SYK Bilocal Collective Fields as Near-AdS2 Data

At large $N$, the quartic Sachdev–Ye–Kitaev (SYK) model replaces $N$ interacting fermions by two functions of two Euclidean times. $G$ records the collective two-point function, while $\Sigma$ enforces its definition and becomes its self-energy at the saddle. Their stationary equations are closed and can be solved numerically.

This page derives that replacement, computes a finite-temperature $q=4$ benchmark, and states exactly which disorder average and order of limits the answer represents. The resulting saddle and fluctuation kernel are useful near-AdS₂ comparison data. They are not local bulk fields, a unique microscopic Hamiltonian, or evidence for a complete gravity dual.

Required background. SYK Models and Local Quantum Criticality fixes the Majorana normalization, disorder variance, and no-minus Green-function convention used below.

Helpful background. Vector Models, Auxiliary Fields, and Large-N Saddles develops collective-field saddle methods; Replica and Supersymmetry Methods for Disorder separates exact averaging, analytic continuation, and saddle assumptions; Near-AdS₂, SYK, and the Duality Interface explains what the gravity comparison does and does not identify.

Scope. We study the standard even-$N$, Gaussian, quartic Majorana ensemble on a Euclidean thermal circle. The explicit saddle calculation is the one-copy annealed problem at leading order in $N$. Replica and fixed-realization statements are labeled separately.

Evidence cutoff: 29 August 2026.

The quartic ensemble and its thermal circle

Let $N$ be even and normalize the Majoranas by

$$\{\chi_i,\chi_j\}=\delta_{ij}, \qquad \chi_i^2=\frac12.$$

For $q=4$, take

$$H_J=\sum_{1\le i<j<k<l\le N} J_{ijkl}\chi_i\chi_j\chi_k\chi_l,$$

with independent Gaussian couplings

$$\overline{J_{ijkl}}=0, \qquad \overline{J_{ijkl}^{2}}=\frac{3!J^2}{N^3}.$$

The frequently used general-$q$ phase $i^{q/2}$ changes only an overall sign at $q=4$; the symmetric Gaussian distribution absorbs that sign. The displayed convention therefore agrees with the prerequisite page.

Define the no-minus Euclidean correlator

$$G_J(\tau_1,\tau_2) =\frac1N\sum_{i=1}^N \left\langle T_\tau\chi_i(\tau_1)\chi_i(\tau_2)\right\rangle_J.$$

It is antisymmetric in its two arguments. At a translation-invariant thermal saddle, write $G(\tau_1,\tau_2)=G(\tau_1-\tau_2)$ and use

$$\begin{aligned} G(i\omega_n) &=\int_0^\beta d\tau\, e^{i\omega_n\tau}G(\tau),\\ G(\tau) &=\frac1\beta\sum_{n\in\mathbb Z} e^{-i\omega_n\tau}G(i\omega_n),\\ \omega_n&=\frac{(2n+1)\pi}{\beta}. \end{aligned}$$

Before doing any calculation, distinguish the operations often compressed into the phrase “the SYK average.”

Which quantity is being computed?

Quantity or ansatz	Operation	Collective-field route	Strongest immediate statement
Fixed realization ZJ	Keep every coupling fixed	Not produced by disorder averaging	One sample-specific quantum system
Annealed log partition function	Average ZJ, then take the logarithm	The one-copy action derived below	Annealed thermodynamics of the declared ensemble
Quenched log partition function	Take log ZJ, then average	Integer replicas followed by a formal n → 0 continuation	Quenched thermodynamics, if the continuation and saddle are controlled
Replica-diagonal saddle	Set off-diagonal saddle fields to zero	An ansatz inside the replicated theory	A candidate saddle, not a proof of dominance
Replica-nondiagonal sector	Allow Gab and Σab for a ≠ b	Fluctuations and possibly additional saddles	Existence, stability, dominance, and the n → 0 limit are separate tests

Disorder contraction produces the G–Σ action

For one fixed coupling realization,

$$Z_J=\int\mathcal D\chi\, \exp\!\left[ -\frac12\sum_i\int_0^\beta d\tau\, \chi_i\partial_\tau\chi_i -\int_0^\beta d\tau\,H_J(\tau) \right].$$

