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arXiv:2601.04330 · Journal of High Energy Physics

Four-point function of the complex Sachdev-Ye-Kitaev model at finite chemical potential

Can Onur Akyuz, E. Cruz, Ludo Fraser-Taliente, Grigory Tarnopolsky

Published Sep 2, 2026• 0 citations
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Abstract

A bstract It is known that, for a range of chemical potentials, the infrared behavior of the complex Sachdev-Ye-Kitaev (cSYK) model is governed by a 1D Nearly Conformal Field Theory (NCFT 1 ), thereby realizing a continuous line of NCFTs. A finite chemical potential μ introduces an asymmetry parameter $$ \mathcal{E} $$ E into the cSYK fermion two-point function in the conformal limit. In this work, we compute the cSYK four-point function in the conformal limit for an arbitrary value of $$ \mathcal{E} $$ E at leading order in 1/ N . We show that the result is fully consistent with the NCFT 1 structure of the cSYK model and use it to extract the structure constants for correlation functions of two complex fermions with bilinear operators.

DOI
10.1007/jhep09(2026)039

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A bstract It is known that, for a range of chemical potentials, the infrared behavior of the complex Sachdev-Ye-Kitaev (cSYK) model is governed by a 1D Nearly Conformal Field Theory (NCFT 1 ), thereby realizing a continuous line of NCFTs. A finite chemical potential μ introduces an asymmetry parameter E\mathcal{E}E E into the cSYK fermion two-point function in the conformal limit. In this work, we compute the cSYK four-point function in the conformal limit for an arbitrary value of E\mathcal{E}E E at leading order in 1/ N . We show that the result is fully consistent with the NCFT 1 structure of the cSYK model and use it to extract the structure constants for correlation functions of two complex fermions with bilinear operators.
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Abstract We describe numerous properties of the Sachdev-Ye-Kitaev model for complex fermions with N ≫ 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.
The Sachdev-Ye-Kitaev (SYK) model is a quantum-mechanical model of NNN Majorana fermions which displays a number of appealing features---solvability in the strong coupling regime, near-conformal invariance and maximal chaos---which make it a suitable model for black holes in the context of the AdS/CFT\mathrm{AdS}/\mathrm{CFT}AdS/CFT holography. In this paper, we show for the colored SYK model and several of its tensor model cousins that the next-to-leading order in the large-NNN expansion preserves the conformal invariance of the two-point function in the strong-coupling regime, up to the contribution of the pseudo-Goldstone bosons due to the explicit breaking of the symmetry which are already seen in the leading-order four-point function. We also comment on the composite field approach for computing correlation functions in colored tensor models.
We describe numerous properties of the Sachdev-Ye-Kitaev model for complex fermions withN≫\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.
Abstract We study the complex Sachdev–Ye–Kitaev (cSYK) model numerically and investigate the thermodynamic behavior of the cSYK model across varying chemical potentials. We discover that the cSYK model remarkably mirrors the first-order phase transition seen in the van der Waals–Maxwell system, culminating at a non-mean-field critical point with distinctively different critical exponents. We analyze in detail the similarity between the van der Waals phase transition and the cSYK model, and further explore the mechanism by which the chemical potential drives the phase transition in the system. Exact diagonalization at finite N resolves the conserved U(1) charge sectors, showing that the chemical potential reshapes the density of states. Within each charge sector we find a strong level repulsion, while mixing sectors shows a Poisson distribution. The normalized spectral form factor displays a clear dip-ramp-plateau at low temperature for the neutral case, whereas for non-vanishing chemical potential the ramp is weakened by finite N reweighting of small edge sectors and only becomes visible at relatively high temperatures. Real-time dynamics, analyzed via analytical continuation of Schwinger–Dyson equations, show rapid decay in the gapless phase and prolonged oscillation lifetimes in the gapped regime. Spectral functions imply a shift from a continuous to a discrete energy level distribution, emphasizing the critical role of chemical potential in shaping spectral properties.
We study the complex Sachdev-Ye-Kitaev (cSYK) numerically and investigate thermodynamic behavior of cSYK model across varying chemical potentials. We discover that the cSYK model remarkably mirrors the first-order phase transition seen in the van der Waals-Maxwell system, culminating at a non-mean-field critical point with distinctively different critical exponents. We analyze in detail the similarity between the van der Waals phase transition and the cSYK model, and further explore the mechanism by which the chemical potential drives the phase transition in the system. Exact diagonalization for the cSYK model reveals the significant impact of chemical potential on energy distribution, with observable energy gaps in the gapped phase. Quantum chaos indicators, including spectral form factors, suggest more stable energy states in the neutral case. Real-time dynamics, analyzed via analytical continuation of Schwinger-Dyson equations, show rapid decay in the gapless phase and prolonged oscillation lifetimes in the gapped regime. Spectral functions imply a shift from a continuous to a discrete energy level distribution, emphasizing the critical role of chemical potential in shaping spectral properties.