CM In the literature: open

Exponentially small gaps inside the quantum spin-glass phase

In plain words

Below the critical sideways field, the two lowest energy levels of a spin glass can approach each other and swap order as the field is reduced, and each near-crossing forces a quantum annealer to slow down. Whether such near-crossings occur in typical large samples, and how small their gaps become, decides whether annealing is exponentially slow even after the critical point is passed.

Precise statement

For the transverse-field Edwards-Anderson model in $d = 2\ \text{and}\ 3$, and its infinite-range (Sherrington-Kirkpatrick) limit, at $0 < \Gamma < \Gamma_c$, determine the probability that the instantaneous ground state has an avoided crossing as $\Gamma$ decreases to 0, and the scaling of the minimum gap $\Delta_{\mathrm{min}}$ with N: $\operatorname{exp}(-c N^a)$ with the value of a, or a power law. An answer gives the typical $\Delta_{\mathrm{min}}(N)$ and its distribution over disorder samples.

What would settle it

Exact diagonalization and parity-resolved quantum Monte Carlo or tensor-network tracking of the two lowest levels along $\Gamma$ for $N$ up to a few hundred spins over many disorder samples.

Status in the literature

Knysh (Nature Communications 2016) argued that such bottlenecks are generic in the spin-glass phase and give gaps exponentially small in N; tests in finite dimensions remain limited.