Krylov Complexity in Free and Interacting Scalar Field Theories with Bounded Power Spectrum

Motivation

• K-complexity measures operator spread in Krylov space and was conjectured to grow exponentially in chaotic finite-dimensional quantum systems, with its rate related to bounds on operator growth.The quantity depends on the operator and Hamiltonian, unlike Nielsen complexity’s additional choices of reference state, cost function, gate set, and penalties.Source: §1 · pp. 3–4
• The cited result could represent a tighter bound than the universal Maldacena–Shenker–Stanford (MSS) chaos bound [19].Source: §1 · p. 3

Research question

• The paper addresses the limited understanding of K-complexity beyond quantum many-body systems, CFTs, and highly symmetric models by studying more general QFTs.This question is motivated by the observation that exponential growth occurs in both chaotic holographic and non-chaotic rational CFTs [41].Source: §1 · p. 4

Approach

• Rather than discretizing the theory, the authors impose a momentum cutoff on finite-temperature continuous QFTs, providing a complementary route to the lattice-based ultraviolet cutoff proposed in.The cutoff is applied to momentum integrals in different spacetime dimensions.Source: §1 · pp. 4–5

Scope

• The analysis covers free massive scalar theories in d dimensions, hard and smooth ultraviolet cutoffs, and interacting four-dimensional scalar theories with relevant or marginally irrelevant deformations.The free-theory study emphasizes the heavy-scalar regime βm ≫ 1, while the interaction study examines cubic and quartic deformations.Source: §1 · p. 5

2 The Lanczos Algorithm

• The Lanczos algorithm maps Heisenberg operator evolution to a particle hopping on a one-dimensional Krylov chain, whose Lanczos coefficients encode operator growth.The Krylov basis is obtained by Gram–Schmidt orthogonalization in the GNS operator Hilbert space with the thermal Wightman inner product.Source: §2 · pp. 6–7
• Operator-growth information is equivalently encoded in the thermal autocorrelation function, its moments, the Wightman power spectrum, and the Lanczos coefficients through a nonlinear moment relation.Krylov complexity is determined by the amplitudes generated from the autocorrelation function and therefore by its moments.Source: §2 · p. 8

2.1 Krylov Complexity and Quantum Chaos

• At finite temperature, thermal poles give ω0 = 2/β and λK = 2π/β for smoothly growing Lanczos coefficients, but this exponential Krylov growth cannot by itself signify QFT chaos.The same analysis connects the Krylov and Lyapunov growth rates to the MSS bound, while nonsmooth Lanczos coefficients prevent a direct λL–λK inequality.Source: §2.1 · pp. 10–11

2.2 Krylov Complexity and Spectral Statistics

• The Wightman power spectrum resolves energy transitions through thermal weights and operator matrix elements, while continuous spectra require density-of-states integrals and depend nontrivially on spectral pair correlations.The discrete spectrum produces delta-function contributions at energy differences, whereas the continuous formulation introduces IR and UV energy bounds and ensemble-averaged densities.Source: §2.2 · pp. 11, 12, 14, 15
• Energy-transfer cutoffs bound the moments and consequently affect Lanczos coefficients and Krylov complexity, with infrared effects visible at small n and ultraviolet effects emerging only at sufficiently large n.For lattice systems, the UV cutoff is tied to the lattice spacing; in the continuum limit, divergent energy-transfer support explains why linear Lanczos growth and exponential Krylov growth are not uniquely chaotic signatures.Source: §2.2 · pp. 12–13
• Energy-transfer cutoffs {ω_I, ω_F} must be distinguished from energy-spectrum bounds {E_IR, E_UV}, which are implicit in the discrete double sum.Evaluating the relevant integral by saddle point is left beyond the discussion, under assumptions that the ω and E integrals can be exchanged and 0 < ω_I < ω_F.Source: §2.2 · p. 16
• Power-spectrum cutoffs influence the Lanczos coefficients b_n and Krylov complexity K_O(t) through the cutoff sensitivity of the moments μ_2n.The analysis focuses on a momentum cutoff in continuous QFT power spectra without averaging over H; a microcanonical construction provides another bounded-spectrum approach [90].Source: §2.2 · p. 16
• The study extends Krylov analysis beyond highly symmetric theories by breaking conformal symmetry with a massive scalar, with the mass acting as an infrared cutoff ω_I ∼ m affecting early-time complexity.It also examines hard ω_F ∼ Λ and smooth exponential ultraviolet modifications motivated by discretized lattice models [74].Source: §2.2 · pp. 16–17
• For interacting theories, perturbative deformations of a four-dimensional massless scalar by gϕ^3 or gϕ^4 are expected to leave different Lanczos-coefficient signatures because they modify ultraviolet physics differently.The hard ultraviolet cutoff is implemented as a momentum cutoff in free QFTs, while interacting theories are treated afterward using perturbation theory.Source: §2.2 · p. 17

