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Comments on the Sachdev-Ye-Kitaev model

Juan Maldacena, Douglas Stanford

arXiv:1604.07818v1hep-thcond-mat.str-el

TL;DR

The paper studies how a solvable large-N model of randomly interacting Majorana fermions realizes emergent nearly conformal dynamics and what its correlators reveal about its excitations and holographic interpretation. It analyzes the bilocal large-N theory, ladder-kernel spectrum, and reparametrization modes, finding that explicit symmetry breaking enhances the four-point function and produces maximal chaos. The results support universal near-AdS2-like dynamics while suggesting additional low-tension string states, within stated low-energy and semiclassical limits.

  • Problem

    The paper asks whether a simple solvable quantum-mechanical model can capture features relevant to holography, including strongly interacting low-energy dynamics, black-hole-like chaos, and a spectrum of physical excitations.

    Source: §1 · p. 3

  • Method

    The authors analyze large-N bilocal equations, two- and four-point functions from ladder kernels, reparametrization fluctuations, and the associated Schwarzian effective theory.

    Source: §1.1 · p. 6

  • Results

    The explicitly lifted reparametrization modes produce an enhanced four-point contribution that dominates the chaos limit and saturates the bound with λL = 2π/β, while finite conformal pieces reveal a tower of composite states.

    Source: §1.1 · pp. 6–8, §6.3 · p. 56

  • Takeaways & Limitations

    The symmetry-breaking pattern is expected to be universal among large-N systems with emergent reparametrization symmetry and is consistent with near-extremal dilaton-gravity dynamics, while the spectrum suggests low-tension strings.

    Source: §1.1 · p. 8, §7 · pp. 58–59

    The analysis cannot be trusted at sufficiently small energies where the Schwarzian action ceases to be semiclassical, and the analytic continuation treats the enhanced h = 2 contribution separately from the remaining correlator.

    Source: §3.2.7 · p. 30, §5 · p. 51

Abstract

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We study a quantum mechanical model proposed by Sachdev, Ye and Kitaev. The model consists of $N$ Majorana fermions with random interactions of a few fermions at a time. It it tractable in the large $N$ limit, where the classical variable is a bilocal fermion bilinear. The model becomes strongly interacting at low energies where it develops an emergent conformal symmetry. We study two and four point functions of the fundamental fermions. This provides the spectrum of physical excitations for the bilocal field. The emergent conformal symmetry is a reparametrization symmetry, which is spontaneously broken to $SL(2,R)$, leading to zero modes. These zero modes are lifted by a small residual explicit breaking, which produces an enhanced contribution to the four point function. This contribution displays a maximal Lyapunov exponent in the chaos region (out of time ordered correlator). We expect these features to be universal properties of large $N$ quantum mechanics systems with emergent reparametrization symmetry. This article is largely based on talks given by Kitaev \cite{KitaevTalks}, which motivated us to work out the details of the ideas described there.

1 Introduction

The paper develops the Sachdev-Ye-Kitaev model as a solvable large-N quantum system with emergent nearly conformal dynamics, and analyzes its correlators, spectrum, reparametrization modes, and holographic implications. Explicit breaking lifts the reparametrization zero modes, producing an enhanced four-point contribution that dominates chaotic dynamics and saturates the chaos bound.

1 Introduction

  • The SYK model contains N Majorana fermions with random few-fermion interactions and becomes tractable at large N through a classical bilocal field.The q-fermion variant is also analytically tractable at large q, while q = 2 is integrable and lacks exponentially growing out-of-time-order correlators.Source: §1 · p. 3, §1.1 · p. 5
  • At low energies, the model develops an emergent reparametrization symmetry that is spontaneously broken to SL(2, R) and explicitly broken, defining nearly conformal quantum mechanics.This symmetry-breaking pattern parallels the near-AdS2 regime of near-extremal black holes.Source: §1 · p. 4, §1.1 · pp. 5–6

