Krylov complexity and orthogonal polynomials

TL;DR

The paper provides a pedagogical, analytically focused study of Krylov complexity, elucidating its construction via the Lanczos recursion and its deep ties to orthogonal polynomials and spectral measures. It develops a generating-function framework for Krylov complexity, derives generic growth features from spectrum type, and delivers explicit, solvable examples using classical and Hahn-class polynomials, plus special cases with gapped or unbounded spectra. By linking operator growth to tight-binding dynamics on the Krylov chain and to functions of the second kind, the work offers exact expressions and continuum-limit insights across a wide class of models. It also surveys the role of different operator inner products (Frobenius, thermal, microcanonical) in shaping Krylov growth, highlighting both practical calculations and conceptual implications for chaos diagnostics and scrambling in quantum systems.

Abstract

Krylov complexity measures operator growth with respect to a basis, which is adapted to the Heisenberg time evolution. The construction of that basis relies on the Lanczos algorithm, also known as the recursion method. The mathematics of Krylov complexity can be described in terms of orthogonal polynomials. We provide a pedagogical introduction to the subject and work out analytically a number of examples involving the classical orthogonal polynomials, polynomials of the Hahn class, and the Tricomi-Carlitz polynomials.

Figures (9)

• Figure 1: The characteristic curves corresponding to the Lanczos coefficients \ref{['cont:Gegenbauer.b']}.
• Figure 2: The characteristic curves corresponding to the Lanczos coefficients \ref{['cont:Tricomi.b']}.
• Figure 3: Complexity \ref{['ex:geg.K2']} as a function of time for a variety of values of $\beta$. Note the different scales.
• Figure 4: Two examples of sequences $\{\Delta_n\}$ that illustrate the generic behaviour $\Delta_o<\Delta_e$ with parameters $\omega_+=1$, $\omega_-=0.2$. The small dots depict $\Delta_{2n}$, the larger dots $\Delta_{2n+1}$. The recursion starts close to the unstable limit solution at $n=99$ and is performed forwards and backwards. Left: $\alpha=\beta=1/2$, $\Delta_1=0.36$, which corresponds to the special case discussed in subsection \ref{['ex:gap']}. In this case, the numerical error is sufficient to drive the cross-over to the stable solution. Right: $\alpha=-0.3$, $\beta=0.8$, $\Delta_1=0.39312029$.
• Figure 5: Two examples of sequences $\{\Delta_n\}$ that illustrate the convergence towards \ref{['ex:lag.large.n']} for $\omega_-=0$, $\kappa=1$. The small dots depict $\Delta_{2n}$, the larger dots $\Delta_{2n+1}$. Left: $\alpha=3$, $\Delta_1=2.999$. Right: $\alpha=7.5$, $\Delta_1=7.4999$.