Krylov complexity and Berry phase in quantum adiabatic dynamics

Abstract: The connection between Krylov complexity and Berry phase in adiabatic dynamics is investigated under the instantaneous eigenstate basis of a slowly evolving spin system. Adiabatic dynamics force the Krylov complexity to vanish if the initial Krylov basis is an instantaneous eigenstate of the Hamiltonian. Nevertheless, we demonstrate that the Krylov complexity will be nonvanishing if the initial Krylov basis is a superposition state rather than an eigenstate. Time evolution of Krylov complexity will behave periodically or quasi-periodically depending on the controlling parameters. In particular, for a single qubit system with constant parameters, the Krylov complexity will oscillate harmonically in time with the frequency relevant to the combination of the strength of the external field and the geometric Berry phase.

• The paper establishes that a qubit’s Krylov-complexity frequency is ν = h − 2γ_B/T, combining the dynamical field strength with a Berry-phase contribution for closed adiabatic loops.
• The paper shows that nontrivial complexity requires a superposition of instantaneous energy eigenstates, producing a two-dimensional Krylov space, while an eigenstate reference makes the complexity vanish.
• The paper derives K(t) = sin²α sin²([h + φ̇(1 − cosθ)]t/2) for constant parameters and identifies quasi-periodic behavior under selected time-dependent drives, while leaving higher-dimensional and nonunitary cases open.

Summary of the paper

This paper establishes a quantitative connection between Krylov (spread) complexity and the geometric Berry phase in adiabatic quantum dynamics. Working within the $(t,t')$ formalism for time-dependent Hamiltonians (2608.14325), the authors analyze a single spin system with Hamiltonian $\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$ and show that the oscillation frequency of Krylov complexity in a single qubit is $\nu = h + \Omega/T = h - 2\gamma_B/T$, where $\Omega$ is the solid angle traced by the field direction and $\gamma_B$ the Berry phase. To the authors' knowledge, the dependence of Krylov-complexity dynamics on a geometric phase had not been previously identified.

Krylov complexity in adiabatic dynamics

The central observation concerns the choice of the initial Krylov basis. If the initial basis state is taken to be an instantaneous eigenstate $|m(t)\rangle$ of $\hat H(t)$, the generator $(\hat H(t)-i\partial_t)$ annihilates the subspace beyond $|K_0\rangle$: one finds $b_1=0$, the Krylov space is trivially one-dimensional, and the spread complexity vanishes identically. This reflects the absence of inter-level transitions in adiabatic evolution. The paper's key move is to instead choose a time-dependent superposition of two instantaneous eigenstates,

$\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$0

with $\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$1. The Lanczos recursion then terminates at $\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$2, yielding a two-dimensional Krylov subspace with nontrivial coefficients $\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$3 that depend explicitly on the Berry connections $\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$4 and $\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$5, the mixing angle $\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$6, and its derivatives. This is the structural result on which all subsequent dynamics rests: adiabaticity does not force Krylov complexity to vanish once the reference state is not an energy eigenstate.

Dynamical regimes of the qubit

For a spin-$\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$7 system, the time evolution of $\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$8 exhibits qualitatively distinct behavior depending on the control parameters. With constant parameters (constant $\hat H(t)=h(t)\hat{\bf S}\cdot{\bf n}(t)$9, $\nu = h + \Omega/T = h - 2\gamma_B/T$0, $\nu = h + \Omega/T = h - 2\gamma_B/T$1, $\nu = h + \Omega/T = h - 2\gamma_B/T$2, and constant angular velocity $\nu = h + \Omega/T = h - 2\gamma_B/T$3), the complexity oscillates harmonically. Numerically, making $\nu = h + \Omega/T = h - 2\gamma_B/T$4 or $\nu = h + \Omega/T = h - 2\gamma_B/T$5 produces progressively denser oscillations, indicating that $\nu = h + \Omega/T = h - 2\gamma_B/T$6 and $\nu = h + \Omega/T = h - 2\gamma_B/T$7 enter the frequency on an equal footing — a fact confirmed analytically below. A linear-in-time $\nu = h + \Omega/T = h - 2\gamma_B/T$8 destroys periodicity and renders the evolution quasi-periodic, whereas a linear $\nu = h + \Omega/T = h - 2\gamma_B/T$9 preserves the periodic behavior. These regimes are consistent with the general expectation that autonomous systems support periodic Krylov dynamics while time-dependent parameters induce quasi-periodicity.

Closed-form result and the Berry phase

To solve the dynamics analytically, the authors remove the diagonal Lanczos coefficients via the phase redefinition $\Omega$0 and work in the Heisenberg picture. After rescaling, the operators $\Omega$1, $\Omega$2, $\Omega$3 close an exact $\Omega$4 algebra, reducing the Heisenberg equations to a linear system with frequency $\Omega$5, where $\Omega$6. For constant parameters the solution yields the closed form

$\Omega$7

which matches the numerics exactly in the constant-parameter case. The oscillation frequency is therefore

$\Omega$8

When the field precesses at constant polar angle $\Omega$9 through a closed loop of period $\gamma_B$0, the second term becomes the solid angle per period, $\gamma_B$1, and since $\gamma_B$2 for a spin-$\gamma_B$3,

$\gamma_B$4

The $\gamma_B$5-dependent part is directly analogous to Larmor precession; the additive geometric contribution $\gamma_B$6 is the paper's principal new result. Two implications follow immediately. First, Krylov complexity provides a dynamical probe of a topologically invariant quantity: the oscillation frequency encodes the Berry curvature flux through the loop in parameter space. Second, the frequency is measurable in principle through the complexity dynamics, suggesting a route to geometric-phase detection via state-spreading observables. It should be noted that the analytic formula holds only for constant control parameters; the periodic and quasi-periodic regimes with time-dependent $\gamma_B$7, $\gamma_B$8, $\gamma_B$9, or $|m(t)\rangle$0 are characterized only numerically.

Limitations and open questions

The construction is restricted to two-dimensional Krylov subspaces generated by a two-level superposition; the authors do not address whether analogous Berry-phase signatures survive in higher-spin systems or in multi-dimensional Krylov spaces where the recursion does not terminate at $|m(t)\rangle$1. The analytic frequency relation relies on constant parameters, so its robustness under slow but non-constant driving — where the adiabatic theorem itself requires careful control of nonadiabatic corrections — remains open. The paper also leaves the extension to other geometric-phase settings, notably the Aharonov–Bohm effect, explicitly to future work, and does not examine whether the $|m(t)\rangle$2-dependent frequency shift survives in open or nonunitary adiabatic dynamics.

Conclusion

This work demonstrates that Krylov complexity in adiabatic spin dynamics is nontrivial precisely when the Krylov basis is a superposition rather than an eigenstate, and that for a driven qubit the complexity oscillates harmonically with a frequency containing an explicit Berry-phase contribution, $|m(t)\rangle$3. The result ties a state-spreading diagnostic to a geometric, topology-dependent phase within an exactly solvable two-dimensional Krylov subspace, and identifies the conditions — constant parameters, closed loops in parameter space — under which the relation holds exactly.