Stiffness Exponents for Lattice Spin Glasses in Dimensions d=3,...,6

S. Boettcher

doi:10.1140/epjb/e2004-00102-5

paper

Stiffness Exponents for Lattice Spin Glasses in Dimensions d=3,...,6

arXiv:cond-mat/0310698 · doi:10.1140/epjb/e2004-00102-5

Abstract

The stiffness exponents in the glass phase for lattice spin glasses in dimensions $d = 3, ..., 6$ are determined. To this end, we consider bond-diluted lattices near the T=0 glass transition point $p^{*}$ . This transition for discrete bond distributions occurs just above the bond percolation point $p_{c}$ in each dimension. Numerics suggests that both points, $p_{c}$ and $p^{*}$ , seem to share the same $1/ d$ -expansion, at least for several leading orders, each starting with $1/ (2 d)$ . Hence, these lattice graphs have average connectivities of $α = 2 d p ≳ 1$ near $p^{*}$ and exact graph-reduction methods become very effective in eliminating recursively all spins of connectivity $\leq 3$ , allowing the treatment of lattices of lengths up to L=30 and with up to $1 0^{5} - 1 0^{6}$ spins. Using finite-size scaling, data for the defect energy width $σ (Δ E)$ over a range of $p > p^{*}$ in each dimension can be combined to reach scaling regimes of about one decade in the scaling variable $L (p - p^{*})^{ν^{*}}$ . Accordingly, unprecedented accuracy is obtained for the stiffness exponents compared to undiluted lattices ( $p = 1$ ), where scaling is far more limited. Surprisingly, scaling corrections typically are more benign for diluted lattices. We find in $d = 3, ..., 6$ for the stiffness exponents $y_{3} = 0.24 (1)$ , $y_{4} = 0.61 (2), y_{5} = 0.88 (5)$ , and $y_{6} = 1.1 (1)$ . The result for the upper critical dimension, $d_{u} = 6$ , suggest a mean-field value of $y_{\infty} = 1$ .

8 pages, RevTex, 15 ps-figures included (see http://www.physics.emory.edu/faculty/boettcher for related information)

Stiffness Exponents for Lattice Spin Glasses in Dimensions d=3,...,6

Abstract

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