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The paper shows that in the N=2 supersymmetric SYK model, the mechanism of supersymmetry breaking—an irrelevant deformation down to N=1 versus a mass deformation down to N=0—controls whether late-time Krylov complexity fills roughly half th

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2026-08-03 20:10 UTC pith:EXS6Z5GY

load-bearing objection Solid numerics and a new N=2 SYK data point, but the headline claim about the mass deformation rests on normalizing by an upper bound, and the paper's own termination lengths reverse the trend. the 2 major comments →

arxiv 2511.20769 v3 pith:EXS6Z5GY submitted 2025-11-25 hep-th cond-mat.str-el

Krylov Complexity of Supersymmetric SYK Models

James Chryssanthacopoulos , David Vegh This is my paper
classification hep-th cond-mat.str-el
keywords Krylov complexityN=2 supersymmetric SYKsupersymmetry breakingLanczos algorithmoperator growthquantum chaosintegrable deformationBPS degeneracy
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper asks how breaking supersymmetry changes the growth of operators in the N=2 supersymmetric Sachdev-Ye-Kitaev (SYK) model, a random-interaction quantum system used as a toy model of black holes. It compares two deformations: an irrelevant 'UV' deformation that reduces N=2 to N=1 by altering the fermionic algebra, and a mass deformation that adds a free quadratic term and breaks supersymmetry directly to N=0. Using Krylov subspace methods at finite system size, the paper finds that both deformations enlarge the Krylov space and increase the absolute late-time saturation complexity, but the relative measure splits: the UV deformation pushes the saturation complexity to roughly half the Krylov dimension, while the mass deformation lowers it to about 8 percent of the bound over the intermediate range studied. The paper reads this as a signature of the symmetry-breaking mechanism—chaos-preserving in one case, integrability-drifting in the other.

Core claim

On the paper's own terms, the central claim is that the impact of supersymmetry breaking on Krylov complexity depends on the specific mechanism of symmetry breaking. In numerical results for the hopping operator at N=6, the undeformed N=2 model has a saturation fraction K_sat/D_max^K of 0.14; the irrelevant deformation (epsilon = 0.2–0.6) raises the fraction to 0.49–0.51, while the mass deformation (epsilon = 1–3) lowers it to 0.08 even though K_sat itself rises from 8.14 to roughly 340. The mechanism behind the contrast is the way each deformation lifts the BPS (zero-energy) degeneracy: the mass deformation completely lifts degeneracy, producing a much larger Krylov bound, whereas the irrel

What carries the argument

The central object is the Krylov space generated by repeatedly applying the Liouvillian (commutator with the Hamiltonian) to an initial operator, here mainly the hopping operator. The Lanczos algorithm produces the orthonormal Krylov basis and the Lanczos coefficients b_n, which turn operator evolution into a one-dimensional hopping problem; Krylov complexity K(t) is the average position on that chain. The paper's main diagnostic is the saturation fraction K_frac_sat = K_sat / D_max^K, where D_max^K is an upper bound on the Krylov dimension estimated from the average energy degeneracy via D_max^K ~= d/d_E (d/d_E - 1) + 1. The deformation changes the spectrum and degeneracy, which changes the

Load-bearing premise

The paper's central comparison uses D_max^K, an estimated upper bound on the Krylov dimension derived from average energy degeneracy, rather than the actual dimension at which the Lanczos sequence terminates; if the actual chain lengths are used, the mass deformation no longer appears to lower the saturation fraction.

