## Dante Gatteschi, Roberta Sessoli, and Jacques Villain

Print publication date: 2006

Print ISBN-13: 9780198567530

Published to Oxford Scholarship Online: September 2007

DOI: 10.1093/acprof:oso/9780198567530.001.0001

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# (p.329) APPENDIX D BASIC PROPERTIES OF THE MASTER EQUATION

Source:
Molecular Nanomagnets
Publisher:
Oxford University Press

In this appendix, it is shown that all eigenvalues of the ‘master matrix’

(D.1)
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are real and negative, except that which corresponds to equilibrium and is equal to 0. The indices i, j denote the eigenstates of the (time+independent) Hamiltonian and the Θ matrix appears in equation (5.12) which can be written as
(D.2)
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The transition probabilities γji are real and positive and satisfy the principle of detailed balance
(D.3)
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It is appropriate to introduce the real, symmetric, and therefore Hermitian matrix

(D.4)
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Let v i be an eigenvectorof Θ with the eigenvalue λ:
(D.5)
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Substituting (D.4) one obtains
(D.6)
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where
(D.7)
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and
(D.8)
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Thus, any eigenvalue λ of Θ is also an eigenvalue of the matrix Γji − δji αi, which is real, symmetric, and therefore Hermitian. This implies that λ is real. (p.330) To show that these eigenvalues are non+positive, it is sufficient to show that the quadratic form

(D.9)
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cannot be positive. Indeed, it is easily shown that (D.9) reads
(D.10)
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The only non-vanishing contributions are those of coefficients Γij with ij which are positive equal to zero. Therefore, expression (D.10) is negative except for xi = exp(βε i/2), which corresponds to thermal equilibrium.

The desired property is thus proven.