---
title: Complex Sachdev-Ye-Kitaev Model
url: https://www.emergentmind.com/topics/complex-sachdev-ye-kitaev-model
type: topic
---

# Complex Sachdev-Ye-Kitaev Model

The complex Sachdev-Ye-Kitaev (cSYK) model is a quantum many-body system of $N$ complex fermions with random all-to-all $q$-body interactions and global $U(1)$ charge conservation. Distinguished by maximal non-Fermi liquid behavior, emergent conformal symmetry, and quantum chaos, the cSYK model has become a central paradigm in quantum statistical mechanics, random matrix classification, quantum gravity duality (notably AdS$_2$/SYK correspondence), and experimental quantum simulation. It generalizes the original Majorana-based SYK model by allowing for tunable chemical potential and richer symmetry structures.

## 1. Fundamental Definition and Formal Structure

The cSYK Hamiltonian for $N$ complex fermions $c_i, c_i^\dagger$ ($i=1,\dots,N$) at $q=4$ is
\[
H = \sum_{i<j<k<l} (J_{ijkl}\,c_i^\dagger\,c_j^\dagger\,c_k\,c_l + {\rm h.c.}) - \mu \sum_i c_i^\dagger c_i,
\]
where $J_{ijkl}$ are complex Gaussian random variables with mean zero and variance $\langle|J_{ijkl}|^2\rangle = J^2/(8N^3)$; $\mu$ is the chemical potential. The model conserves
\[
Q = \sum_{i=1}^N (c_i^\dagger c_i - 1/2)
\]
and is solvable at large $N$ by disorder-averaged path integral, introducing bilocal fields $G(\tau_1,\tau_2)$ and $\Sigma(\tau_1,\tau_2)$. The Schwinger–Dyson equations in Matsubara frequency are
\[
G(i\omega_n) = \frac{1}{-i\omega_n - \mu - \Sigma(i\omega_n)}, \quad
\Sigma(\tau) = -J^2 G(\tau)^2 G(-\tau),
\]
with fermionic Matsubara frequencies $\omega_n = (2n+1)\pi/\beta$. Additional symmetry and spectral features are accessed by tuning $q$, imposing chemical potential, and specializing to chiral/discrete symmetry points [2103.16270], [1912.09975], [1910.14099].

## 2. Thermodynamic Phases and Criticality

The cSYK model displays an emergent phase structure controlled by temperature $T$ and chemical potential $\mu$. At fixed $\mu$, the system admits two competing thermodynamic branches:

- **Low-$T$ Gapped Phase:** Weakly interacting, perturbative fermions with discrete energy levels (energy gap appears for sufficiently large $\mu$).
- **High-$T$ Gapless (SYK) Phase:** Strongly interacting non-Fermi liquid with continuous spectrum, characterized by power-law Green’s functions and chaotic dynamics.

The transition between these branches is a first-order phase transition, analogous to the van der Waals-Maxwell liquid-gas system:
- The (μ,𝒬) coexistence curve below critical temperature $T_c$ is multivalued, requiring Maxwell equal-area construction.
- The critical endpoint at $(T_c,\mu_c)\approx(0.06828J, 0.3443J)$ is characterized by non-mean-field critical exponents:
  - $\alpha_+\approx 0.639$, $\gamma_+\approx 0.582$, $q_+\approx 0.401$, $s_+\approx 0.520$, $\tilde q_+\approx 0.512$, $\beta_c\approx 0.640$
  - These exponents diverge from classical mean-field (Ising/van der Waals) values, indicating strong-coupling universality [2103.16270].
- Above $T_c$, no transition exists and the system is smoothly connected.

## 3. Symmetry Classification, Zero Modes, and Supersymmetry

The symmetry structure of the cSYK model, especially with chiral symmetry, results in a fourfold Altland–Zirnbauer classification determined by the interplay of chiral operator $\mathcal S$, its square $(\mathcal S^2)$, and its (anti)commutation with fermion parity $P$. The resulting classes and their level statistics are:

| $N \mod 4$ | $\mathcal S^2$ | Cartan class | $q=0$ sector | $q\ne0$ sectors |
|:----------:|:--------------:|:------------:|:------------:|:---------------:|
|     0      |     +1         |     AI       |    GOE       |      GUE        |
|     1      |     +1         |     D        | (absent)     |      GUE        |
|     2      |     –1         |    AII       |    GSE       |      GUE        |
|     3      |     –1         |     C        | (absent)     |      GUE        |

For odd $N$, many-body zero modes exist, constructed as inter-sector operators $\mathcal O^q_{\mu\mu}=|\psi^q_\mu\rangle\langle\psi^{-q}_\mu|$ that commute with $H$ and anti-commute with parity. These "generalized fermion" modes $d^\dagger = \sum_{q>0,\mu} \mathcal O^q_{\mu\mu}$ satisfy $[H,d]=0$, $\{d,P\}=0$. The existence of such zero modes implies emergent $\mathcal N=2$ supersymmetry with supercharges $\mathcal Q,\,\mathcal Q^\dagger$ defined by energy-weighted sums over these zero-modes:
\[
\mathcal Q^\dagger = \sum_{q>0,\mu} \sqrt{\varepsilon^q_\mu} \mathcal O^q_{\mu\mu}, \quad \{\mathcal Q,\mathcal Q^\dagger\} = H.
\]
These structures manifest in the universal plateau structure of long-time retarded correlators, directly distinguishing the Altland–Zirnbauer classes [1912.09975].

