Note on the $q=2$ $R$-para-fermionic SYK model

TL;DR

The paper investigates a quadratic $q=2$ SYK model built from $R$-para-particles to probe how generalized exchange statistics affect thermodynamics and quantum chaos. It develops a coherent-state formalism that yields exact ensemble-averaged thermodynamics and proves self-averaging, and it analyzes the spectral form factor with a cluster-function approach revealing an exponential ramp whose exponent $C_0$ can remain finite or diverge with $N$ depending on the single-mode generating function $z_R(x)$. The work shows that, unlike ordinary SYK$_2$, the ramp exponent can stay constant across large $N$ for several nontrivial $R$-para-particle statistics, while certain cases induce divergent $C_0$, implying a qualitative shift in chaotic dynamics. A key finding is the dependence of ramp behavior on the parameters of $z_R(x)$ (e.g., $z_R(x)=1+mx$ vs $1+mx+x^2$), and a proposed criterion linking divergence of $C_0$ to $z_R(x) \le (1+x)^L$ for $x>0$. These results illuminate how generalized statistics influence chaos diagnostics and thermodynamics, and they provide a framework for future extensions to $R$-PSYK$_{q>2}$ and possible holographic interpretations.

Abstract

We investigate the $q=2$ SYK model with $R$-para-particles ($R$-PSYK$_2$), analyzing its thermodynamics and spectral form factor (SFF) using random matrix theory. The Hamiltonian is quadratic, with coupling coefficients randomly drawn from the Gaussian Unitary Ensemble (GUE). The model displays self-averaging behavior and exhibits an exponential ramp in its SFF dynamics: $\mathcal{K}(t) \sim e^{C_0t}$. The growth rate $C_0$ tends toward either a constant or infinity in the $N\to \infty$ limit, depending on specific statistics of the model. These results provide novel perspectives on quantum systems with unconventional statistics.

Figures (7)

• Figure 1: The plot compares the ensemble-averaged partition function for numerical simulations and theoretical expectations in two cases. For $z_R(x) = 1 + m x$, the theoretical expectation is $\log\mathcal{Z}/N = \log(1+m) + \frac{m}{2(1+m)^2}\beta^2 + \mathcal{O}(\beta^4)$. For $z_R(x) = 1 + m x + x^2$, it becomes $\log\mathcal{Z}/N = \log(1+m) + \frac{1}{2+m}\beta^2 + \mathcal{O}(\beta^4)$. The numerical results are averaged over 200 samples.
• Figure 2: Short-time SFF comparison between numerical simulations (averaged over 2000 samples) and theoretical predictions from the coherent state approach, for $z_R(x)=1+mx$ (left) and $z_R(x)=1+mx+x^2$ (right).
• Figure 3: The plot shows $-B_0$ and $C_0$ computed numerically as functions of $N$ for $z_R(x) = 1 + m x$. The dashed lines represent the exact analytical results derived from Eq. \ref{['eq:Bp-exact']} and Eq. \ref{['eq:exB-C0-m']}. Focusing on the asymptotic regime, we restrict the fitting to data with $N \geq 20$. The numerical results exhibit excellent agreement with the exact theoretical predictions.
• Figure 4: Left: Log-log plot of SFF using the cluster function approach for $z_R(x)=1+mx$ with $m=1,2,3,4,5$ ($N=400$). Time range $t\in[0.02,8N]$. The $m=1$ case (ordinary fermions) shows an exponential ramp with $C_0 \sim \mathcal{O}(\ln N)$, while $m>1$ cases exhibit qualitatively different behavior with $C_0 \sim \mathcal{O}(1)$. Right: Numerical simulation for $N=12$ with 20,000 samples ($t\in[2^{-4},2^8]$). While the small system size prevents observation of the exponential ramp, the plateau location agrees with the theoretical prediction $\mathcal{K}(t\to\infty) = (m^2+1)^{N}/(m+1)^{2N}$.
• Figure 5: The plot of $-B_0,C_0$ with respect to $N$ for $z_R(x)=1+mx+x^2$. Since we are concerned with asymptotic behavior, we use the data $N\ge 10$ to fit. We find that for $m=1,2$, $C_0$ is divergent with a power law in the large $N$ limit. Unlike the log divergence of $z_R(x)=1+mx$. As for $m\ge 2$, $C_0$ is finite.