1.1 Organization of the paper and summary of results

  • The large-N two-point function and bilocal formulation connect the ultraviolet free theory to the strongly interacting infrared regime, while the four-point function is obtained by summing ladder diagrams through a kernel.The kernel spectrum determines the operator dimensions appearing in the fermion-bilinear OPE.Source: §1 · p. 4, §1.1 · pp. 5–6
  • The finite conformal-limit pieces of the four-point function encode a tower of composite operators with approximately integer spacing and O(1) anomalous corrections, suggesting low-tension strings in nearly AdS2.These states do not compete with the enhanced dilaton-gravity contribution, and the bulk interpretation is framed as a single Regge trajectory.Source: §1 · p. 5, §1.1 · pp. 6, 8
  • The four-point function contains an enhanced contribution from reparametrization Goldstone modes, whose explicit symmetry-breaking action adds a factor of βJ over the conformal result.The same contribution is related to energy fluctuations and the specific heat, and is described by a Schwarzian effective action with coefficient of order (βJ)^-1.Source: §1 · p. 4, §1.1 · pp. 6–7
  • The Goldstone contribution dominates the chaos limit as (βJ/N)e^(2πt/β), saturating the chaos bound, whereas a model with only low-energy SL(2, R) symmetry has no pseudo-Goldstone mode and a submaximal chaos exponent.Thus the reparametrization symmetry, rather than SL(2, R) symmetry alone, is essential for maximal chaos in this analysis.Source: §1 · p. 4, §1.1 · p. 7
  • The Schwarzian contribution yields a linear-in-temperature specific heat and, after the one-loop logarithmic free-energy correction, a spectral density proportional to J^-1e^(S0+√(2c(E−E0))) without powers of E−E0 in the trusted regime.The low-temperature partition function contains an extensive ground-state energy, zero-temperature entropy, and specific-heat term.Source: §1.1 · pp. 6–8
  • The paper argues that the enhanced four-point contribution is universal for large-N systems with emergent reparametrization symmetry and agrees with near-extremal dilaton gravity, while the fermion spectrum suggests a low-tension string interpretation.It also speculates that the bulk fermion description involves one fermion attached to a boundary string and views the bilocal field as a field in an additional kinematic-space-like dimension.Source: §1.1 · p. 8

2 Two point functions

The model’s large-N two-point-function equations reduce to a classical bilocal problem, with conformal, large-q, and q=2 limits providing analytic control. These solutions determine thermodynamics and corrections away from the conformal propagator.

2.1 The model

  • The model contains N Majorana fermions with random q-body interactions, characterized by the energy scale J and studied at leading order in 1/N.At fixed temperature in the large-N limit, the analysis accesses exponentially many low-energy states.Source: §2.1 · p. 9

2.2 Summing the leading order diagrams

  • Disorder-averaged diagrams organize into leading large-N equations for a bilocal two-point function G and self-energy Σ.The first correction scales as J^2 without additional N suppression, and the full propagator is obtained by summing these contributions.Source: §2.2 · pp. 10–11, Figure 1 · p. 10, Figure 2 · p. 10

2.3 The conformal limit

  • At strong coupling, the two-point equations become reparametrization invariant, with conformal dimension ∆ = 1/q and finite-temperature solutions generated by reparametrizations.The conformal expressions also determine retarded correlators and quasinormal frequencies.Source: §2.3 · p. 11

2.4 Large q limit

  • The large-q expansion makes the equations analytically tractable across the free ultraviolet and interacting infrared regimes, with boundary conditions fixing the solution parameter v.The solution is valid at zero and finite temperature, and v increases from zero to one as βJ increases.Source: §2.4 · pp. 12–13

2.5 q = 2

  • For q = 2, the Hamiltonian is quadratic and can be diagonalized into fermionic oscillators whose random masses follow a semicircle distribution.The resulting system behaves as free fermions and can be viewed as a single fermion in an extra dimension, making it a useful solvable extreme.Source: §2.5 · pp. 13–14

2.6 Computing the entropy

  • The bilocal functional integral yields a large-N free energy with ground-state energy, zero-temperature entropy, and a temperature-dependent term proportional to 1/β.In the large-q expansion, corrections separately modify the ground-state energy, zero-temperature entropy, and low-temperature entropy, which is linear in T.Source: §2.6 · pp. 14–16

2.7 Correction to the conformal propagator

  • The leading correction to the conformal propagator is obtained away from the conformal limit, while for q = 2 the order 1/J correction vanishes.The correction is derived using the large-q conformal expressions and evaluated at finite temperature.Source: §2.7 · p. 16

3 Four point functions

This section develops the four-point function through its ladder-diagram and kernel-eigenfunction representations, including the continuum and discrete spectra and the role of the h = 2 mode. It analyzes conformal and nonconformal contributions, retarded-kernel growth, and the resulting Lyapunov exponent across large q and general q.

