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.

The paper asks whether Krylov complexity detects domain-wall confinement generated by long-range interactions, rather than a longitudinal field, and how temperature affects its signatures. It studies the transverse-field Ising chain with couplings $J/r^\alpha$, periodic boundaries, and $\alpha>1$, using finite-system spectral calculations and Lanczos evolution. Meson energies are estimated independently through a finite-ring two-kink approximation. Following a quench from a fully $z$-polarized state, state Krylov complexity is suppressed by up to two orders of magnitude at small transverse field and $\alpha<2$, where the two-kink potential grows without bound. Calculations for $L=12,14,16$ show weak size dependence there and increasing complexity with size beyond a crossover at larger $\alpha$. Fourier peaks track the lowest meson masses. In the strongly confined, weak-quench regime, a two-state approximation gives $\omega_d\simeq m_1$ and $\overline{C_{KS}}\simeq2Lh_x^2/m_1^2$, consistent with the numerical trends. For a connected domain-wall-number operator, finite-temperature calculations using the Wightman inner product at $L=13$ find low-temperature oscillations, staggered Lanczos coefficients, and spectral peaks at meson masses and combinations. A single-mode spectrum gives $C_{KO}(t)=\sin^2(m_1t)$; increasing temperature washes out these features and produces approximately linear growth over the simulated window. The state-complexity average uses $t\in[0,200]$ to avoid finite-size tunneling, but tunneling contaminates part of the crossover region. Operator calculations retain at most 5,000 Krylov vectors. The temperature dependence is a spectral-weight crossover, not an established thermodynamic transition; single-mode oscillations are not specific to confinement. Reconstructing long Lanczos sequences from noisy measurements remains ill-conditioned.