Operator Growth & Symmetry-Modified Transitions

• Operator growth is the process where simple operators evolve under Heisenberg dynamics, acquiring exponential complexity through spreading over an expanding basis.
• Symmetry-modified transitions adjust the rate and structure of operator evolution, necessitating refined diagnostics such as symmetry-resolved coefficient entropy.
• By leveraging Krylov complexity and group-theoretic methods, researchers can accurately capture the effects of conserved and hidden symmetries on quantum scrambling.

Operator growth—also termed operator spreading—refers to the process by which a simple operator evolves under Heisenberg time evolution, acquiring complexity by developing nontrivial support over an exponentially growing set of operator basis elements. This phenomenon is central to the study of quantum chaos, scrambling, and thermalization in both single-particle and many-body systems. Symmetry-modified transitions occur when underlying symmetries, either explicit or hidden, alter the structure or pace of operator growth, often necessitating refined diagnostics. Recent advances leverage symmetry-resolved entropy measures and Krylov complexity decompositions to faithfully characterize operator spreading in the presence of symmetries.

1. Diagnostics of Operator Growth: Coefficient Entropy and Krylov Complexity

A primary quantitative diagnostic of operator growth is the coefficient (Shannon) entropy, defined for a time-evolved operator $\hat O(t)$ expanded in an orthonormal basis $\{\hat B_j\}$ of operator Hilbert space (dimension $D$): $$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$ Here, $S_{\mathrm{coeff}}$ measures delocalization in operator space, with maximum $\ln D$ signaling maximal complexity typical of chaotic evolution (Nie, 2021).

Krylov complexity provides an alternative through the Lanczos recursive generation of an orthonormal Krylov basis $\{|K_n)\}$ using repeated action of the Liouvillian superoperator $\mathcal L = [H,\cdot]$ on an initial operator. The time-evolved operator is expanded as

$$|O(t)) = \sum_{n=0}^{\infty} (i)^n \varphi_n(t) |K_n),$$

and the associated Krylov complexity is

$K(t) = \sum_{n=0}^{\infty} n\,|\varphi_n(t)|^2.$

The growth of $\{\hat B_j\}$0 and the underlying Lanczos coefficients $\{\hat B_j\}$1 encode the spread of the operator in complexity space. In integrable and finite-dimensional symmetric systems, $\{\hat B_j\}$2 may saturate; in chaotic or infinite-dimensional representations, exponential growth is often observed (Patramanis, 2021, Beetar et al., 2023).

2. Symmetry Masking and Symmetry-Resolved Diagnostics

Symmetries can substantially alter operator growth, often leading standard diagnostics to over- or underestimate complexity. In quantum maps such as the quantum cat map, robust but "hidden" symmetries cause the unitary propagator to be block-diagonal in symmetry sectors. The standard coefficient entropy $\{\hat B_j\}$3, when computed in a generic basis that mixes symmetry sectors, may falsely indicate full operator delocalization and chaos even when the evolution is confined within smaller invariant subspaces (Nie, 2021).

To correct this, one performs a block-diagonalization—symmetry resolution—without requiring explicit knowledge of the symmetry generators. Operator decomposition then proceeds as

$\{\hat B_j\}$4

where each $\{\hat B_j\}$5 projects onto a distinct symmetry sector of dimension $\{\hat B_j\}$6. The symmetry-resolved coefficient entropy in sector $\{\hat B_j\}$7 is

$\{\hat B_j\}$8

This entropy saturates only to $\{\hat B_j\}$9 and provides a faithful, basis-independent measure of operator growth within that sector, properly revealing the effect of symmetries on complexity and on associated transitions (Nie, 2021).

3. Symmetry-Resolved Krylov Complexity and Equipartition Regimes

In many-body systems with global symmetries such as U(1) (conserved charge $D$0), operators decompose into blocks $D$1, each confined to a fixed-charge sector. The Liouvillian and Krylov construction likewise split into independent sectoral recurrences. This leads to sector-resolved Krylov complexities $D$2, as well as the total complexity $D$3 and its sectoral average $D$4. At early times, $D$5 and $D$6 coincide to leading order, reflecting deconfined, independent sector growth. At later times, inter-sectoral correlations emerge, and generally $D$7.

Krylov complexity equipartition—where all sectoral complexities coincide—occurs when the recursion coefficients $D$8, $D$9 are independent of charge sector. This regime characterizes universal block-diagonal spreading; its breakdown signals a symmetry-modified transition, where symmetry sectors contribute unevenly and mixing becomes nontrivial (Caputa et al., 2 Jul 2025).

Examples ranging from finite spin models to oscillator chains illustrate regimes of equipartition and transitions away from it, offering insight into symmetry-constrained scrambling and thermalization (Caputa et al., 2 Jul 2025).

4. Symmetry-Modified Operator Growth in Quantum Circuits

In random circuit models, imposition of symmetries modifies the stochastic process governing operator spreading, particularly the growth of Pauli-string operators. With local 2-site unitaries drawn from ensembles with anti-unitary symmetries (COE, CSE, O(d), Sp(d)), the transition probabilities for operator updates are altered, increasing the weight on "self-maps" and splitting updates into symmetry-distinguished classes.