3.1 The ladder diagrams

  • The leading 1/N four-point function is a geometric sum of ladder diagrams generated by repeated multiplication with a kernel K.All ladder diagrams, and only these diagrams, contribute at order 1/N; conjugating K by propagator powers yields a symmetric operator with a complete eigenbasis.Source: §3.1 · pp. 17–18

3.2 Using conformal symmetry

  • The conformal kernel cannot directly determine modes with eigenvalue Kc = 1, because they make the ladder geometric series diverge.Those modes must instead be treated perturbatively outside the conformal limit.Source: §3.2 · pp. 18–19
  • In the conformal limit, SL(2) symmetry makes ladder diagrams powers times functions of one invariant cross ratio, while the kernel commutes with the Casimir and is diagonalized by Ψh(χ).The functions Ψh(χ) are particular hypergeometric functions related to conformal blocks of weight h; a complete basis requires determining the allowed set of h.Source: §3.2 · p. 19

3.2.1 The four point function as a function of the cross ratio

  • Cross-ratio symmetry determines the full correlator from 0 < χ < 2 and requires its derivative to vanish at χ = 2.For χ > 1, the correlator obeys F(χ) = F(χ/(χ−1)), which maps 1 < χ < 2 to 2 < χ < ∞.Source: §3.2.1 · p. 20, Figure 5 · p. 21

3.2.2 Eigenfunctions of the casimir

  • The normalizable Casimir basis consists of continuum states h = 1/2 + is and discrete states h = 2n, with distinct oscillatory and monotonic behavior across χ = 1.Continuum solutions are monotonic for χ > 1 and infinitely oscillatory for χ < 1; discrete solutions are monotonic for 0 < χ < 1 and cross zero n times for χ > 1.Source: §3.2.2 · p. 22

3.2.3 The eigenvalues of the kernel kc(h)

  • The kernel eigenvalue kc(h) is real on both sectors, positive for discrete states and negative for continuum states, and satisfies kc(h) = kc(1−h).Special values include kc(0) = −(q−1) and kc(2) = 1, with the latter associated with a linearized reparameterization of Gc(τ).Source: §3.2.3 · p. 24

3.2.4 The inner products ⟨Ψh, Ψh⟩and ⟨Ψh, F0⟩

  • The continuum eigenfunctions are exactly orthogonal with delta-function normalization, while discrete-state norms and overlaps with the zero-rung ladder are computed using Legendre functions and analytic continuation.The continuum inner product contains no additional finite terms because eigenfunctions with different s must be orthogonal.Source: §3.2.4 · p. 25
  • The inner products are evaluated by dividing the integration region and expressing the resulting integrals with the Euler beta function and eigenvalue function kc(h).F0 is interpreted as proportional to the conformal kernel acting on a delta function, explaining the appearance of kc(h).Source: §3.2.4 · p. 25

3.2.5 The sum of all ladders

  • The sum of ladders is recast as a contour integral whose deformation cancels explicit residues and picks up poles where kc(h) = 1, while excluding the h = 2 divergence.For χ < 1, symmetry manipulations produce a form that can be pushed rightward, again selecting the kc(h) = 1 poles.Source: §3.2.5 · pp. 26–27

3.2.6 Operators of the model

  • In the OPE limit, kc(h) = 1 has solutions h0 = 2, h1, h2, ... giving operator dimensions, conformal blocks, and positive coefficients in the operator product expansion.Source: §3.2.6 · pp. 27–28

3.2.7 Analytic continuation to the chaos region

  • Analytic continuation to the chaos region isolates a double h = 2 pole whose Ψ2 term grows as e2πt/β and saturates the chaos bound.The remaining Fh̸=2 contribution is only a small correction, with its t e2πt/β term corresponding to a finite-coupling decrease in the growth exponent.Source: §3.2.7 · pp. 29, 31

3.3 Proper treatment of the h = 2 subspace

  • The h = 2 ladder eigenfunctions have eigenvalue one conformally, causing the four-point divergence that is regulated by corrections of order 1/(βJ).The exact eigenvalue formulation is valid at any coupling, while the conformal limit is smooth for all contributions except h = 2.Source: §3.3 · pp. 31–32