3.1 The Wightman Power Spectrum f W(ω)

• For d-dimensional free massive scalar theories at finite temperature, the Wightman power spectrum is expressed in terms of the spectral function and normalized by C(0) = 1.Source: §3.1 · pp. 17–18
• The mass acts as an infrared cutoff for the zero-mode divergence, while ultraviolet behavior is studied by either hard momentum truncation or smooth modification of the power spectrum.The hard cutoff is analyzed first in the bounded-spectrum treatment, with β = 1 used for numerical computations.Source: §3.1 · p. 20

3.2 Krylov Complexity with Unbounded Power Spectrum

• Massive scalar Lanczos coefficients split into smooth even- and odd-n families, with staggering increasing with mβ and separation Δb_n of order O(m).This behavior persists across d > 4; fits also find α_odd and α_even approaching π/β at larger n, while their separation remains proportional to m.Source: §3.2 · pp. 20–22
• Lanczos staggering arises from the combined presence of a pole at ω_0 proportional to 1/β and a spacetime-dimension parameter proportional to d − 3.The same mechanism applies to the asymptotic power spectrum relevant for higher-dimensional CFTs, while staggering disrupts the continuous-coefficient relation between Krylov entropy and complexity.Source: §3.2 · pp. 23, 27
• For nonzero βm, Krylov complexity oscillates with period set by m, while oscillation amplitudes decrease at late times as contributions from many amplitudes cancel.For βm = 0, the late-time exponential growth rate is λ_K ∼ 2π/β; for nonzero βm, the measured finite-time slope decreases as mβ increases, but the true asymptotic rate remains unresolved.Source: §3.2 · pp. 23–25

3.3 Krylov Complexity for Bounded Power Spectrum with Hard UV Cutoff

• A hard UV cutoff Λ is introduced in the d = 5 momentum integral, and the resulting power spectrum and moments reduce to the unbounded expressions as Λ →∞.The corresponding Lanczos coefficients are compared numerically for finite and infinite cutoffs using Λ = 80, m = 20, and β = 1.Source: §3.3 · p. 27, Figure 6 · p. 28
• Lanczos coefficients saturate at b_sat^± = Λ ± m, with both families approaching b_sat = Λ/2 in the massless case.The odd- and even-n families have distinct massive saturation values, while the massless families coincide.Source: §3.3 · p. 28
• The saturation index n_sat scales linearly with Λ, decreases as m approaches Λ, and becomes O(1) when m is of order Λ.The bounded support m ≤ |ω| ≤ Λ explains saturation, while the continuum hard-cutoff theory exhibits behavior associated with a coarse-grained free lattice theory with a ∼ 1/Λ.Source: §3.3 · pp. 28, 32
• Finite Λ changes K_O(t) from exponential to linear growth and makes S_K(t) logarithmic, with oscillations when βm is nonzero.The linear behavior follows Lanczos-coefficient saturation; the logarithmic entropy behavior is shown for finite βΛ.Source: §3.3 · pp. 29–30, Figure 7 · p. 30, Figure 9 · p. 31
• The hard-cutoff continuum theory retains infinitely many degrees of freedom, but its Lanczos saturation and linear K-complexity resemble a large-N free lattice theory with finite spacing a ∼ 1/Λ.Finite-N compactness corrections become important when the lattice circumference aN is comparable to scales such as β.Source: §3.3 · p. 32

3.4 Lanczos Coefficients with a Smooth UV Cutoff

• A smooth UV cutoff changes the asymptotic behavior of b_n as its parameter δ varies, confirming that Lanczos growth tracks the exponential decay of the power spectrum.The original spectrum is recovered as δ → ∞, while δ → 0 gives the hard-cutoff limit.Source: §3.4 · p. 33, Figure 10 · p. 33

4.1 Marginally irrelevant deformation (ℓ= 4)

• The quartic one-loop self-energy is obtained from the Euclidean diagram using Matsubara sums, with the vacuum divergence removed by a mass counter-term.The remaining self-energy is finite-temperature dependent and controlled by βm.Source: §4.1 · p. 34
• Quartic interactions are incorporated through a thermal-mass shift, leaving Lanczos coefficients and Krylov complexity almost indistinguishable from the free theory except for massless staggering.The thermal mass decreases rapidly with βm and generates staggering when m = 0.Source: §4.1 · pp. 34–35

4.2 Relevant deformation (ℓ= 3)

• Cubic interactions modify the spectral power through a one-loop correction proportional to g2, from which the Lanczos coefficients are computed numerically.The calculation uses the cubic one-loop self-energy and analytic continuation to Lorentzian correlators.Source: §4.2 · pp. 35–36
• The interaction-induced staggering δbn decreases with n, unlike the persistent mass-induced separation of odd and even Lanczos coefficients.The decreasing pattern is associated with bounded spectral support and no mass gap in the analyzed setting.Source: §4.2 · p. 36, Figure 14 · p. 37
• Interactions change the asymptotic slopes oppositely: the odd-coefficient slope increases with g2, while the even-coefficient slope decreases.The slopes are extracted by fitting bn = α(g)n + γ(g) separately for odd and even n.Source: §4.2 · p. 37, Figure 15 · p. 37

