Krylov Winding in Quantum Many-Body Systems

• Krylov winding is defined as the linear phase progression in a Krylov basis generated by the Lanczos algorithm, revealing coherent operator growth in quantum chaotic systems.
• It arises from the operator growth hypothesis and the asymptotically linear Lanczos coefficients, leading to an effective hopping equation with exponential decay in amplitude and uniform phase increase.
• Its mapping to size winding, as demonstrated in models like the SYK and disordered spin systems, underpins practical applications in quantum teleportation and spectral diagnostics.

Krylov winding is a phenomenon in quantum many-body physics where the complex phase of the operator wavefunction, when expanded in the Krylov basis generated by the Lanczos algorithm, grows linearly with Krylov index. This emergent phase coherence reveals subtle structure in the operator growth process underlying quantum chaos, linking the dynamics of operator complexity with thermodynamic and information-theoretic bounds. Krylov winding is closely related to but more general than the previously studied “size winding” in the Pauli string basis. Its realization depends on algebraic, dynamical, and spectral properties of the system, notably the operator growth hypothesis and saturation of the chaos–operator growth (COG) bound.

1. Conceptual Definition and Mathematical Structure

In thermalizing quantum systems, the operator at time $t$ (denoted $O(t)$) can be expanded in an orthonormal Krylov basis $\{|O_n\rangle\}$ built via the Lanczos algorithm from its initial form and the Hamiltonian $H$:

$$|\rho^{1/2} O(t)\rangle = \sum_n \phi_n(t_\beta) |O_n\rangle, \qquad t_\beta = t + i\beta/4$$

Krylov winding manifests when, for large $n$, the expansion coefficient takes the form:

$$\phi_n(t_\beta) = |\phi_n(t_\beta)| e^{i K(t) n + i\theta_0(t)}$$

where $K(t)$ is a winding rate (here, “Krylov momentum”) and $\theta_0(t)$ is an offset phase. This phase grows linearly with the index $n$ in the Krylov basis. Krylov winding thus describes the universal emergence of coherent phase progression in the operator wavefunction as the operator complexity “winds” through increasing Krylov index.

2. Mechanism: Operator Growth Hypothesis and Dynamical Origin

Krylov winding has its microscopic origin in the operator growth hypothesis, which postulates that in quantum chaotic systems the Lanczos coefficients $b_n$ (arising in the recursion)

$$\mathcal{L}|O_n\rangle = b_{n+1} |O_{n+1}\rangle + b_n |O_{n-1}\rangle$$

grow asymptotically linearly:

$b_n \sim \alpha n$

for some growth rate $\alpha$. This linear ramp leads to an effective hopping equation for the Krylov wavefunction, whose formal solution exhibits exponential amplitude decay and linearly growing phase in $n$. For instance, in exactly solvable cases (e.g., $b_n = \sqrt{n(n+2\Delta-1)}$), the solution is

$$\phi_n(t_\beta) \propto \left[\tanh(t_\beta)\right]^n$$

and the phase is determined by $K(t) = \arg[\tanh(t_\beta)]$. Consequently, the operator’s evolution in the Krylov basis encodes the generic mechanism for phase winding.

3. Mapping to Size Winding: Low-Rank Mapping and Chaos-Operator Growth Bound

Size winding refers to coherent phase alignment in the Pauli basis, where Pauli strings of the same operator size $\ell$ all possess the same phase proportional to $\ell$. Krylov winding is a more general phenomenon, becoming equivalent to size winding when two key conditions are met:

1. Low-Rank Krylov-Size Mapping: The overlap matrix $M_{nm}(\ell)$, defined by

$$M_{nm}(\ell) = \langle O_n | P_{(\ell)} | O_m \rangle$$

(with $P_{(\ell)}$ projecting onto operators of size $\ell$), should be approximately rank-one. This ensures that Krylov-wound phases transfer coherently to the Pauli size basis.

