Growth of block diagonal operators and symmetry-resolved Krylov complexity

TL;DR

This work develops symmetry-resolved Krylov complexity for Hermitian operators commuting with a global charge, decomposing operator growth into charge sectors and analyzing how these sectors contribute to overall operator complexity. Using a Lanczos-based framework with symmetry projections and Fourier techniques, the paper shows that early-time growth is captured by the sector-averaged complexity $ar{C}(t)$ while late-time dynamics involve inter-sector correlations, typically yielding $C_K(t)\,\ar{C}(t)$. Through concrete models—two-spin, a general $4\times4$ block operator, and complex harmonic oscillators—it demonstrates both the potential for Krylov equipartition in specific cases and the general nontrivial interplay between sectors. The results illuminate how symmetry structure shapes operator growth, with implications for thermalization, universality, and the interpretation of information spreading in many-body quantum systems. They also identify conditions under which equipartition holds (e.g., energy-space locality) and extend the analysis to many-body settings, suggesting broader applicability of symmetry-resolved complexity in quantum dynamics.

Abstract

This work addresses how the growth of invariant operators is influenced by their underlying symmetry structure. For this purpose, we introduce the symmetry-resolved Krylov complexity, which captures the time evolution of each block into which an operator, invariant under a given symmetry, can be decomposed. We find that, at early times, the complexity of the full operator is equal to the average of the symmetry-resolved contributions. At later times, however, the interplay among different charge sectors becomes more intricate. In general, the symmetry-resolved Krylov complexity depends on the charge sector, although in some cases this dependence disappears, leading to a form of Krylov complexity equipartition. Our analysis lays the groundwork for a broader application of symmetry structures in the study of Krylov space complexities with implications for thermalization and universality in many-body quantum systems.

Figures (3)

• Figure 1: Difference between the Krylov complexity of the operator \ref{['eq:operator4times4']} and the average \ref{['eq:weighted_Complexity']} over the (two) sectors. All the curves are obtained for $E_1=8,\,E_2=5\,,E_3=3.4\,,E_4=1$. Different colors correspond to distinct choices of the parameters $A_{ii}$ ($i=1,2,3,4$), $A_{12}$, $A_{34}$, $B_{12}$, $B_{34}$ (see Table I in the Appendix). In all the instances shown here, $C_K(t)-\bar{C}(t)\geq 0$, supporting our surmise.
• Figure 2: Difference between the Krylov complexity of the operator \ref{['eq:operator4times4']} and the average \ref{['eq:weighted_Complexity']} over the (two) sectors. Different colors correspond to distinct choices of the parameters $A_{ii}$ ($i=1,2,3,4$), $A_{12}$, $A_{34}$, $B_{12}$, $B_{34}$ (see Table I). In all the instances shown here, $C_K(t)-\bar{C}(t)\geq 0$, supporting our surmise. Curves in different panels are obtained for different sets of energies: top-left $E_1=8.00,\,E_2=5.00\,,E_3=1.00\,,E_4=1.00$, top-right $E_1=8.00,\,E_2=2.00\,,E_3=2.50\,,E_4=1.00$, bottom-left $E_1=7.10,\,E_2=2.72\,,E_3=3.14\,,E_4=1.73$, bottom-right $E_1=10.50,\,E_2=2.20\,,E_3=3.00\,,E_4=3.30.$
• Figure 3: Krylov complexity \ref{['c4']} and symmetry resolved complexities \ref{['eq:Cplusminus']} for the two charge sectors plotted as functions of time. The curves have been obtained for $E_1=8.0,\,E_2=5.0\,,E_3=3.4\,,E_4=1.0$ and $A_{ij}$ and $B_{ij}$ in the first line of Table I. We observe that, for some values of time, the symmetry-resolved Krylov complexity $C_K^{(+)}(t)$ is larger than the Krylov complexity of the total $4\times 4$ operator.