The couplings are Gaussian, so their average can be done exactly. Define the off-shell fermion bilinear

$$G_\chi(\tau_1,\tau_2) =\frac1N\sum_i \chi_i(\tau_1)\chi_i(\tau_2).$$

The $O(N^4)$ index choices combine with the variance $O(N^{-3})$ to leave an $O(N)$ interaction:

$$\overline{Z_J} =\int\mathcal D\chi\, \exp\!\left[ -\frac12\int\chi\partial_\tau\chi +\frac{NJ^2}{8} \int d\tau_1d\tau_2\,G_\chi(\tau_1,\tau_2)^4 \right].$$

Insert a functional delta constraint that replaces $G_\chi$ by an independent antisymmetric field $G$ and introduces an antisymmetric multiplier $\Sigma$. Integrating the $N$ Majoranas gives a Pfaffian:

$$\overline{Z_J} =\mathcal N\int\mathcal DG\,\mathcal D\Sigma\, e^{-N\mathcal I[G,\Sigma]},$$

with

$$\boxed{ \mathcal I[G,\Sigma] =-\log\operatorname{Pf}(\partial_\tau-\Sigma) +\frac12\int d\tau_1d\tau_2 \left[ \Sigma G-\frac{J^2}{4}G^4 \right]. }$$

This is equivalently

$$-\log\operatorname{Pf}(\partial_\tau-\Sigma) =-\frac12\operatorname{Tr} \log(\partial_\tau-\Sigma).$$

It is not $-\tfrac12\log\operatorname{Pf}$: that expression would be $-\tfrac14\operatorname{Tr}\log$ and would miss a factor of two in the first saddle equation. The correctly normalized Pfaffian action appears in Maldacena and Stanford 2016, § 2.6, Eqs. (2.25)–(2.26).

The multiplier contour initially enforces the delta functional and is then deformed through an appropriate saddle. The normalization $\mathcal N$, contour choice, regulator, and functional Jacobian do not change the leading $O(N)$ stationary equations, but they matter for the $O(N^0)$ fluctuation determinant. Thus the displayed action is a leading large-$N$ object, not a complete finite-$N$ measure.

Stationarity gives two closed equations

For an antisymmetric operator $A$,

$$\delta\log\operatorname{Pf}A =\frac12\operatorname{Tr}(A^{-1}\delta A).$$

Take $A=\partial_\tau-\Sigma$. In the trace, the kernel order is reversed; antisymmetry then converts the variation of the Pfaffian term into $-\tfrac12\int G\,\delta\Sigma$. It cancels the explicit $+\tfrac12\int G\,\delta\Sigma$ precisely when

$$G=(\partial_\tau-\Sigma)^{-1}.$$

Variation with respect to $G$ gives the second equation. Together,

$$\begin{aligned} (\partial_\tau-\Sigma)\circ G&=\delta,\\ \Sigma(\tau_1,\tau_2)&=J^2G(\tau_1,\tau_2)^3, \end{aligned}$$

where

$$(A\circ B)(\tau_1,\tau_3) =\int_0^\beta d\tau_2\, A(\tau_1,\tau_2)B(\tau_2,\tau_3).$$

At a translation-invariant saddle these become

$$\begin{aligned} G(i\omega_n) &=\frac1{-i\omega_n-\Sigma(i\omega_n)},\\ \Sigma(\tau)&=J^2G(\tau)^3. \end{aligned}$$

Four elementary checks catch most sign and normalization errors:

• Fermion exchange: $G(\tau_1,\tau_2)=-G(\tau_2,\tau_1)$.
• Thermal reflection: antiperiodicity implies $G(\beta-\tau)=G(\tau)$ for $0<\tau<\beta$.
• Equal-time jump: $G(0^+)=\tfrac12$ and $G(0^-)=-\tfrac12$ in the no-minus convention.
• Ultraviolet tail: $G(i\omega_n)=(-i\omega_n)^{-1}+O(\omega_n^{-2})$.

These equations and checks are reviewed with the corresponding diagrammatics in Chowdhury et al. 2022, §§ II.B–II.D.

A reproducible finite-temperature saddle

Set $J=1$. For each $\beta J\in\{5,10,20,50\}$, solve the full equations rather than dropping the derivative term. Some references instead use $\mathcal J=\sqrt q\,J/2^{(q-1)/2}$; at $q=4$, $\mathcal J=J/\sqrt2$. Every temperature in the table is expressed using $J$.