3.3.1 The h = 2 eigenfunctions and reparameterizations

  • Linearized reparameterizations generate h = 2 eigenfunctions of the conformal kernel with eigenvalue one, while the SL(2) modes n = −1, 0, 1 vanish.Nonzero modes exist for |n| ≥ 2 and organize into highest- and lowest-weight SL(2) representations.Source: §3.3.1 · pp. 32–33

3.3.2 The shift in the eigenvalues

  • At large q, the h = 2 eigenvalue problem simplifies because the propagators are simple, allowing its integral equation to be converted into a differential equation.The side-rail propagators are approximated by their leading term, while the rung correction is exponentiated.Source: §3.3.2 · pp. 33–34
  • Near strong coupling, the shifted eigenvalues are h_n = 2 + |n|(1−v), while the kernel coefficient αK stays reasonably close to three for all values of q.Source: §3.3.2 · pp. 35–36

3.3.3 The enhanced h = 2 contribution

  • The h = 2 contribution is enhanced by βJ, depends on the SL(2) descendant index n rather than only the cross ratio, and its alternating continuation saturates the chaos bound.In the non-alternating configuration, the same contribution agrees with energy fluctuations because the conserved operator T is essentially the Hamiltonian and has a constant thermal two-point function.Source: §3.3.3 · pp. 36, 38, 39

3.3.4 Other terms from the h = 2 subspace

  • Up to order (βJ)^0, corrections from the h = 2 subspace combine with double-pole terms into contributions universal in q apart from an overall prefactor.The assumed second-order eigenvalue relation is supported in the q = ∞ model and numerically to roughly percent-level multiplicative precision for several low q and n values.Source: §3.3.4 · p. 40

3.4 More detail on the q = ∞four point function

  • For q = ∞, the four-point function is obtained as a Green’s function of a differential operator, reproducing the enhanced term and showing y-independence in the non-alternating OPE region through order one.The displayed solution gives the full q = ∞ four-point function through order (βJ)^0, while the next order, (βJ)^−1, is also computable.Source: §3.4 · pp. 40–41

3.5 Summary of the four point function

  • The general four-point function combines a nonconformal conserved-Hamiltonian term with a conformal tower whose dimensions satisfy kc(h_m) = 1, and its chaos continuation cancels logarithmic terms.The first term is y-independent for χ < 1, while the second contains the other conformally invariant operators; the logarithm from 1/(1−kc) cancels the derivative contribution.Source: §3.5 · pp. 42–43

3.6.1 The retarded kernel

  • In the out-of-time-order regime, the asymptotic growth rate is determined by real-time ladder diagrams and requires the retarded kernel to have an eigenfunction with eigenvalue one.The retarded propagators include insertions on both sides of the real-time fold, and the kernel analysis uses a growth ansatz.Source: §3.6.1 · p. 43
  • λL = 2π is the only positive growth rate in the conformal limit; the next-largest allowed rate is zero.This explains why all growing terms in the chaos limit share the same exponent, including subleading terms in Fbig.Source: §3.6.1 · p. 44

3.6.2 Large q

  • In the large q model, λL is obtained exactly from the single bound state of a −2/cosh^2 ˜x potential, approaching 2J at weak coupling and 2π/β at strong coupling.The exponent is derived by mapping the retarded-kernel equation to a Schrödinger problem with bound-state energy −1.Source: §3.6.2 · p. 44

3.6.3 General q

  • For general q, finite-coupling corrections to λL are negative; numerical q = 4 results match smoothly to the large-coupling expression.As q approaches two, the correction diverges, consistent with the free q = 2 model having zero chaos exponent for every βJ.Source: §3.6.3 · p. 46

4 The effective theory of reparameterizations

The effective action reformulates the enhanced four-point contribution as an integral over reparameterizations, whose explicit breaking is captured by a Schwarzian action. This framework connects the near-conformal zero modes to finite-temperature thermodynamics and validates the action nonlinearly.

