Stiffness exponents for lattice spin glasses in dimensions $\mathsf{d = 3,\ldots,6}$

Abstract.

The stiffness exponents in the glass phase for lattice spin glasses in dimensions $d = 3,\ldots,6$ 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/(2d). Hence, these lattice graphs have average connectivities of $\alpha = 2dp\gtrsim1$ near p ^* and exact graph-reduction methods become very effective in eliminating recursively all spins of connectivity $\leq3$, allowing the treatment of lattices of lengths up to L = 30 and with up to 10⁵-10⁶ spins. Using finite-size scaling, data for the defect energy width $\sigma(\Delta E)$ over a range of p > p ^* in each dimension can be combined to reach scaling regimes of about one decade in the scaling variable $L(p-p^*)^{\nu^*}$. 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 $d = 3,\ldots,6$ 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).