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The Ising Spin Glass in a Field

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Spin Glasses

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Abstract

This chapter is dedicated to the search of a would-be phase transition in a three-dimensional spin glass. The discussion will focus entirely on the analysis of the data and on the results.

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Notes

  1. 1.

    A short discussion on PT is given in Appendix A.1.

  2. 2.

    In Appendix B.3 we describe how MSC was implemented in the analysis stage. Multi-spin coding MC in the simulations [Seo13] follows roughly the same principles than in the analyses.

  3. 3.

    In [BJ14b] we studied a bimodal field, while in the work we present here h is constant. Notwithstanding, we will make comparisons with the bounds \(T^\mathrm {up}(h)\) by matching \(\overline{h^2}\) in both models.

  4. 4.

    See the short note in Appendix A.1.

  5. 5.

    A quantile is the value of \(\hat{q}\) that separates a fixed part of the pdf (Sect. 2.5 later on).

  6. 6.

    Let us give a simple example. Take 4 \(Z_2\)-symmetric single-spin systems that can assume different values \(s_1=\pm 1, s_2=\pm 2, s_3=\pm 3, s_4=\pm 4\). We can construct 6 overlaps \(q_{ij}(s_1, s_2, s_3, s_4)\). If we explicitate the \(Z_2\) symmetry, taking all the combinations of our random variables, the histogram of q will be symmetric with zero mean. Yet, if we take the histogram of the median overlap, it will be asymmetric with mean \(\langle q_\mathrm {med}\rangle = -3\). This can easily be checked by computing all the possible combinations of the signs of the \(s_i\) and computing the median in each case: \(q_\mathrm {med}(+1, +2, +3, +4) = 5\), \(q_\mathrm {med}(+1, +2, +3, -4) = -1\), \(q_\mathrm {med}(+1, +2, -3, -4) = -3.5\), and so on.

  7. 7.

    We made power law extrapolations to \(L\rightarrow \infty \) of the maxima of the susceptibility, but they were not satisfactory (too large \(\chi ^2/\mathrm {DOF}\)). Only for \(h=0.2,0.4\) were we able to fit the maxima’s heights and obtained \(\eta (h=0.2) \approx 0.6\) and \(\eta (h=0.4)\approx 0.9\).

  8. 8.

    See Appendix B.21, keeping in mind that \(\eta _\mathrm {eff} = -1\) is somewhat trivial in the limit \(h\rightarrow 0\), where \(\chi \) reduces to \(\chi = V E(q^2)\).

  9. 9.

    For each jackknife block we calculated \(\eta _\mathrm {eff}(T)\) and made a cubic spline temperature interpolation.

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  1. Institut de Physique Théorique, DRF, CEA Saclay, Gif-sur-Yvette, France

    Marco Baity Jesi

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Baity Jesi, M. (2016). The Ising Spin Glass in a Field. In: Spin Glasses. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-319-41231-3_2

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