4.1 Bilocal effective action

  • The large-N bilocal formulation reproduces the four-point function by expanding around the saddle and integrating Gaussian fluctuations governed by the symmetric ladder kernel.Integrating out the auxiliary fluctuation yields the previously derived 1/N four-point expression.Source: §4 · pp. 46–47

4.2 Reparameterization zero modes

  • In the conformal limit, reparameterizations of the saddle are zero modes because the infrared action is invariant under reparameterizations while the solution preserves only SL(2,R).These modes are the Nambu-Goldstone modes of spontaneous breaking from full reparameterization symmetry to SL(2,R).Source: §4 · p. 47
  • Explicit ultraviolet breaking lifts the reparameterization modes and produces a local action that vanishes for global SL(2) transformations.The quadratic action is proportional to n^2(n^2−1) in frequency space, reflecting the three excluded SL(2) modes.Source: §4 · pp. 47–48

4.3 Enhanced contribution

  • The enhanced four-point term arises from integrating over reparameterizations weighted by the lifted-mode action, yielding the previously obtained enhanced contribution.This identifies the non-conformal enhancement with the functional integral over reparameterized infrared saddles.Source: §4 · pp. 48–49

4.4 Schwarzian action

  • The finite-temperature effective action is the Schwarzian derivative, whose coefficient reproduces known large-q and q=2 results and agrees with numerical specific-heat determinations.Figure 12 checks the coefficient beyond infinitesimal reparameterizations, while the action explains the near-extremal entropy but not the zero-temperature entropy.Source: Figure 12 · p. 48, §4 · p. 49

5 The density of states and the free energy

The section derives low-temperature thermodynamics and the density of states from the large-N effective action, then compares these predictions with exact diagonalization at finite N. It finds logarithmic near-zero-mode corrections and a smoothly vanishing numerical spectral density, while identifying a low-energy regime where the semiclassical analysis fails.

5.1 Free energy

  • The low-temperature free energy contains extensive ground-state energy, entropy, and specific heat, with c ≈ 0.396 N/J for q = 4.The order-one determinant correction is separate from the leading terms proportional to N.Source: §5 · p. 50
  • The h = 2 near-zero modes generate a finite logarithmic correction, making the low-temperature partition function proportional to β^-3/2.The same β^-3/2 factor follows from integrating over reparameterizations after dividing by the SL(2,R) volume.Source: §5 · pp. 50–51

5.2 Density of states

  • The density-of-states calculation gives a constant low-energy limit in the saddle-point approximation, but this result is not reliable when cE ∼ 1 because the Schwarzian action ceases to be semiclassical.At that crossover the relevant saddle has β ∼ c ∼ N/J.Source: §5 · p. 51

5.3 Exact spectrum

  • Exact diagonalization reaches N = 32 and shows no scale-invariant 1/E divergence or δ(E) at low energy; instead, the density smoothly approaches zero.The zero-temperature entropy is therefore attributed to a large density of states near the ground state rather than exact degeneracy.Source: §5 · pp. 51–52, Figure 13 · p. 52
  • For N between 24 and 32, level counts fit e^0.33N near the spectral center and e^0.24N near the edge, consistent with the expected exponential densities.The Hamiltonian has O(N^4) random elements, unlike a general random matrix on the full Hilbert space with 2^N random elements.Source: §5 · p. 52

6 Towards a bulk interpretation

The paper interprets SYK's bilocal fluctuations as a two-dimensional bulk-like system with a dominant near-AdS2 reparametrization mode, additional string-like excitations, and scrambling behavior resembling near-extremal black holes.

6 Towards a bulk interpretation

  • The bilocal field behaves as a classical two-variable field at large N, with its four point function interpreted as the propagator obtained by inverting the fluctuation operator.The two times can be viewed as coordinates of a two-dimensional field theory, while the Schwinger-Dyson solution supplies the classical background.Source: §6 · pp. 52–53
  • Near-AdS2 geometries reproduce the dominant boundary reparametrization mode and its action through spontaneous breaking to SL(2, R) plus explicit breaking by the dilaton.This mechanism is insensitive to the detailed matter content and applies to near-extremal black holes with an NAdS2 region.Source: §6 · p. 53
  • The propagating bilocal modes form an infinite tower with order-one anomalous dimension shifts, suggesting low-tension strings rather than weakly interacting bulk fermion pairs.The tower resembles states of an O(N) model or a string theory with string scale comparable to the AdS radius, without Hagedorn growth.Source: §6 · p. 53

