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arXiv:2609.16130

Single-Instance Observables in the Sachdev-Ye-Kitaev Model

Brian Swingle

hep-thcond-mat.str-elquant-phSubmitted Sep 14, 2026• Announced Sep 16, 2026• 0 citations
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Abstract

We present a diagrammatic expansion for thermal expectation values of low-weight Majorana strings in a fixed realization of the $N$-fermion Sachdev--Ye--Kitaev (SYK) model. At each order in the expansion, the observable is expressed as a finite sum of polynomials in the couplings multiplied by temperature-dependent kernels built from the large-$N$ melonic propagator. At fixed $βJ$, the expansion is organized in powers of $N^{-1/2}$, and each order can be efficiently evaluated with classical resources polynomial in $N$. The expansion also extends to correlations at arbitrary imaginary times and sufficiently short real times. For the four-body SYK model, we show that the diagrammatic predictions give an excellent account of exact diagonalization results at sizes up to $N=24$. We also apply the method to the Maldacena--Qi (MQ) model formed from two identical copies of a fixed SYK realization. This enables the identification of a nearby SYK-like Hamiltonian whose thermofield double state has higher fidelity with the Maldacena--Qi ground state than the thermofield double of the original SYK Hamiltonian.

Comments
78 pages, produced in part with AI tools as described in acknowledgments, 5 minute video abstract https://youtu.be/TbxvKG-AW2U

