SYK Models and Local Quantum Criticality

The Sachdev–Ye–Kitaev (SYK) model is a solvable large-$N$ quantum system with random all-to-all interactions, no quasiparticles, emergent conformal time dependence, and maximal chaos in its simplest low-temperature regime. It realizes local criticality because it has time but no spatial momentum. Extending those results to a lattice or material requires extra structure and does not preserve every property automatically.

Required background. Schwinger–Dyson Hierarchies and Their Inputs supplies the closed two-point equations; Tensor Large-N and Melonic Dominance supplies the diagrammatic large-$N$ organization. Helpful background. The Eigenstate Thermalization Hypothesis supplies the relation between spectral statistics, chaos, and thermalization.

Majorana normalization and disorder

Take $N$ even Majorana fermions with

$$\{\chi_i,\chi_j\}=\delta_{ij}, \qquad H=\sum_{i<j<k<l}J_{ijkl}\chi_i\chi_j\chi_k\chi_l,$$

and independent Gaussian couplings

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

The $3!$ and the anticommutator convention fix all later numerical constants. Another convention with $\{\chi_i,\chi_j\}=2\delta_{ij}$ must rescale $J$ and $G$.

Use the no-minus Majorana convention

$$G(\tau)=\frac1N\sum_i \langle T_\tau\chi_i(\tau)\chi_i(0)\rangle, \qquad G(i\omega_n)=\int_0^\beta d\tau\,e^{i\omega_n\tau}G(\tau).$$

After disorder averaging and taking $N\to\infty$, melonic diagrams close its Schwinger–Dyson equations:

$$G(i\omega_n)^{-1}=-i\omega_n-\Sigma(i\omega_n), \qquad \Sigma(\tau)=J^2G(\tau)^3.$$

The large-$N$ limit is taken before the deep infrared. At finite $N$, the many-body level spacing and a scale of order $J/N$ eventually invalidate the continuous saddle description.

Conformal solution

For $|\omega|\ll J$, drop the $-i\omega$ term in the saddle equation. A scale-invariant solution is

$$G_c(\tau)= b\,\frac{\operatorname{sgn}\tau}{(J|\tau|)^{1/2}}, \qquad b^4=\frac1{4\pi}.$$

Thus each Majorana has infrared dimension $\Delta_\chi=1/4$. More generally, a $q$-body SYK interaction has $\Delta_\chi=1/q$ with variance convention $(q-1)!J^2/N^{q-1}$. The exact finite-temperature conformal form follows from the reparameterization $\tau\mapsto\tan(\pi\tau/\beta)$, while soft reparameterization modes generate Schwarzian corrections.

Sachdev and Ye first obtained a related large-$N$ locally critical spin liquid Sachdev and Ye 1993, pp. 3339–3342. Maldacena and Stanford developed the conformal four-point function, soft mode, and chaos exponent for the Majorana model Maldacena and Stanford 2016, §§ 2–4.

Entropy, chaos, and order of limits

The large-$N$ saddle has an extensive zero-temperature entropy obtained by first taking $N\to\infty$ and then $T\to0$. This does not imply that a generic finite-$N$ Hamiltonian has an exactly extensive ground-state degeneracy. The limits do not commute, and the low-temperature Schwarzian/level regime restores finite-size spectral discreteness.

The out-of-time-order four-point function grows with Lyapunov exponent

$$\lambda_L=\frac{2\pi T}{\hbar}$$

in the conformal strong-coupling limit, saturating the general chaos bound under its analyticity assumptions. This concerns operator growth, not a transport relaxation time: the zero-dimensional model has no momentum or conductivity.

From dots to lattices

Arrays of SYK dots, models with complex fermions and conserved charge, or dispersing fermions hybridized with SYK islands can produce diffusion and incoherent metals. Their transport depends on intersite coupling, charge conservation, disorder correlations, and the order of the large-$N$, low-$T$, and long-wavelength limits. Song, Jian, and Balents constructed a translationally invariant higher-dimensional non-Fermi liquid inspired by this mechanism Song, Jian, and Balents 2017, pp. 216601-1–216601-6.

“Local criticality” means momentum-independent leading critical correlations in the specified sector. It does not prove an infinite dynamical exponent for every observable, identify a heavy-fermion material, or eliminate spatial collective modes. Random-coupling self-averaging is an assumption to test when importing the model to a clean system.

Current evidence boundary

Primary SYK and quantum-matter literature was checked through 10 August 2026. The large-$N$ dot is analytically controlled, while lattice coherence, finite-dimensional stability, experimental realization, and transport coefficients are model-specific. Ongoing realization and benchmark claims belong in Quantum Matter and Emergence Research.

Exercises

1. Determine the infrared fermion dimension for a $q$-body SYK interaction.

Solution

If $G(\tau)\sim|\tau|^{-2\Delta}$, then $\Sigma(\tau)\sim G^{q-1}\sim|\tau|^{-2\Delta(q-1)}$. The conformal convolution $G*\Sigma=-1$ requires $2\Delta+2\Delta(q-1)=2$, hence $\Delta=1/q$.

2. Why does maximal chaos not imply a Planckian electrical resistivity in the SYK dot?

Solution

The dot has no spatial current or momentum, so no electrical conductivity is defined. $\lambda_L$ measures growth of an out-of-time-order correlator. A lattice extension needs a current operator and a momentum-relaxation mechanism before transport can be compared.

References

• Maldacena, J., and D. Stanford. “Remarks on the Sachdev–Ye–Kitaev Model.” Physical Review D 94 (2016): 106002. DOI.
• Sachdev, S., and J. Ye. “Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet.” Physical Review Letters 70 (1993): 3339–3342. DOI.
• Song, X.-Y., C.-M. Jian, and L. Balents. “Strongly Correlated Metal Built from Sachdev–Ye–Kitaev Models.” Physical Review Letters 119 (2017): 216601. DOI.