Quantum chaos and late-time equipartition of symmetry-resolved Krylov complexity

Technical Summary: Quantum Chaos and Late-Time Equipartition of Symmetry-Resolved Krylov Complexity

Problem Statement
Krylov complexity ($C_K$) has emerged as a diagnostic for operator growth in quantum many-body systems, typically characterized by an initial exponential growth in chaotic systems followed by a linear regime and eventual saturation. However, the interpretation of the late-time saturation value is complicated by the presence of exact symmetries. In systems where the Hamiltonian $H$ and the initial seed operator $O$ commute with a conserved charge $P$, the Hilbert space decomposes into independent symmetry sectors. While it is established that at early times the full Krylov complexity relates to a weighted average of symmetry-resolved complexities (with weights determined by the initial operator's projection), the relationship between the full and resolved complexities at late times (after saturation) remained unclear. A simple expectation might suggest that the saturation value scales with the square of the total Hilbert space dimension ($D^2$), but the presence of symmetries could alter this scaling, potentially obscuring the diagnostic power of $C_K$ for quantum chaos if not properly resolved.

Methodology
The authors employ a combination of analytic derivation and numerical simulation to investigate the late-time behavior of symmetry-resolved Krylov complexity in finite-dimensional chaotic systems.

• Analytic Framework: The study utilizes the Krylov space formalism generated by the Liouvillian superoperator $\mathcal{L} = [H, \cdot]$. By decomposing the Hilbert space $\mathcal{H}$ into symmetry sectors $\mathcal{H}_q$ with dimensions $d_q$, the authors analyze the dimension of the accessible Krylov space within each sector ($K_q$) and the full unresolved space ($K$). They assume that for generic chaotic Hamiltonians, the energy differences are non-degenerate (except for the universal zero mode) and that the seed operator has non-zero overlap with all symmetry-allowed Liouvillian frequencies. Under these assumptions, they derive the dimensions of the Krylov spaces and apply the hypothesis of late-time delocalization, where the Krylov wavefunction becomes uniformly distributed over the accessible Krylov chain.
• Numerical Implementation: The authors implement the Lanczos algorithm (using Arnoldi iteration for numerical stability) to compute Krylov complexity. They test their analytic predictions on four distinct models:
  1. Real SYK Model: A model of Majorana fermions with $Z_2$ fermion-parity symmetry.
  2. Complex SYK Model: A model of complex fermions with global $U(1)$ charge conservation.
  3. Bosonic Spin Model: A chaotic quadratic spin model with $Z_2 \times Z_2$ symmetries.
  4. Mixed-Field Ising Chain: A non-integrable spin chain with reflection symmetry.
     They also analyze an integrable limit of the Ising chain to contrast chaotic and integrable behaviors.

Key Contributions and Results
The paper establishes that in the saturation regime ($t_{sat} \lesssim t \ll t_{rec}$), the full Krylov complexity becomes additive over symmetry sectors:
$$C_K(t) \simeq \sum_q C_K^{(q)}(t)$$
Crucially, the fractional contribution of each sector is not determined by the initial projection weights (as in the early-time regime) nor by equal sharing, but by the dimensions of the dynamically accessible operator spaces. Specifically, the late-time fractional complexity is governed by:
$$\frac{C_K^{(q)}}{C_K} \simeq \frac{d_q(d_q - 1)}{\sum_{q'} d_{q'}(d_{q'} - 1)} \approx \frac{d_q^2}{\sum_{q'} d_{q'}^2}$$
where $d_q$ is the Hilbert-space dimension of sector $q$. This phenomenon is termed dimension-weighted equipartition.

• Numerical Verification:
  • In the Real SYK model ($N=16$), the two parity sectors have equal dimensions ($d_+ = d_-$), resulting in an equal split ($50\%$) of the saturation complexity, consistent with the $d_q^2$ scaling.
  • In the Complex SYK model ($N=7, 8$), charge sectors have unequal dimensions ($d_q = \binom{N}{q}$). The numerical results show that sectors with larger $d_q$ contribute a larger fraction of the total complexity, precisely following the $d_q^2$ weighting.
  • The Bosonic spin model ($N=6$) with four equal sectors confirms the equal split ($25\%$).
  • The Mixed-Field Ising chain ($N=8$) features reflection sectors with unequal dimensions ($d_+ = 136, d_- = 120$). The saturation values deviate from $50/50$ and instead match the predicted ratio $d_+^2 / (d_+^2 + d_-^2) \approx 0.56$.
• Integrable Contrast: In the integrable limit of the Ising chain, the authors demonstrate that Liouvillian degeneracies and selection rules reduce the true Krylov dimensions ($K_q$) far below the generic chaotic bounds ($d_q(d_q-1)+1$). Consequently, the late-time fractional complexity saturates according to the ratio of the true Krylov dimensions $(K_q-1)/(K_{full}-1)$ rather than the $d_q^2$ scaling, highlighting that the dimension-weighted rule is specific to chaotic dynamics where the Krylov space is maximally explored.

Significance
The paper argues that resolving exact symmetries is essential for correctly interpreting the saturation value of Krylov complexity as a diagnostic of chaotic operator growth. Without symmetry resolution, one might incorrectly assume the saturation scales with the square of the total Hilbert space dimension ($D^2$). Instead, the authors show that for chaotic systems, the saturation scales with the sum of the squares of the sector dimensions ($\sum d_q^2$), which is parametrically smaller than $D^2$ if the symmetry sectors are unequal.

The work provides a "simple generic relation" between full and symmetry-resolved Krylov complexities in chaotic systems, distinguishing the late-time regime (governed by accessible operator space dimensions) from the early-time regime (governed by initial state projections). This clarifies the interpretation of Krylov complexity saturation, ensuring that the diagnostic reflects the true extent of operator growth within the dynamically accessible subspaces rather than being obscured by symmetry-induced block structures. The findings suggest that symmetry resolution acts as a refinement of the saturation diagnostic, offering a sharper characterization of chaotic dynamics.