Symmetry-resolved Krylov Complexity and the Uncoloured Tensor Model

Symmetry-Resolved Krylov Complexity in the Uncoloured Tensor Model

Overview

"Symmetry-resolved Krylov Complexity and the Uncoloured Tensor Model" (2604.05630) presents an in-depth investigation of the interplay between symmetries and operator growth, as quantified by Krylov complexity (K-complexity), with a focus on the Uncoloured Tensor Model (UTM)—a non-disordered, highly symmetric cousin of the Sachdev-Ye-Kitaev (SYK) model. The work establishes general algebraic criteria for when symmetry-resolved Krylov complexity in a charge sector coincides with the full Krylov complexity of an operator, and provides a detailed numerical study of K-complexity and its symmetry-resolved variants in the UTM, highlighting cases of both equipartition and non-equipartition behavior among charge sectors.

Krylov Complexity and Symmetry Resolution

Krylov complexity (KC) quantifies the growth of operator complexity under time evolution, specifically capturing how an initially simple operator spreads across increasingly complex nested commutators with the system Hamiltonian via the Lanczos algorithm. The growth profile of the resulting Lanczos coefficients and the time evolution of the KC are tightly linked to quantum chaotic dynamics.

The paper formalizes symmetry-resolved KC by utilizing the presence of a conserved charge $Q$, decomposing the Hilbert space into charge sectors. For an operator $O$ commuting with $Q$ but generally not $H$, the authors define projectors to construct block-diagonal forms of both Hamiltonian and operator in the $Q$-eigenbasis, allowing KC to be computed within these subspaces (i.e., KC restricted to charges $q$). The average over charge sectors recovers the full system's KC at early times, but late-time behavior is nontrivial due to mixing between sectors.

A key contribution is the identification of stringent necessary and sufficient algebraic criteria for "equipartition": when the symmetry-resolved KC in a given charge sector equals the full KC for an invariant operator. These involve detailed relations between the operator’s projections in the combined eigenbases of the Hamiltonian and the conserved charge. The derived conditions require not only the invariance of the operator but also a matching of spectral data (explicitly, the ratios of squared projections in energy and charge subspaces must be independent of the energy difference class), which are shown to be highly restrictive.

Krylov Complexity in the Uncoloured Tensor Model

The UTM, particularly the $n=3$, $d=3$ case, serves as the primary physical system for explicit analysis. The UTM consists of $N=n^D$ Majorana fermions without disorder, with a symmetry group $O(n)^D$. The study implements the Hamiltonian in terms of gamma matrices, yielding a Hilbert space of dimension $O$0 (here $O$1). Despite the large dimension ($O$2), the spectrum exhibits profound degeneracy: only $O$3 distinct energy eigenvalues and $O$4 Liouvillian eigenvalues are found, reflecting the high degree of underlying symmetry.

The numerically determined Lanczos coefficients for simple fermionic operators display initial linear growth—a hallmark of quantum chaotic systems—followed by a plateau and decay, consistent with expectations for chaotic systems with finite Hilbert space dimension. The resulting KC shows a two-stage behavior: an initial exponential rise followed by a quasi-linear regime before expected saturation (which cannot be accessed due to numerical instability for highly degenerate spectra).

At finite temperatures, lowering $O$5 slows both the growth of Lanczos coefficients and K-complexity, and the onset of finite-size plateau effects occurs at larger Krylov indices, as expected from other analyses of many-body chaos.

Symmetry-Resolved Krylov Complexity: Equipartition and Beyond

The UTM, with its abundance of both discrete and continuous (Noetherian) symmetries, offers a fertile setting for studying symmetry-resolved K-complexity. The paper meticulously analyzes several scenarios:

• Discrete Symmetries: Operators such as $O$6 (a product of all gamma matrices) and $O$7 (product over a subset) both commute with $O$8 and with fermion operators of interest. In these cases, the symmetry sectors have equal dimension and satisfy the algebraic equipartition criterion. The KC (and Lanczos coefficients) computed in any symmetry sector exactly match the full KC—verified numerically up to the limits imposed by the stability of the Lanczos procedure.
• Combined Discrete Charges: When constructing charge operators as sums of independent discrete charges (e.g., $O$9), the charge sectors need not be equal in dimension. The equipartition conditions are still satisfied: each block’s KC matches the full KC, and the block sizes are reflected in the probabilistic weights.
• Noether Charges: For continuous $Q$0 charges (e.g., $Q$1), equipartition fails in general. Charge sectors may have distinct spectra, violating the necessary spectral matching conditions. The KC in each charge sector no longer coincides with the full KC, and, notably, some sectors exceed the full KC. Nonetheless, the average over all charge sectors remains strictly bounded above by the KC of the full operator, consistent with conjectures from previous work [9v9v-54zv_blockdiagKrylov].

Numerical Challenges and Methodology

The study underscores substantial technical challenges inherent in computing Krylov complexity for large, degenerate systems—most notably, the instability of the Lanczos algorithm in the presence of high degeneracy, where numerical error can result in leakage out of the canonical Krylov space. The authors employ both Full Orthogonalization (FO) and Partial Re-Orthogonalization (PRO) variants of the Lanczos algorithm but emphasize that for highly degenerate Liouvillian spectra, distinct from random matrix-like systems, instabilities become prohibitive long before KC saturation is achieved.

Implications and Outlook

This paper clarifies the highly restrictive nature of equipartition for symmetry-resolved Krylov complexity: it occurs not for all invariant operators, but precisely for those whose spectral projections satisfy specific algebraic constraints. This result has important implications for computational studies of operator dynamics in strongly correlated and highly symmetric quantum systems: efficient use of symmetry decomposition to access KC is generally possible, but care must be taken to ensure the algebraic criteria are met.

The strong chaos signature detected in the UTM via KC—despite its massive spectral degeneracy—demonstrates the diagnostic power of K-complexity in systems far from classical random matrix universality. The positive semi-definiteness of the difference between the full KC and its symmetry-resolved average is verified numerically, extending earlier theoretical predictions.

Potential Future Directions

Future extensions could include:

• Infinite-Dimensional Systems: Generalizing the equipartition conditions to continuous spectra or to infinite-dimensional Hilbert spaces, as encountered in quantum field theory or integrable models.
• Algorithmic Innovations: Developing Lanczos-type algorithms tailored to highly degenerate spectra to extend KC calculations to longer timescales.
• Operator Subselection: Constructing operators in symmetry subspaces that can efficiently represent dynamics of full-space operators, potentially enabling simulations of even larger systems.

Conclusion

This work provides a rigorous formal and numerical treatment of symmetry-resolved Krylov complexity, illuminating the precise circumstances under which block-symmetry reductions are valid for operator complexity growth, and charting the associated computational and physical landscape in the Uncoloured Tensor Model. The results are directly applicable to quantum chaos diagnostics in highly symmetric quantum systems and offer a framework for exploiting symmetries in many-body operator dynamics.