Krylov Complexity Under Hamiltonian Deformations and Toda Flows

TL;DR

This paper develops a universal Krylov-space framework to study Hamiltonian deformations by mapping the problem to a time-dependent, tridiagonal generator $L(\tau)$ whose spectrum remains tied to the original Hamiltonian. The deformation flow is shown to obey (generalized) Toda equations, yielding an integrable description of how the Lanczos coefficients $(a_n,b_n)$ evolve with deformation parameters $\tau$. The authors apply the formalism to coherent Gibbs states, random matrices, and supersymmetric quantum mechanics, deriving exact solutions, fixed points, and saturation behavior for Krylov-based spread complexity and Krylov entropy, and highlighting the connection between dynamical growth and thermodynamics via the Lanczos data. The results provide a versatile toolkit for analyzing operator/state complexity under deformations and offer insight into spectral form factors, phase transitions, and the universal features of complexity growth in chaotic and integrable settings.

Abstract

The quantum dynamics of a complex system can be efficiently described in Krylov space, the minimal subspace in which the dynamics unfolds. We apply the Krylov subspace method for Hamiltonian deformations, which provides a systematic way of constructing solvable models from known instances. In doing so, we relate the evolution of deformed and undeformed theories and investigate their complexity. For a certain class of deformations, the resulting Krylov subspace is unchanged, and we observe time evolutions with a reorganized basis. The tridiagonal form of the generator in the Krylov space is maintained, and we obtain generalized Toda equations as a function of the deformation parameters. The imaginary-time-like evolutions can be described by real-time unitary ones. As possible applications, we discuss coherent Gibbs states for thermodynamic systems, for which we analyze the survival probability, spread complexity, Krylov entropy, and associated time-averaged quantities. We further discuss the statistical properties of random matrices and supersymmetric systems for quadratic deformations.

Figures (15)

• Figure 1: The spread complexity for systems with SL(2,R) symmetry. We set $\theta(0)=0.5$ and show three possible cases: (a) stable solution in Eq. (\ref{['kstable']}), (b) unstable in Eq. (\ref{['kunst']}), (c) marginal in Eq. (\ref{['kmarg']}). The $t$-dependence of the spread complexity for the SU(2) case and for the Heisenberg-Weyl case has the same form as the function in panel (a).
• Figure 2: The spread complexity for systems described by Eqs. (\ref{['alt-1']}) and (\ref{['alt-2']}). We set $d=100$ and consider several values of $\gamma(\tau_2)/\alpha(\tau_2)$. The result at $\alpha=\gamma$ is given by Eq. (\ref{['alt-eq']}).
• Figure 3: The Lanczos coefficients $a_n(\beta)$ and $b_n(\beta)$ of the two-dimensional Ising model. We take a $6\times 5$ lattice with open boundary conditions, which gives the Krylov dimension $d=48$. At $\beta=0$, the diagonal components $a_n(0)$ are zero and the off-diagonal components $\beta_n(0)$ are plotted in the inset of the right panel. The critical point at the thermodynamic limit is given by $\beta J=\frac{1}{2}\ln (\sqrt{2}+1)\approx 0.4407$.
• Figure 4: The Lanczos coefficients $a_n$ and $b_n$ of the fully-connected Ising model with $N=2000$. The Krylov dimension is given by $d=N/2+1=1001$ and we plot the coefficients at intervals of 10. The point $\beta J=1$ represents the critical point at the thermodynamic limit.
• Figure 5: The left panel represents the rate function of the two-dimensional Ising model at the thermodynamic limit. The function is periodic in $t$ with the period $Jt=\pi/2$. The right panel represents the phase diagram obtained from the Lee-Yang zeros.