5 Discussion and Conclusions

• A nonzero mass produces staggering into odd and even Lanczos families at all n, while the exponential growth rate computed over finite times decreases with mass.The staggering persists at large n and may therefore affect late-time Krylov complexity.Source: §5 · p. 38
• The proposed no-staggering conditions are mathematical criteria on the spectral power, whose physical meaning and implications for the auto-correlation remain unclear.The conditions require finite positive fW(0) and a continuous derivative within the ultraviolet cutoff.Source: §5 · p. 38
• A hard ultraviolet cutoff saturates the Lanczos coefficients and changes Krylov complexity from exponential to linear growth.A smooth cutoff is proposed as a way to deform the ultraviolet structure of continuum theories related to lattice models.Source: §5 · p. 38
• The interaction analysis is restricted to numerical Lanczos-coefficient studies of cubic and quartic scalar theories in four-dimensional flat spacetime.The authors propose extending the analysis to other models and to Krylov complexity in generic interacting theories.Source: §5 · pp. 38–39
• For cubic and quartic interactions, staggering occurs for all n, but a mass gap determines whether it remains constant or decreases with n.Quartic interactions generate an effective gap even when m = 0, whereas cubic interactions do not and their staggering decreases with n.Source: §5 · p. 39
• The paper cautions that the finite-time estimate of the Krylov exponential rate does not determine the true asymptotic rate at t →∞.Determining λK requires a semi-analytic treatment of the probability amplitudes and their late-time series.Source: §5 · p. 39

A Analytic expressions for the Lanczos coefficients

• Moments in the large-mass regime are inserted into the nonlinear Lanczos recursion to derive analytic expressions for bn, including separate even-n behavior.The derivation starts from the reproduced large-mass moment formula.Source: A Analytic expressions for the Lanczos… · pp. 40–41
• The analytic expression reproduces numerically computed Lanczos coefficients up to approximately n ≈35.Squaring the expression and multiplying by β2 yields the corresponding asymptotic expansion.Source: A Analytic expressions for the Lanczos… · p. 41

B Auto-correlation Function in Odd-Dimensional Free QFTs

• The thermal Wightman two-point function C(d)(t) is analyzed for free odd-dimensional QFTs with mβ ≫1.The appendix introduces the associated dimension-dependent functions.Source: B Auto-correlation Function in… · p. 41

• The conformal map from R^d to S1×H^(d−1) makes the hyperbolic curvature scale depend on the Euclidean-time period β.Source: C Krylov complexity in CFTs on a… · p. 42
• The scalar conformal two-point function on S1×H^(d−1) is specified using scaling dimension Δ, Euclidean times, hyperbolic spatial distance, and normalization N.Source: C Krylov complexity in CFTs on a… · p. 42
• Setting τ1−τ2 = it + β/2 and the spatial distance to zero yields the thermal Wightman two-point function.Source: C Krylov complexity in CFTs on a… · p. 42
• For Δ = d/2−1, the resulting Krylov complexity agrees with the earlier expression for α = π/β and η = 2Δ, up to normalization.Source: C Krylov complexity in CFTs on a… · p. 43

D Conditions for staggering

• Absence of staggering is associated with f(ω) finite at ω = 0 and f′(ω) continuous throughout −Λ < ω < Λ.Source: D Conditions for staggering · p. 43

Example 1

• Example 1 provides a no-staggering case whose UV cutoff makes the Lanczos coefficients saturate to a constant and improves visual inspection.Source: Example 1 · p. 43, Figure 16 · p. 44, Figure 17 · p. 44

Example 2

• Example 2 shows staggering in both cases, but it decreases with n for h = 1 and remains constant for h = 0.The h = 0 spectrum violates condition (I), while h = 1 satisfies (I) but violates (II) at ω = m.Source: Example 2 · pp. 43–44

Example 3

• Example 3 exhibits staggering for λ = 2, attributed to violation of condition (I) because f(0) = 0.Source: Example 3 · pp. 44–45

Example 4

• Example 4 exhibits staggering with a UV cutoff, attributed to violation of condition (II) because f′(ω) is discontinuous at ω = 0.The cutoff is introduced to make the staggering visible in the Lanczos coefficients.Source: Example 4 · pp. 45–46

• Condition (I) implies that the time integral of C(t) is finite, so C(t) is positive most of the time when f(0) is finite.Source: Implications for the auto-correlation… · p. 46
• The authors do not yet have a concrete physical interpretation of this positivity property or a clear interpretation of condition (II) for C(t).Source: Implications for the auto-correlation… · p. 46
• For the free massive scalar field, the mass gap produces oscillations in C(t), leading to a vanishing time integral and violation of condition (I).Source: Implications for the auto-correlation… · p. 46
• The massive scalar field can violate condition (II) as well, through effects of the mass gap and spacetime dimensionality when d ≠ 4.Source: Implications for the auto-correlation… · pp. 46–47
• For the massless scalar field, staggering is observed for d > 4, consistent with power spectra resembling Example 3.Source: Implications for the auto-correlation… · p. 47, Figure 22 · p. 47, Figure 23 · p. 47