Abstract
We consider the prevalent phenomenon that a pair of eigenvalues of the Liouville-von Neumann operator (Liouvillian) changes from pure imaginary to complex values with a common imaginary part for resonance states in an extended function space outside the Hilbert space. Such a transition point is an exceptional point, where non-Hermitian degeneracy occurs and both the pairs of eigenvalues and of eigenvectors coalesce. The transition can be attributed to a spontaneous breakdown of a parity and time-reversal (PT)-symmetry. This PT-symmetry in the Liouvillian dynamics results from the microscopic dynamics based on the fundamental physical laws. The kinetic equation of the Boltzmann type for a particle weakly coupled with a bath consists of the collision term, which is similar to a Hermitian operator and has even parity, and the flow term, which is anti-Hermitian and has odd parity. As a result of the competition between the two terms, a pair of PT-symmetric eigenstates of the effective Liouvillian converts to a PT-symmetry related pair as the flow term becomes more dominant than the collision term beyond an exceptional point.
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Every finite-dimensional matrix can be similarly transformed into this form, because every finite-dimensional matrix is similar to a complex symmetric matrix [18].
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The inner product of the linear operators A and B in the wave function space as vectors in the Liouville space is defined by \(\langle \!\langle A|B\rangle \!\rangle \equiv \mathrm {Tr}\left( A^\dagger B\right) \).
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Detailed analysis on the validity of the Boltzmann approximation will be given elsewhere [22].
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References
L.P. Kadanoff, J. Swift, Phys. Rev. 165, 310–322 (1968)
N. Hatano, D.R. Nelson, Phys. Rev. Lett. 77, 570–573 (1996)
N. Hatano, Phys. A 254, 317–331 (1998). and references therein
N. Moiseyev, Non-Hermitian Quantum Mechanics (Cambridge University Press, 2011)
G. Bhamathi, E.C.G. Sudarshan, Int. J. Mod. Phys. B 13 and 14, 1531–1544 (1996)
M.V. Berry, Czech. J. Phys. 54, 1039–1047 (2004)
W.D. Heiss, Czech. J. Phys. 54, 1091–1099 (2004)
T. Kato, Perturbation Theory for Linear Operators, 2nd edn. (Springer, 1976)
N. Moiseyev, S. Friedland, Phys. Rev. A 22, 618–624 (1980)
C.M. Bender, Rep. Prog. Phys. 70, 947–1018 (2007)
A. Mostafazadeh, Int. J. Geom. Meth. Mod. Phys. 7, 1191–1306 (2010)
R. Balescu, Statistical Mechanics of Charged Particles (John Wiley & Sons, 1963)
S. Klaiman, U. Günther, N. Moiseyev, Phys. Rev. Lett. 101, 080402 (2008)
S. Klaiman, L.S. Cederbaum, Phys. Rev. A 78, 062113 (2008)
P. Amore, F.M. Fernández, J. Garcia, Ann. Phys. 350, 533–548 (2014)
A. Mostafazadeh, J. Math. Phys. 43, 205–214, 2814–2816, 3944–3951 (2002)
K. Hashimoto et al., Prog. Theor. Exp. Phys. 2015, 023A02 (2015)
R. Santra, L.S. Cederbaum, Phys. Rep. 368, 1–117 (2002)
P.-O. Löwdin, J. Math. Phys. 6, 1341–1353 (1963)
T. Petrosky, I. Prigogine, Adv. Chem. Phys. 99, 1–120 (1997)
T. Petrosky, Prog. Theor. Phys. 123, 395–420 (2010)
K. Hashimoto, K. Kanki, S. Tanaka, T. Petrosky, Phys. Rev. E. 93, 022132 (2016)
A. Messiah, Quantum Mechanics (Wiley, 1958)
K. Hashimoto, K. Kanki, S. Tanaka, T. Petrosky, to be published in Prog. Theor. Exp. Phys. (2016)
Z.L. Zhang, Doctoral Dissertation, The University of Texas at Austin (1995)
A. Fring, J. Phys. A: Math. Theor. 48, 145301 (2015)
C.M. Bender, R.J. Kalveks, Int. J. Theor. Phys. 50, 955–962 (2011)
K. Kanki et al., Prog. Theor. Phys. Suppl. 184, 523–532 (2010)
K. Kanki, S. Tanaka, T. Petrosky, J. Math. Phys. 52, 063301 (2011)
S. Tanaka, K. Kanki, T. Petrosky, Phys. Rev. B 80, 094304 (2009)
B.A. Tay, K. Kanki, S. Tanaka, T. Petrosky, J. Math. Phys. 52, 023302 (2011)
T. Petrosky et al., Prog. Theor. Phys. Suppl. 184, 457–465 (2010)
R.K.P. Zia, B. Schmittmann, J. Stat. Mech. 2007, P07012 (2007)
M. Am-Shallem, R. Kosloff, N. Moiseyev, New J. Phys. 17, 113036 (2015)
G. Kimura, K. Yuasa, K. Imafuku, Phys. Rev. Lett. 89, 140403 (2002)
T. Murase, Master Thesis, Osaka Prefecture University (2014)
H. Qian, M. Qian, Phys. Rev. Lett. 84, 2271–2274 (2000)
C.M. Bender, M.V. Berry, A. Mandilara, J. Phys. A: Math. Gen. 35, L467–L471 (2002)
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Kanki, K., Hashimoto, K., Petrosky, T., Tanaka, S. (2016). Spontaneous Breakdown of a PT-Symmetry in the Liouvillian Dynamics at a Non-Hermitian Degeneracy Point. In: Bagarello, F., Passante, R., Trapani, C. (eds) Non-Hermitian Hamiltonians in Quantum Physics. Springer Proceedings in Physics, vol 184. Springer, Cham. https://doi.org/10.1007/978-3-319-31356-6_19
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