How the energy gap closes at the 3D quantum spin-glass transition

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.

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.