Krylov Complexity Under Hamiltonian Deformations and Toda Flows

TL;DR

This paper explores Krylov space methods for Hamiltonian deformations, revealing how certain deformations preserve the Krylov subspace structure and lead to generalized Toda equations, with applications to thermodynamics and random matrices.

Contribution

It introduces a systematic approach to construct solvable models via Krylov subspace methods and relates deformed and undeformed theories through generalized Toda equations.

Findings

Krylov subspace remains unchanged under certain deformations.

Generalized Toda equations describe evolution as a function of deformation parameters.

Applications include analysis of Gibbs states, survival probability, and Krylov entropy.

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.