Mesons, Confinement, and Complexity: A Krylov Space Perspective on the Long-Range Ising Chain

Abstract

We investigate the interplay between confinement and Krylov complexity in the long-range transverse-field Ising chain, where power-law interactions $\sim 1/r^α$ generate an effective confining potential between domain walls without an explicit symmetry-breaking field. Using the time-averaged Krylov state complexity after a quench as a diagnostic, we find that the complexity is strongly suppressed and only weakly system size-dependent at small $α$ and small transverse field $h_x$, where confinement is strongest, and crosses over to rapid growth at larger $α$, where it increases with system size. Within the confined regime, the oscillation frequencies of the complexity agree with the meson masses of an effective two-kink model, with the dominant frequency growing linearly with the lightest meson mass and the time-averaged complexity decreasing approximately as its inverse square. We further study operator Krylov complexity at finite temperature using the Wightman inner product. At low temperature, it exhibits bounded oscillations that directly encode the meson spectrum, whereas this structure is progressively washed out as the temperature increases. These results identify Krylov complexity as a spectroscopic probe of interaction-induced confinement and reproduce, in a microscopic lattice model, the hallmarks of Krylov complexity identified in confining field theories.