Krylov complexity and Wightman power spectrum with positive chemical potential in Schr\"odinger field theory

Abstract

We study Krylov complexity in Schr\"odinger field theory in the grand canonical ensemble with chemical potential $\mu$, with an emphasis on the qualitatively new features that arise for $\mu>0$. In this regime the fermionic Wightman power spectrum is effectively single-sided and sharply truncated at $\omega=\mu$, which induces a crossover in the Lanczos coefficients {and signals a dynamical transition from a bulk-dominated regime to a spectral-edge-dominated regime}: $b_n$ displays a two-stage linear growth (from an early-time slope $\pi/\beta$ to an asymptotic slope $2/\beta$), while $a_n$ bends from near-zero values to a linear descent with slope $-4/\beta$. We provide analytic support for the resulting complexity growth from three complementary viewpoints: (i) using an $SL(2,\mathbb{R})$ algebraic construction matched to the asymptotic Lanczos data, we show that the late-time Krylov complexity must grow quadratically, $K(t)\propto t^{2}$; (ii) by analyzing engineered Wightman spectra with controlled decay and truncation, we identify single-sided exponential decay as the key spectral feature responsible for the quadratic asymptotics, while an approximately even two-sided exponential spectrum explains the early-time $K(t)\sim\sinh^{2}(\pi t/\beta)$ behavior at large $\mu$; (iii) we formulate the problem in terms of orthogonal polynomials and estimate the crossover scale separating the early- and late-stage regimes. Overall, our results help clarify the role of chemical potential and spectral truncation in shaping operator growth and Krylov complexity in this non-relativistic quantum field theory setting.