What would settle it

Recompute K_frac_sat using the actual Krylov dimension from where the Lanczos sequence terminates. Using the paper's own N=6 hopping-operator results—chain lengths of about 33 (undeformed), 239 (UV), and 859 (mass)—the saturation fractions become roughly 0.25, 0.51, and 0.40; the mass-deformed value then sits above the undeformed value, which would settle that the claimed decrease is an artifact of the normalization.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Both deformations increase the absolute late-time saturation complexity and delay the Heisenberg time compared with the undeformed model, so supersymmetry breaking does not simply reduce operator growth.
  • The irrelevant deformation raises the saturation fraction to about half the estimated Krylov bound across epsilon=0.2–0.6, while the mass deformation holds it near 0.08 over epsilon=1–3, so the ratio distinguishes the two breaking mechanisms.
  • Early-time Krylov complexity grows quadratically and then linearly, not exponentially; the deformation raises both the quadratic coefficient and the ballistic velocity, with the UV deformation's growth rate tracking gamma=(1+epsilon^2)/(1-epsilon^2)^2 and the mass deformation's growing roughly linearly in epsilon.
  • Level statistics track the same split: the UV deformation increases the average gap ratio toward chaotic values, whereas the mass deformation pushes it toward the integrable, Poisson-like value. The paper connects the mass-deformation behavior to a drift toward integrability.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • A direct check the paper leaves implicit: recompute the saturation fraction with the actual Lanczos chain lengths rather than the degeneracy-based bound. The paper's N=6 data give chain lengths of about 33 (undeformed), 239 (UV), and 859 (mass); those numbers put the mass-deformed fraction near 0.40, not 0.08, so the claimed contrast is sensitive to this normalization choice.
  • If the ratio contrast survives at larger N, it would give a practical diagnostic: the fraction of Krylov space filled at saturation could be used to tell whether a given deformation preserves a residual supersymmetry or breaks it completely.
  • The quadratic-then-linear growth observed at finite N is likely a finite-size effect; testing the same quantities with larger Hilbert spaces or using the large-q analytic control of the Lanczos sequence would show whether the deformation-dependent growth rates persist beyond the small systems simulated.
  • The paper's two deformations affect different symmetries—the UV deformation breaks U(1) and U(1)_R, while the mass deformation preserves U(1). This suggests symmetry-resolved Krylov complexity, in which each charge sector contributes separately, could make the mechanism even more visible, but that analysis is not in the paper.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper numerically studies Krylov complexity in the N=2 supersymmetric SYK model under two deformations: an irrelevant UV deformation that breaks N=2 to N=1, and a mass deformation that breaks N=2 directly to N=0. For finite system sizes, the authors compute Lanczos coefficients, Krylov dimension, Krylov complexity, and Krylov entropy for several initial operators. They report that both deformations enlarge the Krylov space, increase early-time growth rates, and raise absolute saturation complexity, but differ in the late-time ratio of saturation complexity to Krylov-space size: the irrelevant deformation is claimed to increase this ratio, while the mass deformation is claimed to decrease it. The central late-time claim is operationalized through the quantity K_frac_sat, defined as K_sat/D_max^K, with D_max^K an upper-bound estimate from Eq. (4.1).

Significance. The paper's strength is its careful numerical implementation: it uses high-precision arithmetic, checks orthogonality of the Krylov basis, cross-checks two Lanczos algorithms, and compares against large-q analytic predictions. If the central claim about the distinct late-time signatures of the two supersymmetry-breaking mechanisms were correct, it would be a useful contribution to the operator-complexity literature. However, the main qualitative claim is not robust as stated because the denominator D_max^K is an upper bound on, not the actual size of, the Krylov space. The paper's own reported Lanczos termination lengths give different saturation fractions and reverse the claimed trend for the mass deformation. The numerical work itself is self-contained and not circular, but the headline interpretation requires revision.

major comments (2)
  1. [Sec. 4.2, Fig. 9, and Table 2] The central claim that the mass deformation decreases the saturation complexity fraction is based on K_frac_sat = K_sat/D_max^K, where D_max^K is the upper bound from Eq. (4.1), not the actual Krylov dimension. The paper itself reports that for the N=6 hopping operator the Lanczos sequence terminates at n≈33 (ϵ=0), n≈239 (UV deformation), and n≈859 (mass deformation). Using these termination lengths as the true Krylov dimension gives K_frac_sat ≈ 8.14/33 ≈ 0.25 (undeformed), ≈ 123.81/239 ≈ 0.52 (UV, ϵ=0.2), and ≈ 341.68/859 ≈ 0.40 (mass, ϵ=1). Thus the mass deformation does not decrease the ratio relative to the undeformed model; it increases it. The reported decrease from 0.14 to 0.08 is an artifact of dividing by D_max^K=4,033, which is many times larger than the actual chain length ≈859. Since the abstract and Sec. 5 frame the result as a 'fraction of the Krylov dimension,' this norma
  2. [Abstract and Sec. 5, item 3] The manuscript uses 'Krylov dimension' and 'size of the Krylov space' interchangeably with the upper bound D_max^K. Even if the authors intend to report the fraction of the maximum possible Krylov dimension, the abstract and discussion should say so explicitly. As written, the abstract's statement 'the mass deformation decreases it' is not supported by the data when the actual Krylov dimension is used. The paper can still claim an absolute increase in K_sat and a much larger D_max^K for the mass deformation, but the stated decrease in the saturation fraction is not a robust finding.
minor comments (4)
  1. [Sec. 4.3, Table 2 caption] The table's column heading K_frac_sat is defined numerically as K_sat/D_max^K, but the surrounding text sometimes calls it 'fraction of the maximum Krylov dimension' and other times 'fraction of the Krylov dimension.' Please use a single, precise term consistently.
  2. [Data access statement] The data/code availability statement points to the arXiv page (https://arxiv.org/abs/2511.20769) rather than an actual repository. A stable external repository or DOI should be provided for reproducibility.
  3. [Throughout] Minor typos and formatting issues: 'T able 1' in Table 1, 'F ermion Operator' in several figure captions, and inconsistent use of 'N=0 SYK with q=2 interactions' vs 'N=0 SYK with q=2' in the text.
  4. [Sec. 4.1] The degeneracy tolerance of 10^-10 is mentioned in the caption of Fig. 6 but not discussed in Sec. 4.1. A brief explanation of how this tolerance affects d_E and D_max^K would improve transparency.

