The Spectrum in the Sachdev-Ye-Kitaev Model

• SYK is a 0 + 1 dimensional model of N ≫1 fermions with an all-to-all random quartic interaction.Source: §1 · p. 2
• At large N, SYK permits summing all Feynman diagrams and computing correlation functions at strong coupling.Source: §1 · p. 2
• SYK saturates the maximal Lyapunov-exponent bound, with chaos quantified through an out-of-time-order four-point function [7].Source: §1 · p. 2
• At low energies, SYK exhibits emergent conformal symmetry in its two-point function [9] [10].Source: §1 · p. 2
• The combination of solvability, maximal chaos, and emergent conformal symmetry motivates SYK as a potential solvable model of holography.Source: §1 · p. 3
• SYK was introduced by Kitaev, who advanced its connection to holographic chaos and proposed it as a model of holography [5] [6] [11].Source: §1 · p. 3
• The paper studies the four-point function, solves its Schwinger-Dyson equation, finds discrete and continuous two-particle towers, and evaluates the discrete spectral sum.Source: §1 · p. 3

2 Two-point function

• SYK uses Majorana fermions with quenched-random quartic couplings, and its correlators are understood after disorder averaging.Source: §2 · p. 3
• Large-N disorder averaging simplifies the Feynman diagrams, with the 1PI self-energy expressed in terms of the full two-point function.Source: §2 · p. 4
• The Schwinger-Dyson equations fully determine the two-point function, while their known solution applies only in the low-energy limit after dropping iω.Source: §2 · p. 4
• The low-energy two-point-function solution is valid when J∣t∣≫1 and indicates an IR conformal fixed point with fermion dimension ∆= 1/4.Source: §2 · p. 4
• The IR approximation is self-consistent because the interacting propagator decays at late times, unlike the free propagator's constant behavior.Source: §2 · p. 5
• Conformal reparameterization maps the zero-temperature solution to finite temperature, whose real-time continuation has exponential late-time decay.Source: §2 · pp. 5–6

Sachdev-Ye

• The Sachdev-Ye model has random infinite-range spin exchange, admits a fermionic representation, and becomes quartic in fermions.Source: Sachdev-Ye · p. 6
• SYK avoids a spin-glass phase and requires only N →∞, whereas Sachdev-Ye requires both N-analogous and M→∞ limits; the models also differ in coupling-index structure.Source: Sachdev-Ye · p. 7

3 Spectrum

• The leading connected four-point function is generated by ladder diagrams and obeys a Schwinger-Dyson integral equation whose solution reduces to diagonalizing its kernel.The connected contribution scales as 1/N, and the ladder diagrams use dressed propagators.Source: §3 · p. 7, Figure 2 · p. 8

3.2 SL(2,R) and all eigenvectors

• The kernel’s power-law eigenvectors reveal an SL(2,R) symmetry that generates further eigenvectors with the same eigenvalue, subject to boundary-term subtleties.The symmetry is motivated by the AdS2 interpretation of SYK, while boundary terms require separate treatment.Source: §3 · pp. 7–8, §3.2 · pp. 9–10

3.1 Eigenvalues

• The eigenvalue calculation requires evaluating eight sign-defined integration regions, and its initially divergent integrals are interpreted through analytic continuation.The divergence is later clarified by constructing the complete eigenvector set.Source: §3.1 · p. 9

3.3 Directly finding the eigenvectors

• Direct substitution confirms the SL(2,R)-generated eigenfunctions and fixes the Bessel-function combination J_ν and J_−ν required for true kernel eigenvectors.The direct calculation also recovers the earlier eigenvalues after setting ν = 2α−1.Source: §3.3 · pp. 11–12

Normalization

• The eigenfunctions form a complete basis for antisymmetric functions of the two time variables, consistent with fermion antisymmetry.Their normalization distinguishes discrete eigenfunctions from delta-function-normalizable continuous ones.Source: §3.4 · p. 12, Normalization · p. 13

3.4 A complete set of eigenvectors

• The complete basis consists of continuous states with ν = ir, r > 0 and discrete bound states with ν = 3/2 + 2n for n ≥ 0.Boundary conditions eliminate the unwanted solutions and produce a quantum-mechanical problem with both scattering and bound states.Source: §3.4 · pp. 12–13, Summary · p. 14

Summary

• The normalized eigenbasis provides the spectrum used to express the four-point function as an integral over imaginary ν plus a sum over discrete ν values.The resulting expression represents the four-point function as a sum over intermediate two-particle states, analogous to a conformal block decomposition.Source: Summary · p. 14

