Dissipative spectral form factor for elliptic Ginibre unitary ensemble and applications

TL;DR

The paper introduces exact finite-D and large-D expressions for the dissipative spectral form factor of the elliptic Ginibre unitary ensemble, revealing a scaling relation that maps DSFF to the Gaussian unitary ensemble SFF across the GinUE–GUE crossover. This mapping enables precise, τ-dependent predictions of the dip, ramp, and plateau structure, and provides refined estimates for the Thouless and Heisenberg times. The authors validate their results with Monte Carlo simulations and demonstrate universality by capturing the late-time DSFF of crossover models such as cSYK and cPLBRM. They further draw a physical analogy linking eGinUE eigenvalues to the positions of a rotating fermionic gas in a two-dimensional anisotropic trap, underscoring the broader relevance to non-Hermitian quantum chaos and open-system dynamics.

Abstract

We investigate the dissipative spectral form factor (DSFF)--a widely used probe of non-Hermitian quantum chaos--in the elliptic Ginibre unitary ensemble (eGinUE), which interpolates between the non-Hermitian Ginibre unitary ensemble (GinUE) and the Hermitian Gaussian unitary ensemble (GUE) via a symmetry breaking parameter. We derive exact finite-dimensional results and large-dimensional approximations for the DSFF, revealing a scaling relationship that connects the DSFF of eGinUE to that of GinUE and the spectral form factor of GUE. This relation explains the distinct time scales underlying the characteristic \emph{dip-ramp-plateau} structure across GinUE, GUE, and crossover regimes. Additionally, we refine estimates of dip-ramp and ramp-plateau transition times for different symmetry regimes. We validate our results with Monte Carlo simulations and demonstrate applications to paradigmatic quantum-chaotic systems: the crossover Sachdev-Ye-Kitaev model and the crossover Power-law Banded random matrices. We highlight an analogy between eGinUE eigenvalues and the positions of a rotating fermionic gas in a two-dimensional anisotropic trap.

Figures (8)

• Figure 1: Plots of the eGinUE DSFF $\mathcal{F}(t,s)$ as a function of $|T|$ for $D=10$ and $\varphi=\pi/3$. The solid curves represent exact analytical results, Eq. \ref{['DSFFmain_main']}, while symbols are based on Monte Carlo simulations of the eGinUE matrix model. The crossover from GinUE to GUE symmetry is clearly observed with varying symmetry-breaking parameter $\tau$.
• Figure 2: Plots of DSFF, using $1+\mathcal{F}_\mathrm{dis}-\mathcal{F}_\mathrm{conn}$, where $\mathcal{F}_\mathrm{dis}$ and $\mathcal{F}_\mathrm{conn}$ are given by Eqs. \ref{['FdisAsy_main']} and \ref{['FconnAsy_main']}, respectively for $D=100$, $\varphi=0$ and varying $\tau$: (a) $\tau=0.1$, (b) $\tau=0.7$, (c) $\tau=0.99$, and (d) $\tau=0.9995$. The theoretical estimates for Thouless time ($T_\mathrm{Th,1}\approx(\frac{6}{\pi \eta^3 (1-\tau^2)})^{1/5} D^{2/5}$, $T_\mathrm{Th,2}\approx(\frac{3}{2})^{1/4}\frac{D^{1/2}}{\eta}$), and Heisenberg time ($T_{\mathrm{H},1}\approx4\sqrt{D/(1-\tau^2)}$, $\widetilde{T}_{\mathrm{H},1} \approx2\sqrt{\pi}\sqrt{D/(1-\tau^2)}$, $T_{\mathrm{H},2}\approx\pi D/(2\eta)$, $T_{\mathrm{H},3}\approx2D/\eta$) are shown as vertical lines (see Table \ref{['tab:TH_definitions']} in supp). For (a) and (b), where $\tau<\tau_c$, $T_\mathrm{Th,1}$ serves as a better estimate for Thouless time, whereas for (c), (d) for which $\tau\gtrsim \tau_{c}$ the value $T_\mathrm{Th,2}$ serves better. Similarly, for the Heisenberg time $T_\mathrm{H,1}$ and $\widetilde{T}_{\mathrm{H},1}$ serves as a good estimate for (a) and (b). On the other hand, for (c) and (d), $T_\mathrm{H,2}$ and $T_\mathrm{H,3}$ serve as better estimates, respectively. $T_\mathrm{H,1}$ and $\widetilde{T}_{\mathrm{H},1}$ values in panels (c) and (d) lie beyond the time scale shown.
• Figure 3: The DSFF of the $q = 4$ cSYK model (symbols), given in Eq. \ref{['cSYK']}, for $N = 18$, normalized by $2^{N/2-1}$ non-degenerate eigenvalues and averaged over $5 \times 10^4$ matrices. We compare it with the large-$D$ asymptotic eGinUE result (solid curves), $1 - \mathcal{F}_\mathrm{conn}$ [Eq. \ref{['FconnAsy_main']}], evaluated with $D = 2^8$ at $\varphi = \pi/4$ and find excellent agreement. Varying $\tau$ demonstrates the GinUE–GUE crossover, with $\tau_c = 1 - 1/2^8$.
• Figure 4: The DSFF of the cPLBRM model (symbols) given in Eq. \ref{['cPLBRM']} for matrix size $4096$, averaged over $4000$ realizations using the full eigenspectra. This is compared with the asymptotic eGinUE DSFF results (solid curves), $1 - \mathcal{F}_\mathrm{conn}$ [Eq. \ref{['FconnAsy_main']}], evaluated with $D = 4096$. We set $W = 3$ and $p = 0.5$ to place the system well within the chaotic regime, with $b = 1$ fixed for simplicity. Varying $\tau$ demonstrates the GinUE–GUE crossover. Throughout, we set $\varphi = 0$. The inset shows the same comparison for additional values of $\tau$, where analytical results are shown by solid, dashed and dotted lines for the three values of $\tau$.
• Figure S1: Plots of the eGinUE DSFF $\mathcal{F}(t,s)$ as a function of $|T|$ for $D=125$ and $\varphi=0$. The solid curves represent large-$D$ approximate analytical results [Eq. \ref{['FAsy']}], while symbols are based on Monte Carlo simulations of the eGinUE matrix model. The inset provides an enlarged view of the dip region, demonstrating that the large-$D$ approximation accurately captures even the oscillations.