## 4. Green’s Functions, Spectral Properties, and Charge Asymmetry

The disorder-averaged Green’s function and self-energy for cSYK are central, with anomalous infrared scaling. In the conformal regime ($1/J\ll |\tau| \ll \beta$), for $q$-body interactions,
\[
G(\tau) \sim b^\Delta \exp(\pm \pi E) |\tau|^{-2\Delta}
\]
with $\Delta=1/q$ and spectral asymmetry $E$ determined by chemical potential. The universal "charge–$\theta$" relation is
\[
Q = -\frac{\theta}{\pi} - \Bigl(\frac{1}{2}-\Delta\Bigr)\frac{\sin(2\theta)}{\sin(2\pi\Delta)},
\]
where $\theta$ parametrizes the IR Green's function phase, linked thermodynamically to the entropy derivative $\theta = \frac{1}{2\pi}\frac{\partial S}{\partial Q}$ [1910.14099], [2006.13983]. The density of states at fixed charge $Q$ and energy $E$ is
\[
\rho(E,Q) \propto e^{N \mathcal S(Q/N)} \sinh\left(2\pi \sqrt{2C [E-E_0(Q)]}\right)
\]
with Schwarzian coefficient $C$, and the charge compressibility $K = \partial Q/\partial\mu$ accessible by both analytic and numerical means.

Spectral statistics and quantum chaos are probed via the statistics of the level spacing ratio $r_n$, the spectral form factor, and the inverse participation ratio. The cSYK model at $t \approx U$ displays GUE Wigner-Dyson statistics and non-Fermi-liquid scaling in the Green's function, $G^R(\omega) \propto \omega^{-1/2}$, confirming strong quantum chaos and ergodicity [2105.03208].

## 5. Dualities, Quantum Gravity, and the Double Scaling Limit

The large-$N$ cSYK model has a gravitational (holographic) dual in the form of deformed Jackiw-Teitelboim (JT) gravity with Maxwell field. The black hole equation of state matches the SYK phase diagram, including swallowtail structure in thermodynamic potential and mean-field critical exponents near the critical point. In the double scaling limit ($N\to\infty$, $q\to\infty$, $q^2/N$ fixed), the exact solution of the cSYK model is rendered in terms of "chord diagrams," yielding $q$-deformed density of states and analytical control over all spectral and correlation functions. The chemical potential explicitly renormalizes chord crossing-weights and governing asymmetry, providing a direct handle on the charge sector [2006.13983].

Key features in the double scaling regime include:
- Analytic access to the spectrum, 2-point, and 4-point functions at all energy scales via $q$-orthogonal polynomials and transfer matrices.
- Mapping to black hole microstates and spacetime fragmentation in the gravity dual, especially when inserting heavy operators.

## 6. Quantum Simulation and Experimental Realizations

Quantum simulation of the cSYK Hamiltonian is challenged by the requirement of dense, random, all-to-all quartic couplings. A recently proposed scheme employs Trotterized cycling through sparse time-dependent disorder realizations in single-mode cavity quantum electrodynamics (cQED) platforms. The method builds $R \sim N$ low-rank random Hamiltonians and approximates the target cSYK Hamiltonian by rapid Trotterized evolution. The convergence from sparse to dense coupling distributions is quantified by the Kullback–Leibler divergence, decaying as $1/R^2$ [2411.17802].

Experimental parameters such as speckle pattern switch time, cavity detuning, and achievable cooperativity demonstrate that cSYK-like physics with $N\sim10$ can be accessed with current or near-term technology. The method also reproduces the spectral form factor and out-of-time-ordered correlators characteristic of chaos in the cSYK model.

## 7. Extensions, Disorder Averaging, and Sign-Problem-Free Formulations

Extensions of the cSYK model include:
- Variants with non-Gaussian disorder, where a quartic pillow perturbation shifts the effective variance in the leading-order effective action:
  \[
  \sigma^2 \to \sigma^2 - 2\lambda \sigma^6 + O(\lambda^2, 1/N)
  \]
  while preserving melonic dominance and leading $N$ physics [1812.03008].
- Models designed for sign-problem-free quantum Monte Carlo, where a Majorana representation with extra "interaction channels" enables determinant quantum Monte Carlo without sign issues. These models preserve the non-Fermi-liquid, maximally chaotic regime and random-matrix statistics, further confirmed by exact DQMC numerics [2107.13572].

The cSYK framework is further generalized to coupled-cluster models (two-site SYK with $U(1)$ charge), yielding eternal wormhole ground states (thermofield double structure), Schwarzian effective actions with coupled charge modes, and robust first-order transitions between wormhole and black hole phases [2011.10360]. This demonstrates both the model’s versatility and deep connections to the dynamics of quantum gravity in AdS$_2$.

---

**References:**
- [2103.16270], [1912.09975], [2105.03208], [1910.14099], [2006.13983], [2411.17802], [1812.03008], [2107.13572], [2011.10360]

Source: https://www.emergentmind.com/topics/complex-sachdev-ye-kitaev-model