The calculation uses

$$\tau_j=\frac{(j+\tfrac12)\beta}{M}, \qquad n=-\frac M2,\ldots,\frac M2-1,$$

with the matching finite Fourier pair. Start from $G_0(i\omega_n)=(-i\omega_n)^{-1}$. Before transforming to time, subtract this known ultraviolet tail and add its exact value $G_0(\tau)=1/2$ for $0<\tau<\beta$. One iteration is

$$G(i\omega_n) \longrightarrow G(\tau) \longrightarrow \Sigma(\tau) \longrightarrow \Sigma(i\omega_n) \longrightarrow \widetilde G(i\omega_n),$$

followed by

$$G_{k+1}=(1-x)G_k+x\widetilde G_k, \qquad x=0.25.$$

Stop when the relative fixed-point update is below $10^{-13}$. Independently report the Dyson residual

$$R_{\mathrm{SD}} =\max_n\left\lvert \left[-i\omega_n-\Sigma(i\omega_n)\right] G(i\omega_n)-1 \right\rvert.$$

The primary grid has $M=1024$ points. A second calculation with $M=2048$ supplies a discretization estimate. For comparison, the thermal conformal solution for $0<\tau<\beta$ is

$$G_c(\tau) =b\left[ \frac{\pi}{\beta J\sin(\pi\tau/\beta)} \right]^{1/2}, \qquad b^4=\frac1{4\pi}.$$

Finite-temperature q = 4 saddle, J = 1 and M = 2048

βJ	G(β/4)	G(β/2)	Gc(β/2)	Midpoint difference from conformal	1024 → 2048 midpoint shift	RSD
5	0.391691	0.364121	0.421005	−13.51%	2.26 × 10−8	1.8 × 10−14
10	0.313992	0.279188	0.297696	−6.22%	3.02 × 10−8	3.5 × 10−14
20	0.236904	0.204449	0.210503	−2.88%	3.50 × 10−8	5.0 × 10−14
50	0.155164	0.131678	0.133134	−1.09%	3.02 × 10−8	9.3 × 10−14

The four runs converged in 77–82 iterations. On the $M=1024$ grid, Fourier closure was better than $4.4\times10^{-14}$, reflection symmetry better than $1.3\times10^{-15}$, and imaginary leakage below $2.9\times10^{-16}$. The median ultraviolet-tail error over the highest one-eighth of Matsubara modes ranged from $1.0\times10^{-6}$ at $\beta J=5$ to $1.1\times10^{-4}$ at $\beta J=50$.

The largest last-grid shift, $3.6\times10^{-8}$ in $G(\beta/2)$, is a discretization estimate rather than a rigorous error bound. It dominates the iteration error but is far smaller than the physical difference from the conformal approximation. The systematic trend is clear: the full saddle approaches the conformal answer as $\beta J$ grows, while the short-time endpoints remain ultraviolet-sensitive. The algorithm follows Maldacena and Stanford 2016, Appendix G, with every grid, tail, mixing, stopping, and comparison choice stated here.

Bilocal fluctuations organize four-point data

Write fluctuations around the saddle as

$$G=G_\star+N^{-1/2}g, \qquad \Sigma=\Sigma_\star+N^{-1/2}\sigma.$$

The linear term vanishes by stationarity. The quadratic $O(N^0)$ action couples $g$ and $\sigma$. Eliminating $\sigma$ produces the ladder kernel; in index-free notation its action contains the operator $1-K$, with

$$K(\tau_1,\tau_2;\tau_3,\tau_4) =-3J^2 G_\star(\tau_{13})G_\star(\tau_{24}) G_\star(\tau_{34})^2$$

for $q=4$ in this convention. The inverse $(1-K)^{-1}$ sums ladder diagrams and gives the connected four-point function. A nearly unit eigenvalue in the reparametrization channel is the precursor of the Schwarzian soft mode. The derivation and its normalization are given in Maldacena and Stanford 2016, §§ 3.1 and 4; the bilocal fluctuation viewpoint is developed in Jevicki, Suzuki, and Yoon 2016, §§ 2–3.

The pair $(\tau_1,\tau_2)$ can be reorganized into a center time and a relative time, which makes some conformal structures resemble a field in one extra dimension. That is a useful kinematic representation, not a proof of locality: the Pfaffian and the full quadratic kernel remain nonlocal in the bilocal coordinates. The next page, SYK Conformal Regime and Schwarzian Matching, extracts the soft sector without identifying every bilocal fluctuation with a local bulk particle.

