Notes on the complex Sachdev-Ye-Kitaev model (1910.14099v3)

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 $(G,\Sigma)$ 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.

Overview of the Complex Sachdev-Ye-Kitaev Model

The paper "Notes on the Complex Sachdev-Ye-Kitaev Model" by Yingfei Gu, Alexei Kitaev, Subir Sachdev, and Grigory Tarnopolsky presents a thorough investigation into the properties of the Sachdev-Ye-Kitaev (SYK) model with complex fermions. The paper explores the implications of this model in the context of quantum gravity and condensed matter physics, especially in regimes with a large number of fermion flavors ($N \gg 1$) and a global $U(1)$ charge.

The primary focus of this research is the complex SYK model, distinguishing it from the more commonly analyzed Majorana SYK model, which lacks a globally conserved charge apart from the Hamiltonian itself. The authors derive significant results concerning the infrared (IR) behavior and thermodynamics of this model, tracing connections to the low-temperature characteristics of black holes in AdS space.

Key Contributions

1. Charge and Spectral Asymmetry: The paper provides a detailed analysis of the conserved charge in the complex SYK model using the Green function's UV and IR asymmetries. The relation between the charge and the spectral asymmetry parameters ($E$, $\theta$) is derived using renormalization symmetry arguments and is shown to be analogous to the Luttinger relation in Fermi liquids.
2. Effective Action and Low-Temperature Properties: The authors derive an effective Schwarzian action for low-temperature fluctuations, encapsulating both reparameterization and $U(1)$ phase fluctuations. This result facilitates a unified framework for understanding the specific heat and compressibility behavior at low temperatures.
3. Numerical Methods for Compressibility ($K$): Through both exact diagonalization and numerical solutions of the Schwinger-Dyson equations, the paper yields estimates of the zero-temperature compressibility $K$, finding values consistent across multiple approaches. This contrasts with the specific heat calculation, highlighting that $K$ is sensitive to higher-energy contributions not captured by low-energy effective theories alone.
4. Density of States and Entropic Considerations: The paper establishes that for fixed charge sectors, the density of states’ energy dependence aligns with predictions from the Schwarzian theory, with a complex interplay between IR and UV phenomena shaping these outcomes. The zero-temperature entropy, $\mathcal{S}$, is analytically linked to boundary conditions of a massive Dirac fermion on the hyperbolic disk. Moreover, the entropy per particle is shown to be invariant under charge fluctuations.
5. Dual Perspectives on Entropy: The work probes the dual aspects of black hole thermodynamics and SYK model entropy, emphasizing how universal thermodynamic identities are maintained across these apparently disparate physical systems. Contrasts are drawn with traditional AdS/CFT approaches where entropic contributions predominantly arise from geometrical properties.

Implications and Future Directions

The paper underscores the SYK model’s utility as a solvable prototype for examining non-Fermi liquid behavior governed by strong correlations. In particular, understanding how charge and entropy are distributed in these models sheds light on analogous phenomena in higher-dimensional holographic systems such as near-extremal black holes.

Future work inspired by these findings may explore the interplay between quantum chaos, thermalization, and entanglement properties in models that bridge complex SYK systems and AdS/CFT. Further research could extend these results to other types of fractionalized excitations or emergent higher-spin phases within the broader family of SYK-like models.

The insights afforded by this paper enhance our understanding of complex systems where traditional paradigms of solid-state physics and string theory intersect, pointing toward new methodologies for exploring unified theories of quantum gravity and condensed matter physics.