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

TL;DR

This work addresses the infrared behavior of the complex SYK model at finite chemical potential by computing the four-point function in the conformal limit for arbitrary asymmetry parameter $\mathscr{E}$ at leading order in $1/N$. It employs two complementary methods: a direct resummation of ladder diagrams in the large-$N$ limit and a conformal-field-theory approach using NCFT$_1$ OPE data, yielding a consistent four-point function expressed in terms of conformal blocks. The authors extract explicit CFT structure constants for correlators involving two complex fermions and bilinear operators, providing the full OPE data $\{c_h^a, c_h^s\}$ as functions of the asymmetry parameter. These results validate the NCFT$_1$ description along the $\mathscr{E}$-line, and furnish detailed OPE data that can inform holographic interpretations and further studies of the cSYK model away from maximal symmetry.

Abstract

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 $\mathscr{E}$ 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 $\mathscr{E}$ 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.

Figures (12)

• Figure 1: Ladder diagrams contributing to $\mathcal{F}(\tau_{1},\tau_{2};\tau_{3}, \tau_{4})$ can be grouped into $\mathcal{F}_n$, where $n$ labels the number of "rungs". Each line with an arrow represents the large $N$ Green’s function $G(\tau)$, which satisfies the SD equations \ref{['SDequations0']}.
• Figure 2: Lowest scaling dimensions of the $q=4$ cSYK operators as functions of $\theta$ and $\mathscr{E}$. The horizontal dashed lines indicate their values at $\theta=0$ ($\mathscr{E}=0$) and the vertical dashed line marks the critical value $\theta_{\textrm{crit}}\simeq 0.48$ ($\mathscr{E}_{\textrm{crit}}\simeq 0.18$), at which the cSYK model undergoes the first-order phase transition.
• Figure 3: The structure constants $c_{h}^{a}$ and $c_{h}^{s}$ for the first two scaling dimensions $h_{2}$ and $h_{3}$ as functions of $\theta$ for $q=4$ cSYK model. The vertical dashed line marks the critical value $\theta_{\textrm{crit}}\simeq 0.48$, at which the cSYK model undergoes the first-order phase transition.
• Figure 4: Diagrammatic representation of the basic propagators. We introduced a separate dashed line, which represents the asymmetry factor, and it is assumed that it is always $\mathscr{E}$. For $\mathscr{E}=0$ the asymmetry propagator disappears.
• Figure 5: Diagrammatic representation of the identity (\ref{['AssymPropRel']}).