Circularity Check

0 steps flagged

No circularity: Krylov data are computed directly from Hamiltonian realizations; fitted parameters are descriptive outputs, not inputs.

full rationale

The paper's numerical workflow is self-contained: Lanczos coefficients are computed by directly applying the Liouvillian to explicit Hamiltonian realizations, and Krylov complexity/entropy are obtained by integrating the resulting hopping equation and summing over the Krylov basis. The only fitted quantities (alpha, v_K, delta) are fits to already-computed complexity curves and are not fed back into the Lanczos computation. The large-q formulas from [23] are used as comparisons for b_1 and growth rates, not as input data generating the reported saturation values. No step in the derivation chain reduces by construction to its own inputs: the saturation fraction is defined as K_sat/D_max^K, a normalization convention, not a quantity whose value is inserted into the algorithm. The skeptic's concern about using the upper bound D_max^K instead of the actual Lanczos termination length (n≈859 for the mass deformation, n≈239 for UV, n≈33 for the undeformed model) is a legitimate robustness/correctness critique of the qualitative claim in Sec. 5 and Table 2, but it is not circularity: the same computed K_sat values are simply divided by a different denominator. The data/code link pointing to the arXiv page is an accessibility issue, not a circular-derivation issue. No load-bearing self-citations or imported uniqueness theorems appear; citations to prior work [23, 31, 46] provide external models, definitions, and benchmarks rather than the paper's own results. Accordingly, no specific circular step can be exhibited, and the appropriate score is 0.

Axiom & Free-Parameter Ledger

6 free parameters · 6 axioms · 0 invented entities

The paper introduces no new entities. Its free parameters are descriptive fits to the computed complexity curves. The main load-bearing assumption is the normalization of saturation complexity by the maximum Krylov-dimension bound rather than the actual Krylov dimension.

free parameters (6)
  • Growth-rate proportionality constants α for UV deformation = α ≈ 0.95γ (hopping), 0.78γ (fermion), 0.74γ (number), where γ=(1+ϵ²)/(1-ϵ²)²
    Fitted to the early-time quadratic growth of K(t) in Fig 12; presented as operator-dependent calibrations, not predictions.
  • Ballistic velocity v_K for UV deformation = v_K ≈ 3.4γ (hopping/fermion), 2.0γ (number)
    Fitted to the late-time linear growth of K(t) in Fig 12.
  • Growth-rate proportionality constants α for mass deformation = α ≈ 0.52ϵ (hopping), 0.40ϵ (fermion/number)
    Fitted to the early-time quadratic growth of K(t) in Fig 12.
  • Ballistic velocity v_K for mass deformation = v_K ≈ 4.4ϵ (hopping/fermion), 2.0ϵ (number)
    Fitted to the late-time linear growth of K(t) in Fig 12.
  • Saturation thresholds for K_sat and t_H = K_sat = average of last 5% of complexity values; t_H = time to reach 95% of K_sat
    Chosen by hand in Sec 4.3; these thresholds affect the reported saturation values, though the qualitative trends are insensitive to the exact choice.
  • Degeneracy tolerance for gap ratio and d_E = 10^-10
    Spacings less than 10^-10 are discarded when computing ⟨r⟩ and degeneracies; this affects D_max^K and the level statistics reported in Sec 2.3 and 4.1.
axioms (6)
  • standard math Lanczos algorithm constructs an orthonormal Krylov basis and the Lanczos coefficients determine the dynamics (Eqs. 3.3-3.6)
    Unproved background result from the Krylov subspace literature, used throughout Sec 3 and 4.
  • standard math Operator growth hypothesis: chaotic systems have asymptotically linear Lanczos coefficients b_n ~ αn (Eq. 3.7)
    Adopted from Parker et al. [6] to interpret the Lanczos sequences.
  • domain assumption N=2 SYK Hamiltonian is H={Q,Q†} with the deformations as defined in Sec 2.3 (Eqs. 2.16, 2.21)
    The model is the object of study; the deformations are taken from prior literature [23,30,31].
  • domain assumption Random couplings with Gaussian statistics; quenched disorder averaging over realizations
    Standard SYK setup used in all numerics; required for the level-statistics and Lanczos averages.
  • domain assumption Infinite-temperature inner product (Frobenius) for operator evolution
    The Lanczos algorithm uses Eq. (3.4) with β=0; finite-temperature extensions are deferred.
  • ad hoc to paper Use of D_max^K (Eq. 4.1) as a proxy for 'Krylov space size' in the saturation fraction
    K_frac_sat is defined via an upper bound on the Krylov dimension, not the actual termination index of the Lanczos sequence; this normalization drives the central mass-deformation claim.