Consequently, both the butterfly velocity $$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$0 (propagation speed of operator front) and the diffusion constant $$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$1 (front broadening) are suppressed compared to generic $$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$2 circuits. This slows ballistic front propagation and reduces noise due to enhanced step–step anticorrelations. The ranking of symmetric circuit classes for local qubits yields

$$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$3

with the CSE ensemble exhibiting the slowest scrambling (Hunter-Jones, 2018).

These results establish that even minimal models of quantum dynamics are sharply affected by symmetry, emphasizing the need for symmetry-aware diagnostics in interpreting operator growth.

5. Symmetry and Universal Operator Growth Bound Violations

The universal operator growth hypothesis (OGH) posits an upper bound on the growth of Lanczos coefficients, typically linear in the chain index: $$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$4. In theories with extended symmetries such as $$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$5 (spin-3 current) in 2d CFTs, this bound can be violated. The inclusion of $$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$6 generators in the Liouvillian leads to 23 distinct classes of Lanczos coefficients, 10 of which exhibit superlinear—ranging up to quadratic ($$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$7)—growth with descendant level $$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$8.

The mechanism is rooted in the quadratic nature of the commutator $$S_{\mathrm{coeff}}(t) = -\sum_{j=1}^D |c_j(t)|^2 \ln |c_j(t)|^2, \quad \hat O(t) = \sum_{j=1}^D c_j(t) \hat B_j.$$9, which, via partition counting of oscillator descendants, induces multiplicities scaling as high as $S_{\mathrm{coeff}}$0. The associated operator growth features a sequence of crossovers between linear, fractional power, and ultimately quadratic growth plateaus—markedly enriched compared to the traditional Virasoro-only case (Jatkar et al., 2 Jun 2025).

More generally, CFTs with $S_{\mathrm{coeff}}$1 symmetry (spin $S_{\mathrm{coeff}}$2) are expected to exhibit maximal Lanczos coefficient growth as $S_{\mathrm{coeff}}$3, controlled by the highest-spin generator. This extends to any system with a similar extended symmetry structure, necessitating refinements of the OGH (Jatkar et al., 2 Jun 2025).

6. Symmetry-Driven Phase Transitions in Operator Complexity

Operator growth diagnostics such as Krylov complexity reveal and distinguish physical phase transitions controlled by symmetries, including non-Hermitian symmetries such as $S_{\mathrm{coeff}}$4-symmetry. In coupled oscillator models, bounded oscillatory complexity indicates an unbroken $S_{\mathrm{coeff}}$5-symmetric (Rabi) phase, where operator spreading is reversible and confined to finite Krylov subspaces. In contrast, both weak- and ultra-strong-coupling regimes with broken $S_{\mathrm{coeff}}$6-symmetry exhibit unbounded exponential growth in Krylov complexity, signifying irreversible scrambling and information spreading into an unbounded complexity space.

This diagnostic cleanly demarcates dynamical regimes and phase boundaries: real Liouvillian eigenfrequencies (unbroken symmetry) correspond to bounded complexity, while complex eigenfrequencies (broken symmetry) induce divergence (Beetar et al., 2023).

7. Group-Theoretic and Analytical Approaches Leveraging Symmetry

In systems with Lie group symmetries (e.g., SU(1,1) or SU(2)), operator growth and its entanglement diagnostics can be computed exactly, bypassing the need for the Lanczos procedure, by exploiting coherent-state representations and algebraic properties. Group data alone determines wavefunctions, hopping amplitudes, and complexity measures.

For SU(1,1) (chaotic regime), scaling is exponential in time, with linearly growing Lanczos coefficients; for SU(2) (integrable), growth saturates due to finite operator Hilbert space dimension. This correspondence further establishes universality classes of operator growth conditioned by the character of the underlying symmetry (Patramanis, 2021).

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Summary Table of Key Symmetry-Modified Diagnostics

Diagnostic	With Symmetry	Without Symmetry
Coefficient Entropy	Saturates at $S_{\mathrm{coeff}}$7 in each sector; basis independent (Nie, 2021)	Saturates at $S_{\mathrm{coeff}}$8, possibly overestimates complexity
Krylov Complexity Equipartition	Holds if recursion data identical in all sectors (Caputa et al., 2 Jul 2025)	Unpartitioned, universal growth
Lanczos Coefficient Growth	Linear for pure Virasoro; superlinear/quadratic for $S_{\mathrm{coeff}}$9 (Jatkar et al., 2 Jun 2025)	Linear ($\ln D$0)
Butterfly Velocity ($\ln D$1)	Reduced, symmetry-dependent (Hunter-Jones, 2018)	Maximal for $\ln D$2 circuits
Boundedness of Complexity	Phase-dependent; bounded in unbroken symmetry, diverges in broken phases (Beetar et al., 2023)	Phase-independent; generic growth

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Operator growth in the presence of symmetries exhibits qualitatively distinct behaviors compared to symmetry-free cases. Faithful diagnostics require symmetry resolution, both at the level of operator basis entropies and Krylov complexity, to capture true complexity dynamics and symmetry-modified transitions. These methods provide a refined lens for probing chaos, integrability, and universality in quantum systems.