6.1 Comments on kinematic space.

  • Kinematic-space variables make the infrared Casimir resemble the AdS2 Laplacian, while the nonlocal quadratic action encodes many tower states through multiple solutions of 1/kc(h) −1 = 0.The Euclidean bulk is expected to be H2, whereas the kinematic space retains Lorentzian character and can be viewed as dS2 or periodically identified AdS2.Source: §6.1 · pp. 54–55

6.2 The fermions

  • The fermion interpretation favors one bulk fermion with a boundary string or O(N) color-electric flux, rather than N independent bulk fermions.For q = 2, diagonalizing the random mass matrix produces a nearly continuous mass distribution that behaves like an extra dimension; the string picture remains speculative.Source: §6.2 · pp. 55–56

6.3 Scrambling for near extremal black holes and its stringy corrections

  • Near-extremal black-hole scrambling receives a βJ enhancement for the gravitational contribution, while stringy corrections to the chaos exponent can remain small, with SYK corrections of order 1/(q^2βJ).The authors interpret gravity as dominating despite light strings, keeping the effective Regge spin near two.Source: §6.3 · pp. 56–58

7 Brief Conclusions

The conclusions identify spontaneously and explicitly broken reparametrization symmetry as the source of SYK's universal low-energy behavior, while its operator spectrum suggests a model-specific low-tension string dual.

7 Brief Conclusions

  • Spontaneously and explicitly broken reparametrization symmetry produces linear specific heat and an enhanced four point function that saturates the chaos bound in the out-of-time-ordered regime.The authors expect these properties to be universal among large-N systems with emergent reparametrization symmetry, or NCFT1s.Source: §7 · p. 58
  • The spectrum of fermion-bilinear dimensions suggests a single Regge trajectory of low-tension strings in nearly AdS2, while the model's precise holographic interpretation remains unresolved.The paper also analyzes the non-enhanced parts of the four point function and identifies several open questions about the bulk dual.Source: §7 · p. 59

A The Schwinger-Dyson equations and the kernel

The appendix derives corrections around the infrared Schwinger-Dyson solution by perturbing the conformal propagator and reusing the kernel that governs the four point function.

A The Schwinger-Dyson equations and the kernel

  • The iω term is treated as a source perturbing the conformal solution, with G written as Gc + δG and the self-energy varied through the Schwinger-Dyson equations.Expanding to first order and eliminating δΣ yields an equation for the propagator correction.Source: A The Schwinger-Dyson equations and… · p. 59
  • A formal h = −1 mode supplies the correction, but ultraviolet divergences near coincident times regulate it and determine the full coefficient through the complete UV-to-IR flow.The conformal kernel alone cannot fix the coefficient because the relevant mode is affected by UV divergences.Source: A The Schwinger-Dyson equations and… · pp. 59–60
  • The same kernel appearing in the four point function also governs corrections around the infrared solution.This links the perturbative Schwinger-Dyson analysis to the ladder-kernel treatment of four point functions.Source: A The Schwinger-Dyson equations and… · p. 60

B The kernel as a function of cross ratios

The appendix exploits conformal symmetry and cross-ratio invariance to reduce the ladder-kernel problem to a closed one-dimensional equation.

B The kernel as a function of cross ratios

  • The kernel maps an n-rung ladder diagram to the corresponding (n+1)-rung diagram.Conformal transformations fix three times and replace an integration variable with a cross ratio, reducing the original integral structure.Source: B The kernel as a function of cross… · p. 60
  • Cross-ratio symmetry restricts the problem to 0 ≤ χ ≤ 2, where changing variables produces a closed integral equation.The relevant symmetry is F(χ) = F(χ/(χ −1)).Source: B The kernel as a function of cross… · p. 60
  • The resulting bracketed expression, divided by α0, is the conformal kernel Kc(χ, χ̃).This identifies the reduced cross-ratio equation with the kernel used in the main four point-function analysis.Source: B The kernel as a function of cross… · p. 61

C Representing F0 in terms of Ψh

The appendix expresses F0 through residue sums over poles of kc(h), with separate formulas for χ > 1 that can be checked numerically against the original expression.

C Representing F0 in terms of Ψh

D Writing Ψh(χ) in terms of Ψh,n(θ1, θ2)

This appendix relates the χ-space and angle-space representations of Ψh,n by solving the Casimir equation, normalizing continuum modes, and matching their discrete limits.

