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Statistical mechanics of Ising chains in random magnetic fields
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26.In Ref. 15, for instance, it was stated that the model studied there was “similar” to that of Ref. 5 while in fact it is precisely equivalent to a special case of the model studied in 5.
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29.For instance, in the case of copolymeric DNA, A may represent a base pair adenine‐thymine and B a base pair guanine‐cytosine.1
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34.Related methods have been termed “Monte Carlo method” in the literature 1,6 but should not be confused with what one usually understands as Monte Carlo method 27,28 in statistical physics.
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36.The case of an antiferromagnetic chain can be handled as well if we calculate rather than [cf. Eqs. (20),(25)] in Eq. (26), or equivalently replace u by v, v by u in Eq. (12), and hence the matrix elements replaced by in as well as in the final result, Eq. (40). Furthermore we note that the method is trivially generalized to a case where in Eq. (12) is replaced by a as long as the randomness is small enough, such that have the same sign. The case where have different sign can be treated also, however, by transforming the to new pseudopsin variables
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39.This idea is also the basis of the related techniques of Fan and McCoy12 and Lehman. 4,5
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45.Monte Carlo data for are taken from Landau and Blume (Ref. 15).
46.Since with the Monte Carlo method we use a finite N rather than taking the thermodynamic limit, we use periodic boundary conditions rather than the free boundary conditions of Eq. (5) in order to make finite‐size effects small. For further details see Refs. 27, 28.
47.An outline of an algorithm to calculate these coefficients can be found in Ref. 37. See also S. Katsura, Progr. Theor. Phys. 55, 1049 (1976);
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48.We expect that on the average there should be no correlation in length between a and the adjacent cluster, which is rigorously correct for
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50.E. Stoll, K. Binder, and T. Schneider, Phys. Rev. B 8, 3266 (1973).
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