SYK Bilocal Collective Fields as Near-AdS2 Data

The Sachdev–Ye–Kitaev model becomes tractable at large $N$ because its disorder-averaged dynamics can be rewritten in terms of a two-point function $G$ and self-energy $\Sigma$. These bilocal fields are collective variables of a specified ensemble; they are not local bulk fields and do not by themselves establish a gravity dual.

Required background. SYK Models and Local Quantum Criticality supplies the Hamiltonian, disorder normalization, and large-$N$ limit.

Helpful background. Vector Models, Auxiliary Fields, and Large-N Saddles develops collective-field saddle methods; Replica and Supersymmetry Methods for Disorder distinguishes quenched and annealed averages; Near-AdS2, SYK, and the Duality Interface states what the gravity comparison does and does not identify.

Evidence cutoff: 25 July 2026.

From random couplings to bilocal variables

Take $N$ Majorana fermions with even interaction order $q$,

$$H=i^{q/2}\!\!\sum_{1\le i_1<\cdots<i_q\le N} J_{i_1\cdots i_q}\,\chi_{i_1}\cdots\chi_{i_q}, \qquad \overline{J_{i_1\cdots i_q}^{\,2}} =\frac{(q-1)!\,J^2}{N^{q-1}} .$$

The overbar is a Gaussian disorder average. Introducing

$$G(\tau_1,\tau_2)=\frac1N\sum_i\chi_i(\tau_1)\chi_i(\tau_2)$$

with a Lagrange multiplier $\Sigma$, then integrating out the fermions, gives the replica-diagonal Euclidean action

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

The factor $1/2$ reflects Majorana antisymmetry. Varying the action yields the Schwinger–Dyson equations

$$(\partial_\tau-\Sigma)\circ G=\delta, \qquad \Sigma(\tau_1,\tau_2)=J^2G(\tau_1,\tau_2)^{q-1},$$

where $(A\circ B)(\tau_1,\tau_3)=\int d\tau_2\,A(\tau_1,\tau_2)B(\tau_2,\tau_3)$. This derivation, including the large-$N$ counting and fluctuation kernel, follows Maldacena and Stanford 2016, §§2–3.

First application: solve the translation-invariant saddle

For an equilibrium saddle, $G(\tau_1,\tau_2)=G(\tau_1-\tau_2)$. Fourier transformation reduces the convolution equation to

$$G(i\omega_n)=\frac{1}{-i\omega_n-\Sigma(i\omega_n)}.$$

A reproducible numerical solution alternates this algebraic update with $\Sigma(\tau)=J^2G(\tau)^{q-1}$, transforms between time and frequency, and mixes successive iterates. At fixed $q$, $J$, inverse temperature $\beta$, and antiperiodic Matsubara grid, the output is the ensemble-averaged two-point saddle. The conformal power law appears only for

$$\frac1N\ll \frac{|\omega|}{J},\,\frac{T}{J}\ll1;$$

the ultraviolet derivative controls the crossover at $\lvert\omega\rvert\sim J$, while finite-$N$ fluctuations eventually invalidate the saddle. Bilocal fluctuations around this solution organize the four-point kernel and the soft reparametrization sector Jevicki, Suzuki, and Yoon 2016.

Quenched, annealed, and replica assumptions

The thermodynamic free energy is quenched,

$$F_{\rm q}=-\beta^{-1}\overline{\log Z} =-\beta^{-1}\lim_{n\to0}\frac{\overline{Z^n}-1}{n},$$

whereas the single-copy collective action directly computes an annealed object $-\beta^{-1}\log\overline Z$. Replica-diagonal dominance can make the two agree at leading large $N$ in a stated regime, but that is a saddle statement, not an identity. Replica-nondiagonal saddles and exponentially small spectral effects matter precisely where ensemble and fixed-realization questions become sharp.

Adversarial control: change the average

Repeat the calculation for several fixed coupling realizations before averaging, or permit replica-nondiagonal $G_{ab}$. The smooth large-$N$ saddle remains useful for short-time and coarse spectral observables, but sample-specific levels fluctuate and the quenched free energy need not equal its annealed counterpart. Thus the bilocal derivation licenses an ensemble-averaged large-$N$ effective description. It does not license a unique Hamiltonian, a factorized nonperturbative completion, or exact finite-$N$ late-time dynamics.

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

• Jevicki, Antal, Kenta Suzuki, and Junggi Yoon. “Bi-Local Holography in the SYK Model.” Journal of High Energy Physics 2016, 7 (2016): 7. DOI.
• Maldacena, Juan, and Douglas Stanford. “Remarks on the Sachdev–Ye–Kitaev Model.” Physical Review D 94, 106002 (2016). DOI. Open PDF.