Universal Growth of Krylov Complexity Across A Quantum Phase Transition

Abstract

We study the statistical properties of the spread complexity in the Krylov space of quantum systems driven across a quantum phase transition. Using the diabatic Magnus expansion, we map the evolution to an effective one-dimensional hopping model. For the transverse field Ising model, we establish an exact link between the growth of complexity and the Kibble-Zurek defect scaling: all cumulants of complexity exhibit the same power-law scaling as the defect density, with coefficients identical to the mean, and the full distribution asymptotically becomes Gaussian. These results yield general scaling arguments for the growth of complexity across arbitrary second-order quantum phase transitions.

Figures (6)

• Figure 1: Off-diagonal Lanczos coefficients for different system sizes and driving times, all of them growing approximately as $\sim\sqrt n$, and collapsing onto a single curve by the proper rescaling given in Eq. \ref{['eq:a_n_b_n']}.
• Figure 2: Statistics of Krylov complexity for a system of $L=400$ spins with an effective size set to $L_\mathrm{eff}=48$, capturing the low-energy spectrum that governs the dynamics. The solid lines show the agreement with the Gaussian approximation for $\tau=3$ and $34$ within the KZ scaling regime, while for $\tau=750$, slight deviations emerge as the system approaches the onset of adiabaticity.
• Figure 3: First three cumulants for the complexity as a function of $\tau$. With small fluctuations, they all follow the universal power laws predicted by Eq. \ref{['eq: cumulants']}, matching the KZ scaling of defect cumulants $(L=400,\,L_\mathrm{eff}=46)$.
• Figure 4: Time evolution of the Krylov complexity for different quench times. The sharp increase around $g_c=1$ captures the interplay between the QPT and complexity growth, followed by a non-universal and oscillatory regime that smoothly converges to the final value, as described by universal power laws. $(L=200,\,L_\mathrm{eff}=46)$.
• Figure 5: Krylov complexity histograms for driving times $\tau=4,20,50$ for various intermediate values of $g$. In the paramagnetic phase, a peaked distribution is observed in line with adiabatic dynamics, which for faster processes starts to broaden earlier, once the critical point is approached. Inside the broken symmetry phase, the final Gaussian shape sets in earlier for slower processes $(L=400,\,L_\mathrm{eff}=46)$.