D Writing Ψh(χ) in terms of Ψh,n(θ1, θ2)

E Direct approach to the shift in eigenvalue

The appendix computes the eigenvalue shift by evaluating rail and rung corrections using conformal representations and a large-frequency, large-temperature limit.

E Direct approach to the shift in eigenvalue

  • Corrections to the rail propagators enter through a distinct kernel change, whereas rung-propagator corrections provide the simpler eigenfunction-based reduction.Source: E Direct approach to the shift in… · p. 62
  • The rail correction to the h = 2 eigenvalue is shown, using conformal symmetry, to be proportional to |n|.The calculation represents the relevant propagator and eigenfunction corrections as integrated conformal three-point functions.Source: E Direct approach to the shift in… · pp. 62–63
  • The seven-variable integral for the rail shift is reduced using SL(2) covariance, analytic prescriptions, and residues at a fifth-order pole.The θc integral produces a cosine numerator, while the θa residue contributes a factor proportional to n^2(n^2−1).Source: E Direct approach to the shift in… · p. 63
  • The large-n, large-β limit with fixed Ω makes the ultraviolet region dominant and permits comparison of rung and rail integrals in proper-time coordinates.The frequency scales out, allowing Ω = 2 to be chosen for the ratio calculation.Source: E Direct approach to the shift in… · pp. 63–64
  • The explicitly evaluated rail contribution agrees with the result obtained from the normalized rung contribution and their integral ratio.Source: E Direct approach to the shift in… · p. 64

F The first order change in h = 2 eigenvectors

The first-order h = 2 eigenvector correction is analyzed through reparameterization covariance and exact Schwinger-Dyson identities, with UV terms separated from the infrared contribution.

F The first order change in h = 2 eigenvectors

  • The corrected h = 2 eigenvector remains independent of q up to an overall multiple at first order.This follows from the relation between reparameterizations of Gc and of the universal correction δG.Source: F The first order change in h = 2… · pp. 64–65
  • Reparameterization covariance is applied to an integral with weight Δ = 1/q, using the exact kernel and propagator together with Schwinger-Dyson equations.Source: F The first order change in h = 2… · pp. 64–65
  • The nominal order-(βJ)^−1 contribution vanishes because the reparameterized conformal propagator is a unit-eigenvalue mode of the conformal kernel.The leading infrared terms therefore begin at order (βJ)^−2 and cancel between the relevant corrections.Source: F The first order change in h = 2… · p. 65
  • The calculation defines the UV-divergent integral by retaining only the cutoff-independent (βJ)^−2 term and discarding the power divergence.In the exact theory, UV terms can scale as (βJ)^−h and (βJ)^−h−1 and may dominate the infrared term for h ≠ 2.Source: F The first order change in h = 2… · p. 65
  • Figure 15 compares the exact G(θ) with the conformal approximation and its first correction at βJ = 10 and βJ = 50.Source: Figure 15 · p. 66

G Numerical solution of the SD equations

The section develops numerical Euclidean and real-time solutions of the Schwinger-Dyson equations at finite βJ, comparing them with conformal predictions and leading corrections.

Euclidean equations

  • The Euclidean equations are solved by iterating between Matsubara frequency and Euclidean time representations, with weighted updates stabilizing convergence.Exact solutions are obtained for different βJ values, and the resulting correlators are used to study finite-coupling corrections.Source: G Numerical solution of the SD equations · p. 66

Finite-coupling corrections

  • For large βJ, the difference between exact and conformal correlators is fitted in the long-time regime to determine αG(q), including αG(4) ≈ 0.1872.The fitted values are extrapolated in the cutoff and J for small q, and a Padé approximant stays within approximately one percent of the numerical answer.Source: G Numerical solution of the SD equations · pp. 66–67

Free energy

  • The finite-temperature free energy is evaluated from the Schwinger-Dyson solution, and subtracting S0 enables extraction of the specific-heat coefficient a3.The numerical evaluation agrees well with the zero-temperature entropy density S0(q), and the resulting data are used for figure 12.Source: G Numerical solution of the SD equations · p. 67

Numerical solutions

  • Numerical solutions at large βJ agree well with the conformal correlator plus its leading correction, subject to finite-cutoff effects.The real-time computation requires both infrared and ultraviolet frequency cutoffs, while the Euclidean iteration uses weighted updates for convergence.Source: G Numerical solution of the SD equations · pp. 66, 68

Real-time equations

  • Real-time continuation uses the spectral function to obtain retarded and Wightman correlators, while the Lyapunov exponent is found by imposing a unit eigenvalue condition on a discretized integral equation.The numerical iteration converges despite the additional infrared and ultraviolet cutoffs required in real time.Source: G Numerical solution of the SD equations · pp. 67–68

H A model without the reparametrization symmetry

This section studies a deformation that removes the reparametrization symmetry, regularizing the four-point function and shifting its chaotic growth below the maximal bound.