Three infrared boundaries are physically different

The shorthand “finite-$N$ cutoff” hides distinct mechanisms. For a low-energy state with entropy $S(E)=O(N)$, the relevant hierarchy is parametrically

$$J \gg \frac JN \gg \delta E(E), \qquad \delta E(E)\sim J e^{-S(E)},$$

up to model- and energy-dependent prefactors.

Do not collapse the three boundaries into one cutoff

Boundary	Parametric scale	What changes	What remains useful
Ultraviolet crossover	Energy of order J	The derivative term cannot be dropped	The full bilocal Schwinger–Dyson equations
Quantum Schwarzian crossover	Energy parametrically of order J/N	The classical conformal saddle is insufficient; the soft mode fluctuates strongly	The Schwarzian effective theory in its stated window
Sample-specific discreteness	Mean level spacing of order J e−S(E)	A smooth density cannot resolve individual levels or recurrences	Coarse or averaged spectral observables above the spacing scale

Thus $J/N$ is not the many-body level spacing. The classical conformal window may be summarized as $J/N\ll\max(T,|\omega|)\ll J$ only when one is specifically discussing the leading conformal saddle. Below $J/N$, the Schwarzian sector is the next effective description; exponentially farther into the infrared, a fixed sample’s discrete spectrum becomes unavoidable.

Quenched and annealed claims need separate evidence

The one-copy derivation computed $\overline{Z_J}$. The quenched quantity begins instead with an integer replica number:

$$\overline{Z_J^n} =\mathcal N_n \int\mathcal DG_{ab}\,\mathcal D\Sigma_{ab}\, e^{-N\mathcal I_n[G,\Sigma]},$$

where

$$\begin{aligned} \mathcal I_n ={}&-\log\operatorname{Pf} \left(\partial_\tau\delta_{ab}-\Sigma_{ab}\right)\\ &+\frac12\sum_{a,b=1}^n \int d\tau_1d\tau_2 \left[ \Sigma_{ab}G_{ab} -\frac{J^2}{4}G_{ab}^{4} \right]. \end{aligned}$$

Here $G_{ab}(\tau_1,\tau_2)=-G_{ba}(\tau_2,\tau_1)$, and likewise for $\Sigma_{ab}$. The quenched free energy is formally

$$F_{\mathrm q} =-\frac1\beta\overline{\log Z_J} =-\frac1\beta \lim_{n\to0} \frac{\overline{Z_J^n}-1}{n}.$$

The replica-diagonal choice $G_{ab}=\delta_{ab}G$ and $\Sigma_{ab}=\delta_{ab}\Sigma$ reduces the saddle to $n$ copies of the one-copy equations. Calling this an ansatz keeps the logic honest: dominance, analytic continuation, and the absence of another saddle are additional questions.

At every finite $N$, Jensen’s inequality gives

$$\overline{\log Z_J} \le \log\overline{Z_J}, \qquad F_{\mathrm q}\ge F_{\mathrm a}.$$

Equality of leading free-energy densities is nevertheless much better supported for the standard fermionic ensemble than an appeal to replica diagonality alone:

• In the normalization studied by Anschuetz, Chen, Kiani, and King, the intensive annealed–quenched log-partition gap is bounded by $O_q(\beta^2N^{-q/2})$. It therefore vanishes for $\beta=o(N^{q/4})$—for $q=4$, $\beta=o(N)$—but the result does not reach exponentially low temperatures or identify finite-$N$ samples. Constants must be translated before combining their normalization with the $J$ convention on this page Anschuetz et al. 2025, § IV, Eq. (9), and Appendix C, Corollary C.1.
• For every fixed even $q$, Gamarnik, Pernice, Schmidhuber, and Zlokapa prove the existence and compute the common annealed and quenched free-energy density at sufficiently small but constant $\beta$ for their Rademacher-coupling ensemble and normalized trace. Their $q=4$ numerical comparison agrees with the physics bilocal equations over the tested interval $\beta\in[0,3]$; that interval is not itself the theorem’s full rigorously established range. Applying the theorem literally to the Gaussian $J$ convention above requires a separate universality and normalization translation Gamarnik et al. 2026, § 2, Theorem 5, and § 5, Eqs. (40)–(41) and Figure 3.
• Replica-nondiagonal solutions do exist in controlled ansätze. Aref’eva and collaborators found finite-integer-replica candidates that are subleading and singular as $n\to0$, alongside strong-coupling conformal candidates whose regulated continuation requires separate interpretation Aref’eva et al. 2019, §§ 3–5 and Appendix B.
• Numerical long-time correlators support the replica-diagonal saddle plus Gaussian fluctuations—including off-diagonal fluctuation components—on times shorter than the inverse many-body level spacing Wang et al. 2019, §§ 3–5. Independent analytic and spectral tests found no spin-glass phase in the standard model within their examined regimes Gur-Ari, Mahajan, and Vaezi 2018, §§ 2–3.

