Notes on the complex Sachdev-Ye-Kitaev model

TL;DR

This work analyzes the complex SYK model with a large number of flavors and a global U(1) charge, formulating the problem in the (G,Σ) framework to connect UV charge with IR spectral asymmetry through a universal relation between the charge and the IR Green function asymmetry. It derives a covariant low-energy effective action that includes a Schwarzian mode and a compact U(1) degree of freedom, linking thermodynamics, density of states, and compressibility to both IR data and UV perturbations. The authors develop a renormalization theory to reproduce the charge–asymmetry relation from intermediate-scale dynamics and perform three independent numerical methods to compute the charge compressibility, all yielding consistent results around $K\approx 1.04/J$ for $q=4$, thereby highlighting the non-universal UV contributions to compressibility. A two-dimensional bulk construction with Dirac fermions on hyperbolic space reproduces the zero-temperature entropy $\mathcal{S}(\mathcal{Q})$ and its dependence on the spectral asymmetry, forging a deep link between SYK physics and AdS$_2$ holography via a bulk determinant subtraction (the spooky propagator) and the Plancherel structure of $\widetilde{SL}(2,\mathbb{R})$.

Abstract

We describe numerous properties of the Sachdev-Ye-Kitaev model for complex fermions with $N\gg 1$ flavors and a global U(1) charge. We provide a general definition of the charge in the $(G,Σ)$ formalism, and compute its universal relation to the infrared asymmetry of the Green function. The same relation is obtained by a renormalization theory. The conserved charge contributes a compact scalar field to the effective action, from which we derive the many-body density of states and extract the charge compressibility. We compute the latter via three distinct numerical methods and obtain consistent results. Finally, we present a two dimensional bulk picture with free Dirac fermions for the zero temperature entropy.

Figures (14)

• Figure 1: (a) Conservation law: the total current through a closed (dashed) circle is zero; (b) Flow $\mathcal{Q}$, as the total current through a cross section $\tau_0$ (blue dashed line), is independent of the position $\tau_0$. In general, there are contributions from all time scales; longer scale currents are shown in red.
• Figure 2: RG flow of a perturbation $\sigma$ (solid line), generating the response $\delta G$ (dashed line) at larger time scales.
• Figure 3: (a) The ground energy $E_{0}(Q)$ as a function of charge $Q$ in units $J$ ($q=4$). The number of samples for each charge are 250000 ($N=10$), 120000 ($N=11$), 50000 ($N=12$), 10000 ($N=13$), 2000 ($N=14$), 1000 ($N=15$), 500 ($N=16$), 200 ($N=17$). Dashed lines are fit by $E_{0}(Q)=E_{0}+a Q^{2}$. (b) Plot for the ground energy $E_{0}$ at zero charge. The dashed line is a fit by $E_{0}=- 0.079 N-0.479 + 1.6/N$. The leading large $N$ term to be compared with $2\epsilon_{0}$, where $\epsilon_{0}\approx -0.0406$Cotler:2016fpe
• Figure 4: Plot of numerical solution for $G(\tau)$ at $q=4$, $\beta J =200$ and $\beta \mu=20$. The dashed line is conformal solution (\ref{['Gconf']}) with $\theta$ found from numerical $\mathcal{Q}$ using the formula (\ref{['charge theta']}).
• Figure 5: (a) Plot of ${\mathcal{Q}}/ \mu$ for different temperatures $T$ and chemical potentials $\mu$ for $q=4$. (b) Plot of $K$ for different $\mu$.