Krylov complexity and Wightman power spectrum with positive chemical potential in Schrödinger field theory

Abstract

We study Krylov complexity in Schrödinger field theory in the grand canonical ensemble with chemical potential $μ$, with an emphasis on the qualitatively new features that arise for $μ>0$. In this regime the fermionic Wightman power spectrum is effectively single-sided and sharply truncated at $ω=μ$, 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 $π/β$ to an asymptotic slope $2/β$), while $a_n$ bends from near-zero values to a linear descent with slope $-4/β$. 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 $K(t)\sim\sinh^{2}(πt/β)$ behavior at large $μ$; (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.

Figures (7)

• Figure 1: The Lanczos coefficients $\{a_n\}$ (panel (a)) and $\{b_n\}$ (panel (b)) for chemical potentials $\mu = 1, 10, 20, 50, 100, 200$. All numerical moments entering the Lanczos construction are evaluated with the series truncation $\sum_{k=0}^{200}(\cdots)$ in Eq. \ref{['2.3']}. The red dashed line in panel (b) shows the early-stage linear growth $\beta b_n = \pi n$. The insets in panels (a) and (b) show the residuals of the corresponding linear fits, plotted using the numerical fit parameters reported in Table \ref{['tab:lanczos_fits_app']}, with the linear fits performed over the window $301\le n\le 450$. Panels (c) and (d) display the (numerical) local slopes of $\beta a_n$ and $\beta b_n$ as functions of $n$; the red dashed lines at $-4$ and $2$ indicate the expected asymptotic constants.
• Figure 2: Time evolution of the Krylov complexity with chemical potentials $\mu = 1, 10, 20, 50$, $100, 200$. All numerical moments entering the Lanczos construction are evaluated with the series truncation in Eq. \ref{['2.3']}. We plot $1+K(t)$ for numerical convenience. The vertical axis is plotted on a logarithmic scale.
• Figure 3: Comparison of Krylov complexities with different chemical potentials $\mu = 1, 10, 20, 50,$$100$ and $200$. The blue dashed line represents the asymptotic behavior $1+K(t) = 1 + \sinh^2 (\pi t / \beta)$. The curve for the complexity with $\mu = 200$ approaches the asymptotic behavior closely.
• Figure 4: Lanczos coefficients $a_{n}$ and $b_{n}$ for different Wightman power spectra $f^{W}(\omega)$, highlighting the role of frequency cutoffs/truncations. The red and blue curves correspond to a single-sided truncation (fit/truncation window $\omega\le 100$, i.e. $\omega\in(-\infty,100]$), while the black curves correspond to a symmetric two-sided truncation (window $|\omega|\le 100$, i.e. $\omega\in[-100,100]$). All numerical moments are computed with the same series truncation $\sum_{k=0}^{200}(\cdots)$.
• Figure 5: Lanczos coefficients $a_n$ and $b_n$ versus $n$ for various $b$ (with fixed $a=\beta/2$). All numerical moments entering the Lanczos construction are evaluated with the series truncation $\sum_{k=0}^{200}(\cdots)$ like Eq. \ref{['2.3']}.