Time evolution of spread complexity and statistics of work done in quantum quenches

TL;DR

The paper establishes a general link between the statistics of work done in a sudden quantum quench and the Lanczos coefficients from the Krylov (Lanczos) construction, using the work distribution's cumulants and the autocorrelation-derived moments. By expressing cumulants in terms of Bell polynomials of the moments, it shows that the first two Lanczos coefficients encode physically measurable quantities: the mean work $raket{W}$ and its variance. The authors demonstrate this connection in two concrete settings—the harmonic chain and a mass quench in a bosonic field—where the mean and variance match $a_0$ and $b_1^2$, and discuss how zero modes can drive divergences in critical quenches. In general, the approach provides a thermodynamic interpretation of spread complexity and offers a framework to relate information-theoretic growth to experimentally accessible work statistics, with potential extensions to more complex interacting systems. The results imply that, for Gaussian work distributions, spread complexity grows quadratically in time, while non-Gaussian distributions tie this growth to higher cumulants through Bell polynomial relations.

Abstract

We relate the probability distribution of the work done on a statistical system under a sudden quench to the Lanczos coefficients corresponding to evolution under the post-quench Hamiltonian. Using the general relation between the moments and the cumulants of the probability distribution, we show that the Lanczos coefficients can be identified with physical quantities associated with the distribution, e.g., the average work done on the system, its variance, as well as the higher order cumulants. In a sense this gives an interpretation of the Lanczos coefficients in terms of experimentally measurable quantities. Consequently, our approach provides a way towards understanding spread complexity, a quantity that measures the spread of an initial state with time in the Krylov basis generated by the post quench Hamiltonian, from a thermodynamical perspective. We illustrate these relations with two examples. The first one involves quench done on a harmonic chain with periodic boundary conditions and with nearest neighbour interactions. As a second example, we consider mass quench in a free bosonic field theory in $d$ spatial dimensions in the limit of large system size. In both cases, we find out the time evolution of the spread complexity after the quench, and relate the Lanczos coefficients with the cumulants of the distribution of the work done on the system.

Figures (1)

• Figure 1: Time evolution of SC the zero mode (red) and the sum of the non-zero mode contributions (brown). Here $\lambda_0=5, k=2, N=350$. At large times the contribution of the zero mode always dominates.