How the energy gap closes at the 3D quantum spin-glass transition
In plain words
At the critical sideways field the energy needed to create the first excitation (the gap) shrinks as the sample grows. In two dimensions this shrinking has been measured separately for excitations that keep and that flip the global spin-reversal symmetry; in three dimensions it is unsettled.
Precise statement
Model: H = -sum_<xy> J_xy sz_x sz_y - Gamma sum_x sx_x on an L x L x L cubic lattice, $J_{xy} = \pm 1$ or Gaussian, $T = 0$. At $\Gamma_c$ determine the finite-size scaling of the parity-even gap Delta_e(L) = E_1,even - E_0,even (power law L^-$z_e$, and the value of z_e) and of the parity-odd gap Delta(L) = E_0,odd - E_0,even (power law, or activated $\ln(1/\Delta)\sim L^\psi$ as at an infinite-randomness fixed point), including their sample-to-sample distributions. An answer gives $z_e$, $\nu$ and the form and exponent of the odd-gap scaling with error bars.
What would settle it
Parity-resolved quantum Monte Carlo or tensor-network simulations of the 3D model at $\Gamma_{c}$ for $L$ up to about 16 to 20, with the $T = 0$ limit checked by periodic and antiperiodic imaginary-time boundaries.
Status in the literature
Unverified note
Bernaschi et al. (Nature 2024) measured these quantities in $d = 2$; no 3D study of comparable precision was found as of 2026.