pith-pipeline@v1.3.0-alltime-deepseek · 29049 in / 16030 out tokens · 159673 ms · 2026-08-03T20:10:19.358858+00:00 · methodology

0 comments
read the original abstract

We study the effect of supersymmetry breaking on Krylov complexity in the $\mathcal{N}=2$ SYK model under irrelevant and mass deformations of the Hamiltonian. The irrelevant deformation breaks $\mathcal{N}=2$ supersymmetry down to $\mathcal{N}=1$, while the mass deformation breaks supersymmetry completely. Using Krylov subspace methods, we analyze the Lanczos sequence, Krylov dimension, complexity, and entropy of the undeformed model and both deformations at finite system size. Both deformations enlarge the Krylov space and raise the saturation complexity as the BPS degeneracy is lifted. For the system sizes explored, the irrelevant deformation drives the saturation complexity to roughly half the Krylov dimension, while the mass deformation decreases it over an intermediate range of deformation strengths as the system drifts toward integrability. These distinct behaviors reveal how the mechanism of supersymmetry breaking leaves an imprint on quantum complexity.

Figures

Figures reproduced from arXiv: 2511.20769 by David Vegh, James Chryssanthacopoulos.

Figure 1
Figure 1. Figure 1: The spectral density ρ(E) for different SYK models. Figure (a) resembles the Wigner semicircle distribution, typical of chaotic systems, while figure (b) resembles a Gaussian, which is characteristic of integrable systems. Figure (c) exhibits a large zero-energy ground state degeneracy and a gap to the first excited state. Each plot corresponds to 500 realizations of the Hamiltonian with N = 10, F = 5, and… view at source ↗
Figure 2
Figure 2. Figure 2: The level spacing density p(s) for different SYK models. Figure (a) is peaked around s > 0 because the energy levels repel each other, while figure (b) is a Poisson distribution peaked at s = 0. Figure (c) has a peak at E = 0 because of the zero-energy degeneracy, but otherwise peaks at s > 0 similar to figure (a). Each plot corresponds to 500 realizations of the Hamiltonian with N = 10, F = 5, and J = 1. … view at source ↗
Figure 3
Figure 3. Figure 3: The spectral density ρ(E) and level spacing density p(s) for N = 2 SYK as the strength ϵ of an irrelevant deformation varies. The BPS degeneracy lifts, while the energy gap closes. The level spacings shift to the right before flattening out. Each plot corresponds to 500 realizations of the Hamiltonian with N = 10, q = 3, and J = 1. non-BPS states. When the deformation is turned on, half of the N = 2 supers… view at source ↗
Figure 4
Figure 4. Figure 4: The average gap ratio ⟨r⟩ as the strength ϵ of both deformations varies. The undeformed model has ⟨r⟩ ≈ 0.48. The irrelevant deformation increases the gap ratio to a level consistent with chaotic systems. The mass deformation decreases the ratio to a value in line with integrable systems. Each curve corresponds to the average of 500 realizations of the Hamiltonian with N = 10, q = 3, and J = 1. The shaded … view at source ↗
Figure 5
Figure 5. Figure 5: The spectral density ρ(E) and level spacing density p(s) for N = 2 SYK as the strength ϵ of a mass deformation varies. The BPS degeneracy and energy gap are eliminated, and the energy is allowed to go negative. The level spacing density resembles a Poisson distribution when the deformation is turned on. Each plot corresponds to 500 realizations of the Hamiltonian with N = 10, q = 3, and J = 1. 3.1 Krylov S… view at source ↗
Figure 6
Figure 6. Figure 6: The average energy degeneracy dE and bound on the Krylov dimension Dmax K for N = 2 SYK, with and without deformation. The undeformed model has a high degeneracy and low bound. After deformation, the degeneracy decreases, while the bound increases. The mass deformation has a more significant impact than the irrelevant deformation. The N = 2 model used q = 3, while N = 0 used q = 4. The coupling J = 1 was u… view at source ↗
Figure 7
