Complex Sachdev-Ye-Kitaev Model

• The cSYK model is a quantum many-body system characterized by N complex fermions with random all-to-all q-body interactions and U(1) charge conservation.
• It exhibits dual thermodynamic phases, transitioning from a low-temperature gapped phase with discrete energy levels to a high-temperature chaotic non-Fermi liquid phase.
• The model provides deep insights into emergent conformal symmetry, quantum chaos, and holographic duality, with implications for quantum simulation and gravity duals.

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 (Cao et al., 2021, Behrends et al., 2019, Gu et al., 2019).

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 (Cao et al., 2021).
• 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 (Behrends et al., 2019).

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 $$\theta = \frac{1}{2\pi}\frac{\partial S}{\partial Q}$$ (Gu et al., 2019, Berkooz et al., 2020). 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 (Dieplinger et al., 2021).

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 (Berkooz et al., 2020).

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 $1/R^2$ (Baumgartner et al., 2024).

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 (Krajewski et al., 2018).

• 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 (Kang et al., 2021).

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 (Zhang, 2020). This demonstrates both the model’s versatility and deep connections to the dynamics of quantum gravity in AdS$_2$.

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References:

• (Cao et al., 2021, Behrends et al., 2019, Dieplinger et al., 2021, Gu et al., 2019, Berkooz et al., 2020, Baumgartner et al., 2024, Krajewski et al., 2018, Kang et al., 2021, Zhang, 2020)