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    In fact, frustration can arise with purely anti-ferromagnetic interactions, for example via different neighbours, and even nearest-neighbour ferromagnetism can be frustrated if there are other constraints, such as the ‘spin ice rule’.
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    The reader is referred to the literature, such as cited above (e.g. ()), for further details of the theory of SK-type spin glasses. We do, however, note again that its solution has exposed many new concepts which have had significant consequences in several branches of the science of complex systems (e.g. ) and in mathematics (e.g. ()).
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    inline image (), inline image.
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    Note that strain coupling is required to get the tetragonal crystal structure distortion of the ferromagnetic phase at larger x. However, the present interest is in the relaxor phase which has neither overall polarization nor change in average crystal structure.
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    Note that an antiferromagnetic interaction of longer than nearest neighbour range can already be frustrated. As an example of spin glass behaviour with purely dipolar interaction in combination with site-dilution, see ().
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    As noted earlier, the discussion above has excluded the strain coupling and absorbed the Ba and O ion effects into an effective BB interaction. To deal with the ferroelectric phase more completely these need to be included.
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    The so-called Burns temperature presumably corresponds to the onset of nanodomains, given by when the density of states of the Anderson-like equation cross zero, while the onset of non-ergodicity and the relaxor transition occurs when the lower Anderson mobility edge crosses zero.
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    D. Sherrington, Phys. Rev. B 89, 064105 (2014).
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    The ionic radii of Pb++ and Ba++ are, respectively, 133 and 149 pm. That of O −−is 126 pm.
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    Note that even pure TiNi is often referred to as an alloy, as in ‘shape-memory alloy’, but in fact it is a compound. When the expression ‘alloy’ is used in this paper it refers to systems that deviate in a random fashion from a periodic structure.
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    A. Guiliani, J. L. Lebowitz, and E. H. Lieb, Phys. Rev. B 74, 064420 (2006).
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    Note, however, that actually the V of Eq. (19) (and its extension to higher dimension) decays with an inverse power law inline image just beyond the realm of applicability of the proof of (), together with an extra angular factor. Hence boundary effects are more important in this pure case. But the main interest here is in the disordered alloy extension.
  • 78
    Note that for inline image the twinned phase will have defects.
  • 79
    The SK model, exact for infinite-ranged interactions, has a ‘mixed’ phase for an x-region above inline image, but such a phase in short-ranged spin glasses is contentious. In the martensitic alloys the interactions are power-law, which may suffice.
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    D. Sherrington, Prog. Theor. Phys. Suppl. 87, 180 (1986).
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    X. Ren, Y. Wang, Y. Zhou, Z. Zhang, D. Wang, G. Fan, K. Otsuka, T. Suzuki, Y. Ji, Y. Tian, S. Hou and X. Ding, Philos. Mag. 90, 141 (2010).
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    Y. Wang, X.-B. Ren, K. Otsuka, and A. Saxena, Phys. Rev. B 76, 132201 (2007).
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    J. Zhang, Y. Wang, X. Ding, Z. Zhang, Y. Zhou, X.-B. Ren, K. Otsuka, J. Sun, and M. Song, Phys. Rev. B 84, 214201 (2011).
  • 84
    Note that the phase lines in the strain glass predictions shown in the figures are schematic. In particular, the sign of the slope between the martensitic phase and the mixed phase in the figure for a continuous inline image is not calculated and hence its prediction is currently uncertain.
  • 85
    D. Sherrington, in: Heidelberg Colloquium on Spin Glasses, edited by I. Morgenstern and L. van Hemmen (Springer, Berlin, Heidelberg, 1983), pp. 125136.
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    D. Sherrington, J. Phys. Soc. Jpn. Suppl. 52, 229 (1983).
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    D. J. Elderfield and D. Sherrington, J. Phys. C 16, L497 (1983).
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    P. M. Goldbart and D. Sherrington, J. Phys. C 18, 1923 (1985).
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    D. M. Cragg, D. Sherrington, and M. Gabay, Phys. Rev. Lett. 49, 158 (1982).
  • 95
    For low fields the GT shift goes as inline image and the AT crossover as inline image, where H is the applied field.
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    D. Elderfield and D. Sherrington, J. Phys. A 15, L513 (1982); J. Phys. A 15, L785 (1982).
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    This paper does not, however, cite these predictions.
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    By ‘non-trivial’ we refer to a situation where the random fields are essential for the phase transition. Thus, the (soluble) infinite-ranged ferromagnet with random fields is ‘trivial’ in that the only phases are paramagnet and ferromagnet.
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    A. Sharma and A. P. Young, Phys. Rev. E 81, 061115 (2010).
  • 103
    Note, however, that the spin glass phase is not induced by the random fields, so this system is still ‘trivial’ in the sense of footnontrivial although the solution of this model requires subtle mathematics.
  • 104
    When Ti are displaced in relaxor BZT they do provide extra quasi-random fields but these are secondary to the interaction-driven terms discussed above.
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    S. Nambu and K.  Sugimoto, Ferroelectrics 198, 11 (1997).