Operator growth in open quantum systems: lessons from the dissipative SYK

TL;DR

This work extends Krylov (K-) complexity to open quantum systems governed by Lindblad dynamics, using the dissipative q-body SYK model as a testbed. It uncovers an operator size concentration in the large-q limit that makes the open-system Lanczos data tractable: a_n obtains an imaginary linear dependence on n, while b_n matches the closed-system growth, leading to a symmetric tridiagonal Lindbladian in the Arnoldi basis. The authors show Krylov complexity grows exponentially with time but saturates at a scale set by dissipation, with the saturation time scaling as t_* ~ (1/2α) log(1/μ) in the weak-dissipation limit, and they bound the OTOC by the Krylov complexity, suggesting a universal open-system scrambling behavior. Finite-q numerics and holographic considerations (Keldysh wormholes) support and contextualize the large-q results, pointing to a coherent open-system operator-growth framework. These insights contribute a quantitative, cross-validated picture of scrambling and complexity in dissipative quantum dynamics.

Abstract

We study the operator growth in open quantum systems with dephasing dissipation terms, extending the Krylov complexity formalism of Phys. Rev. X 9, 041017. Our results are based on the study of the dissipative $q$-body Sachdev-Ye-Kitaev (SYK$_q$) model, governed by the Markovian dynamics. We introduce a notion of ''operator size concentration'' which allows a diagrammatic and combinatorial proof of the asymptotic linear behavior of the two sets of Lanczos coefficients ($a_n$ and $b_n$) in the large $q$ limit. Our results corroborate with the semi-analytics in finite $q$ in the large $N$ limit, and the numerical Arnoldi iteration in finite $q$ and finite $N$ limit. As a result, Krylov complexity exhibits exponential growth following a saturation at a time that grows logarithmically with the inverse dissipation strength. The growth of complexity is suppressed compared to the closed system results, yet it upper bounds the growth of the normalized out-of-time-ordered correlator (OTOC). We provide a plausible explanation of the results from the dual gravitational side.

Figures (4)

• Figure 1: (a) Operator size distribution of the Krylov basis operators $\mathcal{O}_n$ in large $N$, $q=4$ SYK model. In the main plot, we plot the standard deviation of the operator size divided by the average size of $\mathcal{O}_n$ as a function of $n$. The inset shows the whole operator size distribution for a few Krylov basis operators (the value of $n$ is given by the color code which is the same as in the main). The dashed lines are a guide to the eye. The distribution is strongly peaked at the largest operator size $s = (q-2) n + 1$ present in $\mathcal{O}_n$, so that the standard deviation is $\le 0.016$ times the mean for all $n \le 17$, and appears to tend to a constant as $n$ increases. (b) Magnitude of the matrix elements of the matrix $h_{m,n}$ resulting from the Arnoldi iteration in the large $N$, $q=4$ dissipative SYK model with $\tilde{\mu} = 1$ and $J = 1$. The presence of the secondary off-diagonal elements in the upper Hessenberg form indicates the finite $q$ effect.
• Figure 2: (a) The diagonal and primary off-diagonal elements in the matrix $h_{m,n}$ resulting from the Arnoldi iteration in the large $N$, $q=4$ dissipative SYK model with $\Tilde{\mu} = 1, 0.1, 0.01$ and $J = 1$. The markers indicate the value of $\tilde{\mu}$. We also plot the quantity $\varepsilon_n$ defined in \ref{['eq:errdef']} which measures the distance to the ideal scenario (it is multiplied by $100$ for visibility). The diagonal elements and $\varepsilon_n$ are rescaled by $\tilde{\mu}$ to display the collapse. (b) $\mathrm{SYK}_4$ (i.e., $q=4$) Lindbladian matrix (single realization) in Krylov (Arnoldi) basis for dissipation $\mu = 0.01$. We fix $J = 1$ and the system size $N=18$. The presence of the secondary off-diagonal elements in the upper Hessenberg form is due to both $q$ and $N$ being finite, and to the absence of disorder averaging; compare with Fig. \ref{['fig:opsize_syk4']} (b).
• Figure 3: Behavior of the (a) primary diagonal and (b) off-diagonal elements respectively, for two different dissipation $\mu = 0.01$ and $\mu = 0.02$. The dotted lines in (a) are the fitted straight lines \ref{['fithnn']}. The inset in (b) shows the difference $|h_{n,n-1} - h_{n-1,n}|$ grows with the dissipation. (c) The behavior of the diagonal elements (magnitude) for different system sizes. The black dotted line denotes the straight line \ref{['fithnn']} (with an offset introduced for visualization). (d) The saturation value (averaged for $n=10$ to $n=40$) of the diagonal elements for different system sizes. In all cases, our system size is $N = 18$ (single realization) and we choose $q = 4$, $J=1$, and the fixed dissipation $\mu = 0.01$ in (c) and (d).
• Figure 4: (a) Snapshots of the wave-function on the Krylov chain, from the exact solution \ref{['eq:anbnexact']}, \ref{['eq:phin_exact']} with $\eta = 1.5$ and $u = 0.01$. For display, we rescaled the wavefunctions by a $t$-dependent constant; we also interpolated $\varphi_n$ to non-integer values of $n$. At $t\to\infty$, a stationary profile is reached with an exponential tail $\propto e^{-n/\xi}$ with $\xi = 1/u$, indicated by the dashed curve. (b) Growth of the K-complexity for various dissipation strength ($u$), from the exact solution \ref{['eq:Ktex']}. The exponential growth, which lasts indefinitely in the closed system ($u=0$), saturates at $t \sim \ln (1/u)$, or $K(t) \sim 1/u$, in the presence of dissipation.