-symmetrically broken. ReF = 0 implies that Hamiltonian H 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 CPT
$\phi_n $ (x) =$\phi_n $ (x). 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.-
Keywords:
- Hamiltonian /
- PT-symmetry /
- commutator /
- normalized feature functions
[1] Bender C M, Brody D C, Jones H F 2002 Phys. Rev. Lett. 89 270401
Google Scholar
[2] Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243
Google Scholar
[3] Bender C M 2005 Contemp. Phys. 46 277
Google Scholar
[4] Bender C M, Boettcher S, Meisinger P N 1999 J. Math. Phys. 40 2201
Google Scholar
[5] Wu T T 1959 Phys. Rev. 115 1390
Google Scholar
[6] Brower R C, Furman M A, Moshe M 1978 Phys. Lett. B 76 213
Google Scholar
[7] Fisher M E 1978 Phys. Rev. Lett. 40 1610
Google Scholar
[8] Bender C M 2007 Rep. Prog. Phys. 70 947
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[9] Bender C M, Gianfreda M, Ozdemir S K, Peng B, Yang L 2013 Phys. Rev. A 88 062111
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[10] Croke S 2015 Phys. Rev. A 91 052113
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[11] Konotop V V, Yang J, Zezyulin D A 2016 Rev. Mod. Phys. 88 035002
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[12] El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S, Christodoulides D N 2018 Natural. Phys. 14 1
Google Scholar
[13] Pi J H, Sun N N, Lü R 2020 Commun. Theor. Phys. 72 4
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[14] Yu S, Meng Y, Tang J S, Xu X Y, Wang Y T, Yin P, Guo G C 2020 Phys. Rev. Lett. 125 240506
Google Scholar
[15] Nielsen M A, Chuang I 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) p76
[16] Mostafazadeh A 2002 J. Math. Phys. 43 205
Google Scholar
[17] 梁昆淼 2010 数学物理方法(第四版) (北京: 高等教育出版社) 第82—89页
Liang K M 2010 Methods of Mathematical Physics (Vol. 4) (Beijing: Higher Education Press) pp82–89
[18] Samsonov B F, Roy P 2005 J. Phys. A Math. Gen. 38 L249
Google Scholar
[19] Bender C M, Brody D C, Jones H F 2004 Phys. Rev. Lett. 93 251601
Google Scholar
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表 1 比较刻画哈密顿量H PT对称性的两种方法
Table 1. Compare two depiction methods of PT-symmetry of Hamiltonian H
第一种方法 第二种方法 H是全局PT对称 J(CPT, F) ReF H是局部PT对称 J(CPT, F), I(CPT, F) ImF H是PT对称完全破缺 I(CPT, F) ImF 
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[1] Bender C M, Brody D C, Jones H F 2002 Phys. Rev. Lett. 89 270401
Google Scholar
[2] Bender C M, Boettcher S 1998 Phys. Rev. Lett. 80 5243
Google Scholar
[3] Bender C M 2005 Contemp. Phys. 46 277
Google Scholar
[4] Bender C M, Boettcher S, Meisinger P N 1999 J. Math. Phys. 40 2201
Google Scholar
[5] Wu T T 1959 Phys. Rev. 115 1390
Google Scholar
[6] Brower R C, Furman M A, Moshe M 1978 Phys. Lett. B 76 213
Google Scholar
[7] Fisher M E 1978 Phys. Rev. Lett. 40 1610
Google Scholar
[8] Bender C M 2007 Rep. Prog. Phys. 70 947
Google Scholar
[9] Bender C M, Gianfreda M, Ozdemir S K, Peng B, Yang L 2013 Phys. Rev. A 88 062111
Google Scholar
[10] Croke S 2015 Phys. Rev. A 91 052113
Google Scholar
[11] Konotop V V, Yang J, Zezyulin D A 2016 Rev. Mod. Phys. 88 035002
Google Scholar
[12] El-Ganainy R, Makris K G, Khajavikhan M, Musslimani Z H, Rotter S, Christodoulides D N 2018 Natural. Phys. 14 1
Google Scholar
[13] Pi J H, Sun N N, Lü R 2020 Commun. Theor. Phys. 72 4
Google Scholar
[14] Yu S, Meng Y, Tang J S, Xu X Y, Wang Y T, Yin P, Guo G C 2020 Phys. Rev. Lett. 125 240506
Google Scholar
[15] Nielsen M A, Chuang I 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press) p76
[16] Mostafazadeh A 2002 J. Math. Phys. 43 205
Google Scholar
[17] 梁昆淼 2010 数学物理方法(第四版) (北京: 高等教育出版社) 第82—89页
Liang K M 2010 Methods of Mathematical Physics (Vol. 4) (Beijing: Higher Education Press) pp82–89
[18] Samsonov B F, Roy P 2005 J. Phys. A Math. Gen. 38 L249
Google Scholar
[19] Bender C M, Brody D C, Jones H F 2004 Phys. Rev. Lett. 93 251601
Google Scholar
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