Figure 7. Figure 7: shows the Lanczos sequences as the deformation strength ϵ varies. The horizontal axis employs a log scale to better visualize the initial growth. Because the irrelevant deformation breaks the various symmetries of the model, including the U(1) symmetry, it is not possible to work in smaller subsectors of the Hamiltonian corresponding to a fixed charge. From a computational perspective, this effectively res… view at source ↗
Figure 8
Figure 8. Figure 8: The first Lanczos coefficient b1 for the hopping, fermion, and number operators for N = 2 SYK as the deformation parameter ϵ is varied. To first approximation, b1 provides an estimate for the growth rate α of the Lanczos coefficients, bn ∼ α √ n. Results are reported separately for UV and mass deformations. The full orthogonalization version of the Lanczos algorithm was used to generate the Lanczos sequenc… view at source ↗
Figure 9
Figure 9. Figure 9: The complete sequence of Lanczos coefficients bn of N = 2 SYK for the hopping operator. Each curve corresponds to the average of ten realizations of the Hamiltonian, with shaded regions indicating the standard deviation. The first value of ϵ in the legend corresponds to the UV deformation, and the second value to the mass deformation. The values N = 6, q = 3, and J = 1 were used throughout. Dmax K = 241. W… view at source ↗
Figure 10
Figure 10. Figure 10: An example collapse of the Lanczos coefficients of N = 2 SYK for the hopping operator, in which oscillations between two branches collapse to a single branch. This pattern was observed for single realizations of the Hamiltonian under the UV deformation for N = 6 with ϵ = 0.6. 19 [PITH_FULL_IMAGE:figures/full_fig_p021_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: shows Krylov complexity for the different initial operators as the deformation strength is varied. The shaded area around each curve represents the standard deviation from the quenched disorder averaging. For both types of deformations, the complexity grows initially quadratically, then starts growing linearly. In the notation of [PITH_FULL_IMAGE:figures/full_fig_p022_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of Krylov complexity growth parameters for the hopping, fermion, and number operators for N = 2 SYK as the deformation parameter ϵ is varied. Prior to t ∼ t∗, the growth is given by a power law K(t) ∼ α 2 t 2 . After t∗, complexity grows linearly, K(t) ∼ υKt. Results are reported separately for UV and mass deformations. symmetry [51, 52]. Quadratic growth is also a generic consequence of workin… view at source ↗
Figure 13
Figure 13. Figure 13: The complete profile of Krylov complexity K(t) of N = 2 SYK for the hopping operator. The first value of ϵ in the legend corresponds to the UV deformation, and the second value to the mass deformation. The values N = 6, q = 3, and J = 1 were used throughout. undeformed model saturates at a value of 8.14 after 18 s. Turning on the irrelevant deformation delays the Heisenberg time, while increasing the satu… view at source ↗
Figure 14
Figure 14. Figure 14: Krylov entropy S(t) of N = 2 SYK for different initial operators: the hopping operator hN−1,N , the fermion operator c1, and the number operator n1 = c † 1 c1. The effects of UV and mass deformations on the entropy are shown as the deformation strength ϵ is varied. The values N = 12, q = 3, and J = 1 were used throughout. (3.23). The behavior of Krylov entropy at early times is shown in [PITH_FULL_IMAGE:… view at source ↗
Figure 15
Figure 15. Figure 15: The complete profile of Krylov entropy S(t) of N = 2 SYK for the hopping operator. The first value of ϵ in the legend corresponds to the UV deformation, and the second value to the mass deformation. The values N = 6, q = 3, and J = 1 were used throughout. 5 Discussion This paper explored the effect of supersymmetry breaking on Krylov complexity in the N = 2 SYK model using two types of deformations of the… view at source ↗
Figure 16