These statements concern the standard fermionic SYK ensemble. Bosonic or spin variants can behave differently; the comparison in Baldwin and Swingle 2020, §§ II–IV is a warning against transferring the conclusion by name alone.

Adversarial control: change the averaging hypothesis

Now change one assumption at a time.

Surviving claim under an adversarial change of average

Change	What can survive	What no longer follows	Reason for the downgrade
Annealed → quenched	Leading free-energy density and coarse correlators in regimes covered by concentration or a controlled saddle	Finite-N equality or sample-specific spectral data	The logarithm and disorder average do not commute
Add off-diagonal Gaussian fluctuations	Replica-diagonal leading saddle and corrected four-point functions	A strictly diagonal fluctuation theory	Off-diagonal fluctuations need not imply an off-diagonal saddle
Admit a dominant nondiagonal saddle	The exact replicated functional integral	The one-copy saddle as the full quenched answer	The saddle and possibly its n → 0 continuation have changed
Ensemble → one fixed realization	Self-averaging coarse observables in a demonstrated regime	Exact late-time recurrences, individual levels, or a unique nonperturbative completion	The smooth collective saddle has discarded realization-specific information

The strongest claim that survives all of these tests is deliberately limited: a normalized large-$N$ SYK saddle, its fluctuation kernel, and their averaging convention can supply coarse infrared data that match selected near-AdS₂ observables. Annealed–quenched agreement of an intensive quantity does not turn an ensemble into one Hamiltonian; replica-nondiagonal saddle existence does not establish dominance; and melonic bilocal dynamics does not by itself prove a gravity dual.

Common pitfalls

Halving the Pfaffian twice. A Majorana Gaussian integral gives a Pfaffian, and $\log\operatorname{Pf}A=\tfrac12\operatorname{Tr}\log A$. Writing both the Pfaffian and an additional factor $1/2$ loses a factor of two.

Calling the one-copy action replica diagonal. The one-copy action has no replica indices and computes an annealed object. “Replica diagonal” describes an ansatz made only after constructing $\overline{Z^n}$.

Equating existence with dominance. A stationary point can be subleading, unstable, or singular under $n\to0$. Its action, Hessian, contour, and continuation must be checked.

Using $1/N$ as a level spacing. The scale $J/N$ governs the quantum soft-mode crossover. A typical many-body level spacing is exponentially smaller, of order $Je^{-S(E)}$.

Turning bilocal coordinates into a local bulk. Reorganizing two times as center and relative coordinates is kinematically useful. It does not remove the nonlocal Pfaffian or prove a local two-dimensional bulk action.

Exercises

1. Pfaffian versus determinant

Show why replacing $-\log\operatorname{Pf}A$ by $-\tfrac12\log\operatorname{Pf}A$ changes the saddle normalization.

Solution

For antisymmetric $A$,

$$\operatorname{Pf}(A)^2=\det A,$$

so

$$-\log\operatorname{Pf}A =-\frac12\operatorname{Tr}\log A.$$

An extra factor $1/2$ would instead give $-\tfrac14\operatorname{Tr}\log A$. Its variation would be half as large, while the explicit $\Sigma G$ term would be unchanged. The resulting stationarity condition could not be $G=A^{-1}$ with the displayed interaction normalization.

2. Vary the collective action

Derive both Schwinger–Dyson equations from $\mathcal I[G,\Sigma]$.

Solution

Use

$$\delta\log\operatorname{Pf}A =\frac12\operatorname{Tr}(A^{-1}\delta A), \qquad A=\partial_\tau-\Sigma.$$

Keeping the reversed kernel order inside the trace and using antisymmetry, the $\Sigma$ variation gives

$$G=(\partial_\tau-\Sigma)^{-1}.$$

The $G$ variation is

$$\frac12\int d\tau_1d\tau_2\, \left(\Sigma-J^2G^3\right)\delta G,$$

so $\Sigma=J^2G^3$. Multiplying the first equation by $\partial_\tau-\Sigma$ gives the convolution form.

3. Equal-time jump and ultraviolet tail

Find $G(0^\pm)$ and infer the leading large-frequency behavior.