Deformation

  • The deformation is defined by time-dependent couplings with a factor |τ|−2α, while α → 0 recovers the original model to leading orders in 1/N.The modified Schwinger-Dyson equations admit a scale-invariant low-energy ansatz with dimension ˆ∆.Source: H A model without the… · p. 69

Modified kernel

  • For 0 < α < 1, the modified kernel has g < 1, so the h = 2 contribution is finite and the conformal four-point sum is regular.The deformation changes the scaling dimension to ˆ∆ < 1/q and moves the first pole to hp < 2.Source: H A model without the… · p. 70

Chaotic growth

  • The retarded kernel always has a solution with −1 < hchaos < 0, producing growth e^(−hchaos)2πt/β that is slower than the chaos bound.This follows because ˆkR(0) = q−1 > 1 while ˆkR(−1) = g < 1.Source: H A model without the… · p. 70

Small-deformation limit

  • As α approaches zero, the normalizable h = 2 contribution diverges like 1/α, with crossover controlled by the relative sizes of α and 1/(βJ).The associated effective reparametrization action contains explicit symmetry-breaking terms that distinguish the two regimes.Source: H A model without the… · pp. 70–71

Holographic interpretation

  • The construction is motivated by holographic settings in which a higher-dimensional conformal field theory induces an AdS2 sector without purely dynamical two-dimensional gravity.The discussion connects this perspective to Kondo models and related holographic examples.Source: H A model without the… · p. 71

I Further coments on Kinematic space

The section maps the finite-temperature bilocal theory to wave equations in Lorentzian black-hole regions and expresses the bulk-boundary relation in SL(2)-covariant embedding coordinates.

Exterior region

  • The bilocal two-point function determines the metric of the space in which the finite-temperature wave equation is defined outside the Lorentzian black hole.The construction begins by changing coordinates so the Casimir becomes the relevant wave operator.Source: I Further coments on Kinematic space · p. 71

Interior region

  • A complex thermal shift of the two times produces the interior-region wave operator and connects to particle creation accessible from both thermofield-double sides.The shift is t1 − t2 → t1 − t2 + iβ/2.Source: I Further coments on Kinematic space · p. 71

Embedding coordinates

  • Embedding coordinates represent boundary points projectively, and pairs of boundary points define bulk points obeying the AdS2 hyperboloid constraint.The resulting formulas are manifestly covariant under SL(2).Source: I Further coments on Kinematic space · p. 72

I.0.1 The kinematic space in the full model at q = ∞

At q = ∞, the kinematic-space kernel becomes an AdS2 Laplacian with a Dirichlet cutoff, providing a simple but qualified geometric interpretation of the full model.

q = ∞ limit

  • At q → ∞, the singlet spectrum retains only the h = 2 state while other states decouple without becoming heavy, unlike in large-’t Hooft-coupling gauge theories.This motivates analyzing the kernel using the bulk coordinates introduced earlier.Source: §I.0.1 · p. 72

Kernel geometry

  • After changing variables to t and z = σ + ϵ, the kernel becomes twice the AdS2 Laplacian with Dirichlet boundary conditions at z = ϵ.The corresponding wave equation has a cutoff at z = ϵ, and normalizable functions vanish at the cutoff boundary.Source: §I.0.1 · p. 72

Regularization

  • The resulting quadratic term has a simple local form, and the construction realizes the same type of cutoff used to regularize AdS2 computations.At finite temperature, the geometry becomes dS2 or AdS2 with periodic time, with an additional rescaling of circle sizes.Source: §I.0.1 · p. 72

Limitation

  • The geometric interpretation remains tentative because it is unclear whether the model admits a local AdS description, and the construction may regulate kinematic rather than actual bulk space.This qualification limits how directly the cutoff should be interpreted holographically.Source: §I.0.1 · p. 72
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