Krylov Complexity in the Schrödinger Field Theory

Abstract: We investigate the Krylov complexity of Schr\"odinger field theories, focusing on both bosonic and fermionic systems within the grand canonical ensemble that includes a chemical potential. Krylov complexity measures operator growth in quantum systems by analyzing how operators spread within the Krylov space, a subspace of the Hilbert space spanned by successive applications of the superoperator $[H,\cdot]$ on an initial operator. Using the Lanczos algorithm, we construct an orthonormal Krylov basis and derive the Lanczos coefficients, which govern the operator connectivity and thus characterize the complexity. Our study reveals that the Lanczos coefficients ${b_{n}}$ are independent of the chemical potential, while ${a_{n}}$ exhibits a dependence on it. Both ${a_{n}}$ and ${b_{n}}$ show linear relationships with respect to $n$. For both bosonic and fermionic systems, the Krylov complexities behave similarly over time, especially at late times, due to the analogous profiles of the squared absolute values of their autocorrelation functions $\abs{\varphi_{0}(t)}^{2}$. The Krylov complexity grows exponentially with time, but its asymptotic scaling factor $\lambda_{K}$ is significantly smaller than the twice of the slope of the ${b_{n}}$ coefficients, contrasting to the relativistic field theories where the scaling aligns more closely with the twice of the slope of ${b_{n}}$.