Solution

The no-minus convention and $\chi_i^2=1/2$ give

$$G(0^+)=\frac12.$$

Fermionic time ordering exchanges the operators across the origin, so $G(0^-)=-1/2$. The unit discontinuity is the Green function of the first-order kinetic operator, hence

$$G(i\omega_n)\sim\frac1{-i\omega_n}.$$

This is also obtained directly by neglecting $\Sigma(i\omega_n)$ relative to $\omega_n$ at large frequency.

4. Read the numerical control

At $\beta J=50$, compare the $M=2048$ midpoint value with the conformal value. Which numerical uncertainty dominates?

Solution

The fractional difference is

$$\frac{0.131678-0.133134}{0.133134} \simeq-1.09\%.$$

The last grid shift is $3.02\times10^{-8}$, whereas the Dyson residual is $9.3\times10^{-14}$. Discretization therefore dominates iteration error. The 1.09% physical difference from the conformal approximation is still much larger than either numerical diagnostic.

5. Jensen’s inequality and free energies

Determine the ordering of annealed and quenched free energies.

Solution

Because $\log$ is concave,

$$\overline{\log Z_J}\le\log\overline{Z_J}.$$

Multiplying by $-1/\beta$ reverses the inequality:

$$F_{\mathrm q}\ge F_{\mathrm a}.$$

If their difference divided by $N$ vanishes in a large-$N$ regime, the leading free-energy densities agree there. The finite-$N$ operations remain distinct.

6. Separate the infrared scales

Suppose $N=100$ and a relevant energy window has entropy $S(E)=0.2N$. Compare $J/N$ with the rough level-spacing estimate $Je^{-S(E)}$.

Solution

The soft-mode scale is

$$\frac JN=10^{-2}J.$$

The spacing estimate is

$$Je^{-S(E)}=Je^{-20}\simeq2.1\times10^{-9}J.$$

They differ by almost seven orders of magnitude. Crossing $J/N$ calls for quantum Schwarzian dynamics; resolving $Je^{-S(E)}$ calls for the individual spectrum of a fixed finite-$N$ sample.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Anschuetz, Eric R., Chi-Fang Chen, Bobak T. Kiani, and Robbie King. “Strongly Interacting Fermions Are Non-Trivial Yet Non-Glassy.” Physical Review Letters 135, 030602 (2025). DOI and open article. Open PDF.
• Aref’eva, Irina, Mikhail Khramtsov, Maria Tikhanovskaya, and Igor Volovich. “Replica-Nondiagonal Solutions in the SYK Model.” Journal of High Energy Physics 2019, 7 (2019): 113. DOI and open article. Open PDF.
• Baldwin, Christopher L., and Brian Swingle. “Quenched versus Annealed: Glassiness from SK to SYK.” Physical Review X 10, 031026 (2020). DOI and open article. Open PDF.
• Chowdhury, Debanjan, Antoine Georges, Olivier Parcollet, and Subir Sachdev. “Sachdev–Ye–Kitaev Models and Beyond: A Window into Non-Fermi Liquids.” Reviews of Modern Physics 94, 035004 (2022). DOI. Open PDF.
• Gamarnik, David, Francisco Pernice, Alexander Schmidhuber, and Alexander Zlokapa. “The Free Energy Limit of the SYK Model at High Temperature.” arXiv:2605.02768 [cond-mat.dis-nn, hep-th, math.PR, quant-ph] (2026). Open PDF.
• Gur-Ari, Guy, Raghu Mahajan, and Abolhassan Vaezi. “Does the SYK Model Have a Spin Glass Phase?” Journal of High Energy Physics 2018, 11 (2018): 070. DOI and open article. Open PDF.
• Jevicki, Antal, Kenta Suzuki, and Junggi Yoon. “Bi-Local Holography in the SYK Model.” Journal of High Energy Physics 2016, 7 (2016): 007. DOI and open article. Open PDF.
• Maldacena, Juan, and Douglas Stanford. “Remarks on the Sachdev–Ye–Kitaev Model.” Physical Review D 94, 106002 (2016). DOI. Open PDF.
• Wang, Hanteng, Dmitry Bagrets, Alexander L. Chudnovskiy, and Alex Kamenev. “On the Replica Structure of the Sachdev–Ye–Kitaev Model.” Journal of High Energy Physics 2019, 9 (2019): 057. DOI and open article. Open PDF.