4 Four-point function

• The four-point function is expressed as a sum over intermediate two-particle states, combining an imaginary-ν integral with discrete real values ν_n = 3/2 + 2n.The decomposition is described as reminiscent of a conformal block decomposition.Source: §4 · pp. 14–15
• The decomposition contains a ν = 3/2 pole, a remaining discrete tower, and a continuous integral over ν = ir with r > 0.The pole arises because g(3/2) = 1.Source: §4 · p. 15

Divergence

• The ν = 3/2 divergence is an artifact of the infrared limit, and ultraviolet corrections to the two-point function shift eigenvalues and remove it.The same shifts also remove eigenvalue degeneracy by introducing frequency dependence.Source: Divergence · pp. 15–16

Chaos

• At strong coupling βJ ≫ 1, SYK has Lyapunov exponent κ = 2π/β, obtained from the out-of-time-order four-point function.The Euclidean correlator is mapped to finite temperature and then continued to a Lorentzian out-of-time-order correlator; the stated time regime is κ^-1 ≪ t ≪ κ^-1 log N.Source: Chaos · p. 17

Discrete Sum

• The discrete tower sum is evaluated using generating functions and Hilbert transforms for Legendre functions, yielding an expression involving the dilogarithm.The continuous contribution is reduced by performing the ω integral, but the remaining r integral is left for future work.Source: Discrete Sum · pp. 19–20, Continuous Sum · p. 20

A Integrals in eigenvalue computation

• The eigenvalue integral is split into eight integration regions, with t3 > t4 chosen without loss of generality.Beta-function representations are used to evaluate the resulting pieces.Source: A Integrals in eigenvalue computation · pp. 20–21
• The calculation labels integration ranges according to the signs of the relevant factors and uses tij = ti − tj notation for time differences.One region is absent because the ordering assumption t3 > t4 excludes it.Source: A Integrals in eigenvalue computation · pp. 21–22

B Eigenvectors and boundary terms

• The SL(2,R) Casimir is related to the Lorentzian Laplacian and, in frequency space, to a Bessel operator, providing candidate kernel eigenfunctions.The construction applies the symmetry even for Euclidean four-point functions.Source: B Eigenvectors and boundary terms · pp. 22–23
• Dropping the boundary term is valid only for particular Bessel-function combinations, and the eigenvectors used in the main text have the required vanishing boundary term.A second seemingly possible Bessel solution would not automatically satisfy this condition.Source: B Eigenvectors and boundary terms · pp. 22–23

C Integral in eigenvector computation

• The appendix evaluates the integral used in the direct eigenvector calculation and represents the eigenvector as a combination of Bessel functions.The Fourier-transformed expressions impose ranges of validity on the Bessel integrals.Source: C Integral in eigenvector computation · pp. 23–24
• The cited passage attributes the factor 3 to Feynman-diagram combinatorics and 1/(4π) to two-point-function normalization, immediately before presenting the eigenvalues.Source: C Integral in eigenvector computation · p. 25
• Setting ν = 2α − 1 gives (3.8), and the eigenfunctions (C.12) agree with (3.18).Source: C Integral in eigenvector computation · p. 25

• The appendix reviews integral identities involving products of Bessel functions.Source: D Integrals of products of Bessel… · p. 25
• Its subject is the evaluation of integrals containing products of Bessel functions.Source: D Integrals of products of Bessel… · p. 25
• The material collects identities for integrals built from Bessel-function products.Source: D Integrals of products of Bessel… · p. 25

• The coefficient functions a(ν) and b(ν) are arbitrary periodic functions with period one, while Jν and Yν denote Bessel functions.Source: Laplace transform of JνJν and JνJ−ν · p. 25
• The derivation uses the Bessel addition formula and a hypergeometric 2F1 transformation relation.Source: Laplace transform of JνJν and JνJ−ν · pp. 25–26
• The resulting identity reproduces formula (13-22, 2) of.Source: Laplace transform of JνJν and JνJ−ν · p. 26
• A chosen Bessel-function combination evaluates to a combination of hypergeometric functions of type 3F2.Source: Laplace transform of JνJν and JνJ−ν · p. 26

• The transform is expressed as a discontinuity of Q_{ν−1/2} across its branch cut, with an iϵ prescription.Source: Fourier sine and cosine transform of… · p. 27
• For 0 < a < γ−β, corresponding to z > 1, the transform vanishes because the argument lies away from the branch cut.Source: Fourier sine and cosine transform of… · p. 27
• For γ−β < a < γ+β, the relevant range is −1 < z < 1, where the continued Legendre function is used.Source: Fourier sine and cosine transform of… · p. 27
• For γ+β < a, corresponding to z < −1, the branch contribution arises from the z^{−ν−1} term and matches Eq. 6.672 of.Source: Fourier sine and cosine transform of… · p. 27