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We present a diagrammatic expansion for thermal expectation values of low-weight Majorana strings in a fixed realization of the NNN-fermion Sachdev--Ye--Kitaev (SYK) model. At each order in the expansion, the observable is expressed as a finite sum of polynomials in the couplings multiplied by temperature-dependent kernels built from the large-NNN melonic propagator. At fixed βJβJβJ, the expansion is organized in powers of N−1/2N^{-1/2}N−1/2, and each order can be efficiently evaluated with classical resources polynomial in NNN. The expansion also extends to correlations at arbitrary imaginary times and sufficiently short real times. For the four-body SYK model, we show that the diagrammatic predictions give an excellent account of exact diagonalization results at sizes up to N=24N=24N=24. We also apply the method to the Maldacena--Qi (MQ) model formed from two identical copies of a fixed SYK realization. This enables the identification of a nearby SYK-like Hamiltonian whose thermofield double state has higher fidelity with the Maldacena--Qi ground state than the thermofield double of the original SYK Hamiltonian.
The Sachdev-Ye-Kitaev (SYK) model, in its simplest form, describes kkk Majorana fermions with random all-to-all four-body interactions. We consider the SYK model in the framework of a many-body Altland-Zirnbauer classification that sees the system as belonging to one of eight (real) symmetry classes depending on the value of kmod8k\phantom{\rule{3.33333pt}{0ex}}mod\phantom{\rule{0.28em}{0ex}}8kmod8. We show that, depending on the symmetry class, the system may support exact many-body zero modes with the symmetries also dictating whether these may have a nonzero contribution to Majorana fermions, i.e., single-particle weight. These zero modes appear in all but two of the symmetry classes. When present, they leave clear signatures in physical observables that go beyond the threefold (Wigner-Dyson) possibilities for level spacing statistics studied earlier. Signatures we discover include a zero-energy peak or hole in the single-particle spectral function, depending on whether symmetries allow or forbid zero modes to have single-particle weight. The zero modes are also shown to influence the many-body dynamics, where signatures include a nonzero long-time limit for the out-of-time-order correlation function. Furthermore, we show that the extension of the four-body SYK model by quadratic terms can be interpreted as realizing the remaining two complex symmetry classes; we thus demonstrate how the entire tenfold Altland-Zirnbauer classification may emerge in the SYK model.
The Sachdev-Ye-Kitaev (SYK) model is a model of qqq interacting fermions whose large N limit is dominated by melonic graphs. In this review we first present a diagrammatic proof of that result by direct, combinatorial analysis of its Feynman graphs. Gross and Rosenhaus have then proposed a generalization of the SYK model which involves fermions with different flavors. In terms of Feynman graphs, these flavors can be seen as reminiscent of the colors used in random tensor theory. Applying modern tools from random tensors to such a colored SYK model, all leading and next-to-leading orders diagrams of the 2-point and 4-point functions in the large NNN expansion can be identified. We then study the effect of non-Gaussian average over the random couplings in a complex, colored version of the SYK model. Using a Polchinski-like equation and random tensor Gaussian universality, we show that the effect of this non-Gaussian averaging leads to a modification of the variance of the Gaussian distribution of couplings at leading order in NNN. We then derive the form of the effective action to all orders.
We derive an approximate analytical formula for the spectral density of the qqq-body Sachdev-Ye-Kitaev (SYK) model obtained by summing a class of diagrams representing leading intersecting contractions. This expression agrees with that of QQQ-Hermite polynomials, with QQQ a nontrivial function of $q\ensuremath{\ge}2$ and the number of Majorana fermions NNN. Numerical results, obtained by exact diagonalization, are in excellent agreement with this approximate analytical spectral density even for relatively small $N\ensuremath{\sim}8$. For $N\ensuremath{\gg}1$ and not close to the edge of the spectrum, we find that the approximate analytical spectral density simplifies to ${\ensuremath{\rho}}_{\mathrm{asym}}(E)=\mathrm{exp}[2{\mathrm{arcsin}}^{2}(E/{E}_{0})/\mathrm{log}\ensuremath{\eta}]$, where $\ensuremath{\eta}(N,q)$ is the suppression factor of the contribution of intersecting Wick contractions relative to nested contractions and E0{E}_{0}E0​ is the ground-state energy per particle. This spectral density reproduces the known result for the free energy in the large-qqq and large-NNN limit at arbitrary values of the temperature. In the infrared region, where the SYK model is believed to have a gravity dual, the analytical spectral density is given by $\ensuremath{\rho}(E)\ensuremath{\sim}\mathrm{sinh}[2\ensuremath{\pi}\sqrt{2}\sqrt{(1\ensuremath{-}E/{E}_{0})/(\ensuremath{-}\mathrm{log}\ensuremath{\eta})}]$. It therefore has a square-root edge, as in random matrix ensembles, followed by an exponential growth, a distinctive feature of black holes and also of low-energy nuclear excitations. Results for level statistics in this region confirm the agreement with random matrix theory. Physically this is a signature that, for sufficiently long times, the SYK model and its gravity dual evolve to a fully ergodic state whose dynamics only depends on the global symmetry of the system. Our results strongly suggest that random matrix correlations are a universal feature of quantum black holes and that the SYK model, combined with holography, may be relevant to modeling certain aspects of the nuclear dynamics.
We solve numerically the large-NNN Dyson-Schwinger equations for the Sachdev-Ye-Kitaev (SYK) model utilizing the Legendre polynomial decomposition and reaching ${10}^{\ensuremath{-}36}$ accuracy. Using this we compute the energy of the SYK model at low temperatures $T\ensuremath{\ll}J$ and obtain its series expansion up to T7.54{T}^{7.54}T7.54. While it was suggested that the expansion contains terms T3.77{T}^{3.77}T3.77 and T5.68{T}^{5.68}T5.68, we find that the first noninteger power of temperature is T6.54{T}^{6.54}T6.54, which comes from the two-point function of the fermion bilinear operator ${O}_{{h}_{1}}=\ensuremath{\chi}{\ensuremath{\partial}}_{\ensuremath{\tau}}^{3}\ensuremath{\chi}$ with scaling dimension ${h}_{1}\ensuremath{\approx}3.77$. The coefficient in front of T6.54{T}^{6.54}T6.54 term agrees well with the prediction of the conformal perturbation theory. We conclude that the conformal perturbation theory appears to work even though the SYK model is not strictly conformal.
We discuss a supersymmetric generalization of the Sachdev-Ye-Kitaev (SYK) model. These are quantum mechanical models involving NNN Majorana fermions. The supercharge is given by a polynomial expression in terms of the Majorana fermions with random coefficients. The Hamiltonian is the square of the supercharge. The N=1\mathcal{N}=1N=1 model with a single supercharge has unbroken supersymmetry at large NNN, but nonperturbatively spontaneously broken supersymmetry in the exact theory. We analyze the model by looking at the large NNN equation, and also by performing numerical computations for small values of NNN. We also compute the large NNN spectrum of ``singlet'' operators, where we find a structure qualitatively similar to the ordinary SYK model. We also discuss an N=2\mathcal{N}=2N=2 version. In this case, the model preserves supersymmetry in the exact theory and we can compute a suitably weighted Witten index to count the number of ground states, which agrees with the large NNN computation of the entropy. In both cases, we discuss the supersymmetric generalizations of the Schwarzian action which give the dominant effects at low energies.
We analyze a simple and efficient experimental protocol to cool the Sachdev-Ye-Kitaev (SYK) model to low temperatures. The protocol utilizes local couplings between two copies of an SYK model to create a gapped adiabatic path, between a high temperature product state and a low temperature thermofield double state. By smoothly varying the coupling strength between these two limits, one efficiently cools the SYK model. We support these predictions-and demonstrate fast cooling to the low-temperature gravitational regime-via exact numerical solutions to the large-N equations of motion that govern the ground state and dynamical properties of the coupled system. Finally, we present a theoretical framework based upon eigenstate thermalization that provides a microscopic explanation for the efficacy of the cooling protocol; intriguingly, this suggests that the protocol may be applicable for cooling strongly-interacting quantum Hamiltonians more broadly.