Figure 16. Figure 16: Lanczos coefficients bn of N = 2 SYK for various system sizes N as the irrelevant deformation strength ϵ is varied. The values q = 3 and J = 1 were used throughout. 27 [PITH_FULL_IMAGE:figures/full_fig_p029_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: Lanczos coefficients bn of N = 2 SYK for various system sizes N as the mass deformation strength ϵ is varied. The values q = 3 and J = 1 were used throughout. 0 1 2 3 4 0 3 6 9 12 N = 9 K(t) Hopping Operator 0 1 2 3 4 Fermion Operator 0 1 2 3 4 Number Operator 0 1 2 3 4 0 3 6 9 12 N = 10 K(t) 0 1 2 3 4 0 1 2 3 4 0 1 2 3 4 0 3 6 9 12 Jt N = 11 K(t) 0 1 2 3 4 Jt 0 1 2 3 4 Jt ϵ = 0 ϵ = 0.2 ϵ = 0.4 ϵ = 0.6 N … view at source ↗
Figure 18
Figure 18. Figure 18: Krylov complexity K(t) of N = 2 SYK for various system sizes N as the irrelevant deformation strength ϵ is varied. The values q = 3, and J = 1 were used throughout. 28 [PITH_FULL_IMAGE:figures/full_fig_p030_18.png] view at source ↗
Figure 19
Figure 19. Figure 19: Krylov complexity K(t) of N = 2 SYK for various system sizes N as the mass deformation strength ϵ is varied. The values q = 3, and J = 1 were used throughout. 0 1 2 3 0 1 2 3 N = 9 S(t) Hopping Operator 0 1 2 3 Fermion Operator 0 1 2 3 Number Operator 0 1 2 3 0 1 2 3 N = 10 S(t) 0 1 2 3 0 1 2 3 0 1 2 3 0 1 2 3 Jt N = 11 S(t) 0 1 2 3 Jt 0 1 2 3 Jt ϵ = 0 ϵ = 0.2 ϵ = 0.4 ϵ = 0.6 N = 0 [PITH_FULL_IMAGE:figur… view at source ↗
Figure 20
Figure 20. Figure 20: Krylov entropy S(t) of N = 2 SYK for various system sizes N as the irrelevant deformation strength ϵ is varied. The values q = 3 and J = 1 were used throughout. 29 [PITH_FULL_IMAGE:figures/full_fig_p031_20.png] view at source ↗
Figure 21
Figure 21. Figure 21: Krylov entropy S(t) of N = 2 SYK for various system sizes N as the mass deformation strength ϵ is varied. The values q = 3 and J = 1 were used throughout. 30 [PITH_FULL_IMAGE:figures/full_fig_p032_21.png] view at source ↗
Figure 22
Figure 22. Figure 22: The cumulative number of PRO reorthogonalizations Nr as the Lanczos iteration n increases for N = 2 SYK. The parameters were set to ϵP = 10−13 , ϵM = 10−15, and nO = 2. The first value of ϵ in the legend corresponds to the UV deformation, and the second value to the mass deformation. The values N = 6, q = 3, and J = 1 were used throughout. 32 [PITH_FULL_IMAGE:figures/full_fig_p034_22.png] view at source ↗
Figure 23
Figure 23. Figure 23: Comparison of the Lanczos coefficients bn computed using the FO algorithm using two different levels of numerical precision for the UV-deformed N = 2 SYK model with ϵ = 0.6. The low-precision computation used the IEEE 754 double-precision floating-point format, which utilizes between 15 and 16 decimal places, or dps. The high-precision computation used 1024 dps. The values N = 6, q = 3, and J = 1 were use… view at source ↗
Figure 24
Figure 24. Figure 24: Krylov exponent δ(t) of N = 2 SYK for different initial operators: the hopping operator hN−1,N , the fermion operator c1, and the number operator n1 = c † 1 c1. The exponent is estimated using the smoothed complexity K(t) and its smoothed first and second derivatives. The effects of UV and mass deformations on the exponent are shown as the deformation strength ϵ is varied. The first value of ϵ in the lege… view at source ↗
Figure 25
Figure 25. Figure 25: Krylov coefficient α(t) of N = 2 SYK for different initial operators: the hopping operator hN−1,N , the fermion operator c1, and the number operator n1 = c † 1 c1. The coefficient is estimated using the smoothed complexity K(t) and its smoothed first and second derivatives. The effects of UV and mass deformations on the coefficient are shown as the deformation strength ϵ is varied. The first value of ϵ in… view at source ↗

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Grand Canonical vs Canonical Krylov Complexity in Double-Scaled Complex SYK Model

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    In double-scaled complex SYK, grand-canonical Krylov complexity is the charge-weighted sum of canonical complexities, saturating a conjectured inequality.

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