Gauging the complex SYK model

TL;DR

This work analyzes gauging a global U$(1)$ symmetry in the complex SYK model by introducing a Wilson line of charge $k$, and studies the theory both with a fixed background gauge field and via a large-$N$ saddle that integrates over the holonomy $u$. The authors compare two routes to compute observables—the direct gauge-field integration and the large-$N saddle—against exact diagonalization, finding good agreement and uncovering how the charge density $\kappa=k/N$ controls IR conformality: the low-energy conformal behavior survives only at $\kappa=0$ and is broken for $\kappa\neq0$, while chaos remains maximal for all $k$. A key technical feature is the necessity to sum over all winding-number sectors (labeled by $n$) to preserve gauge periodicity, which parallels the role of axion-like modes in the low-energy effective action. The operator spectrum and the chaos exponent are analyzed through SD equations and conformal kernels, revealing a $u_{\rm IR}$-dependent spectrum with universal $h=2$ and a gauge-fixed $h=1$ axion that must be excluded from the physical spectrum; nevertheless, chaos saturates the bound with $\lambda=2\pi$ (in the appropriate units). Overall, the paper provides a coherent framework for gauged SYK-like models with Wilson lines, connects to near-AdS$_2$ holography, and demonstrates robust maximal chaos across charge sectors.

Abstract

Motivated by SYK-like models describing near-BPS black holes in string/M-theory, we consider gauging the U$(1)$ symmetry of the complex SYK model in the presence of a Wilson line with charge $k$. At a fixed background gauge field, solutions to the Schwinger-Dyson equations display vastly different properties from those at a fixed real chemical potential. In the partition function and the two-point function, the integral over the gauge field is performed either directly or via a large $N$ saddle point approximation, and both results are consistent with exact diagonalization data. From the behaviour of the two-point function at large $N$, we deduce that the conformal symmetry at low energies is preserved at fixed $κ= k/N = 0$, but broken at $κ\neq 0$. In addition, we find that there is maximal chaos for all $k$.

Figures (12)

• Figure 1: The left and right panels of this plot show the real and imaginary parts of $G(\tau)$ respectively. The points in blue give the numerical solution in the case $q=4$, $\beta J=200$, $u=\pi$, computed with $M=2^{12}$ discretised points and tolerance $\epsilon=10^{-10}$. They are displayed together with plots of the analytic conformal solution \ref{['conf const fixed']} in the dotted black lines. The parameter $u_\text{IR} = 0.04077$ is determined by fitting against the numerical solution. In other words, the numerical solution gives the explicit renormalization of $u$. Notice that there is excellent agreement between the numerical and conformal solutions except for a small window near $\tau = 0, 1$ as we expected.
• Figure 2: The left panel above plots $u_\text{IR}$, obtained by fitting numerical solutions $G_0(\tau;u,n)$ to the conformal ansatz, against $u$. The solutions on different branches $n$ are coloured differently, but distinct solutions on the same branch have the same colour. The right panel plots the fitted values of $u_\text{IR}$ for the gauge equivalent solutions $G_0(\tau;u-2\pi n,0)= e^{-2\pi in\tau}G_0(\tau;u,n)$. Each segment with a uniform colour on the right panel plots the solutions which are gauge equivalent to those in a particular branch with the same colour on the left panel. For instance, in the left panel, we have labelled the $n=-3$ branch of solutions, which are all coloured orange. Their gauge equivalent counterparts lie on the orange coloured segment in the right panel, which is also labelled.
• Figure 3: The black curve plots $u$ against $\mathop{\mathrm{\mathbb{I}m}}\nolimits\mathcal{Q}$ according to \ref{['eos u']}, where $\mathcal{Q} = i\mathop{\mathrm{\mathbb{I}m}}\nolimits\mathcal{Q}$. The integer $n$ in \ref{['eos u']} is chosen such that $u\in (0,2\pi)$. The parameters of the model are set to $q = 40$ and $\beta\mathcal{J} = 100$, for which $v\approx 3.05$. At each value of $u$, say $u = 3$ which is traced by the blue dotted line, there seem to be infinitely many solutions corresponding to the intersections between the blue dotted line and the black curve. However, only the two coloured red are consistent.
• Figure 4: Plots of $\mathop{\mathrm{\mathbb{R}e}}\nolimits G(\tau)$ and $\mathop{\mathrm{\mathbb{I}m}}\nolimits G(\tau)$ against $\tau$ at $q=40$, $\beta\mathcal{J} = 100$, and $u = 3$. The black dotted curves are the analytic predictions according to \ref{['large q sol']}, while the blue solid curves are the numerical solutions obtained via the method of Section \ref{['sec:num']}. In the first row, we pick the case where $n = 0$, and $\mathcal{Q}\approx 0.275$ is the horizontal position of the red point to the right of the vertical axis in Figure \ref{['fig:u vs imQ at large q']}. In the second row, we took $n = -1$, and $\mathcal{Q}\approx -0.303$ is given by the horizontal position of the red point to the left of the vertical axis in Figure \ref{['fig:u vs imQ at large q']}. In the two cases, the original parameter $\beta J$ which appears in the Schwinger-Dyson equations \ref{['SD eqn unit beta']} is determined by \ref{['first deriv pos']}, and it evaluates to $\beta J \approx 671943, 426399$ for $n=0,1$ respectively.
• Figure 5: Contours of integration in the complex $u$ plane. The blue contour is the original contour along the real axis from $-\infty$ to $\infty$, while the red contour is the target contour that is reached via a continuous deformation. The red contour passes through the saddle point for some fixed $\kappa$ on the imaginary axis, indicated by a black cross, and is made up of the imaginary axis and two circular arcs. It should be understood that the arcs are actually infinitely far away from the origin, and are only schematically drawn at a finite distance.