1. Saturation of Chaos-Operator Growth Bound: The chaos-operator growth bound relates the Lyapunov exponent $\lambda_L$ of the system to the Lanczos growth rate $\alpha$, via

$h = \frac{\lambda_L}{2\alpha} \leq 1$

When the bound is saturated ($h=1$), the Krylov winding rate induces linear phase growth in size. Otherwise, phase growth becomes superlinear:

$\arg q(\ell, t) \sim \ell^{1/h}$

with $q(\ell,t)$ being the winding distribution in size space.

4. Physical and Quantum Information Implications

Krylov winding reveals emergent coherence in the state space of chaotic quantum systems—even as operators irreversibly scramble and grow in size, a single phase alignment arises out of complexity. When transferred to the operator size basis via the above mapping, it produces the optimal size winding required for certain quantum information tasks. Specifically,

• Quantum Teleportation Protocols: Size winding enables many-body quantum teleportation schemes that depend crucially on the ability to “reverse the slope” of the phase alignment, a concept directly inherited from Krylov winding. Systems that saturate the chaos-operator growth bound (i.e., $h=1$) yield sharply peaked momentum distributions and maximal teleportation fidelity.
• Spectral Diagnostics: The mathematical features of Krylov winding correspond to particular traits of spectral rigidity and long-ramp behavior in Krylov complexity evolution, serving as diagnostic tools for quantum chaos.

5. Model Implementations and Numerical Results

The phenomenon has been illustrated concretely in:

• Sachdev-Ye-Kitaev (SYK) Model: Both analytic and numerical results in SYK (and its large-$q$ variants, including cases coupled to baths) confirm linear phase winding in Krylov basis, resulting in size winding in Pauli basis under suitable mapping conditions.
• Disordered k-Local Spin Model: Numerical studies of Lanczos coefficients and Krylov momentum spectra show sharp phase alignment and coherent winding, with finite-size effects visible as plateaus in the Lanczos coefficients and broadening of momentum peaks.

The mapping between average operator size and Krylov index is one-to-one when $h=1$, and the Fourier transform of the size distribution $q(\ell, t)$ exhibits a sharp peak at momentum $\mu_K$:

$\mu_K(t) = -2\;\arg[\tanh(\alpha t_\beta)]$

For $h<1$, the momentum peak broadens and the phase advances superlinearly, reflecting degraded teleportation efficacy and modified operator growth dynamics.

6. Comparative Analysis and Relation to Other Winding Concepts

Krylov winding is a generic feature across quantum chaotic models. It is distinct from winding numbers in topological band theory or algebraic curve theory, which quantify global topological invariants and may not involve dynamical phase alignment. Its operational realization in chaotic systems contrasts with the algebraic gradation seen in regular homotopy classes or surface Lie algebras, where winding is a static topological grading.

The phenomenon is also robust to the addition of dissipation and decoherence (as explored in inflationary backgrounds and open systems (Li et al., 17 Jan 2024)), suggesting that Krylov winding may play a structural role in operator complexity even for nonunitary or open dynamics.

7. Future Research Directions

Open questions and likely avenues for future investigation include:

• Environmental Effects: Understanding the modification of Krylov winding under non-unitary, dissipative evolution and environmental decoherence.
• Experimental Realization: Developing quantum simulation and measurement protocols that can directly observe phase winding in Krylov and size bases, leveraging platforms capable of accessing operator wavefunctions.
• Extension to Holographic Systems: Exploring the holographic interpretation and potential bulk duals of Krylov winding, especially vis-à-vis plateau and ramp features in complexity growth and information dynamics.
• Finite-Size Scaling: Systematic characterization of finite-size and non-ideal mapping effects on the winding phenomenon, and their impact on quantum information tasks.

These directions emphasize the foundational role Krylov winding plays in the theory of operator growth and its application to both quantum chaos diagnostics and quantum information science.