Operator K-complexity in DSSYK: Krylov complexity equals bulk length

TL;DR

The paper develops a detailed, chord-diagram-based analysis of Krylov complexity in double-scaled SYK with matter insertions, establishing two complementary notions: operator KC and state KC for O_seed deformed TFD. It demonstrates that, in the semiclassical limit, the Krylov basis aligns with fixed total chord-number states, turning KC into the expectation value of a length operator in the bulk, and connects this to JT gravity via triple scaling. Analytic results for Lanczos coefficients and KC regimes reveal a rich structure, including a quadratic to linear growth transition and a scrambling–like exponential window controlled by operator size, with a gravitational interpretation through Morse/Liouville-type potentials. The work extends prior matterless results to include matter insertions, derives the triple-scaled bulk Hamiltonians, and clarifies the relationship between operator and state KC in this holographic setting, opening paths to explore saturation and more general chord-algebraic structures. Overall, it strengthens the boundary-bulk dictionary by tying explicit boundary Krylov dynamics to bulk geometric length in the gravity dual.

Abstract

In this paper we study the notion of complexity under time evolution in chaotic quantum systems with holographic duals. Continuing on from our previous work, we turn our attention to the issue of Krylov complexity upon the insertion of a class of single-particle operators in the double-scaled SYK model. Such an operator is described by a matter-chord insertion, which splits the theory into left/right sectors, allowing us, via chord-diagram technology, to compute two different notions of complexity associated to the operator insertion: first a Krylov operator complexity, and second the Krylov complexity of a state obtained by an operator acting on the thermofield double state. We will provide both an analytic proof and detailed numerical evidence, that both Krylov complexities arise from a recursively defined basis of states characterized by a constant total chord number. As a consequence, in all cases we are able to establish that Krylov complexity is given by the expectation value of a length operator acting on the Hilbert space of the theory, expressed in terms of basis states, organized by left and right chord number. We find analytic expressions for the semiclassical limit of K-complexity, and study how the size of the operator encodes the scrambling dynamics upon the matter insertion in Krylov language. We furthermore determine the effective Hamiltonian governing the evolution of K-complexity, showing that evolution on the Krylov chain can equivalently be understood as a particle moving in a Morse potential. A particular type of triple scaling limit allows to access the gravitational sector of the theory, in which the geometrical nature of K-complexity is assured by virtue of being a total chord length, in an analogous fashion to what was found in [1] for the K-complexity of the thermofield double state.

Figures (27)

• Figure 1: Summary of results for DSSYK without matter for $\lambda=0.01$ and $J=1$. Left: the Lanczos coefficients as given by \ref{['bn_DSSYK_noMatter']}. Their behavior switches from $\sim \sqrt{n}$ behavior to constant behavior at $n=1/\lambda$. Right: Krylov complexity switches from $\sim t^2$ behavior to linear behavior at a time $t_*=1/J$. The blue line shows the numerical solution gotten from solving the Krylov problem for the Hamiltonian \ref{['H_matterless']}.
• Figure 2: Profile of three Krylov elements $|\psi_n\rangle$, for $n=10,15,20$ (respectively, left, center and right) over the basis $|k,m-k\rangle$; entries are shown in absolute value (from red to purple, in decreasing absolute value). In all cases we fixed $\widetilde{q}=0.56$, and considered two values of $\lambda = 0.05,0.5$ (resp. top and bottom). We can observe that in the case where $\lambda=0.5$ the Krylov elements $|\psi_n\rangle$ develop significant projection over sectors with total chord number $m<n$.
• Figure 3: Results of the numerical implementation of the Lanczos algorithm for a system with $\lambda = 0.05$, $\widetilde{q}=0.577$, using a finite Hilbert space truncation that reaches up to the $N=50$ chord sector. Top left: Lanczos coefficients $b_n$ as a function of $n$. Vertical lines indicate, for reference, the values $n=\frac{1}{\lambda}$ and $n=N$. The coefficients $b_n$ for $n>N$ are subject to finite-truncation effects (in fact, the $b_n$ sequence was found to continue up to termination at $K\sim d_N\sim N^2/2$, even though it is not plotted for the sake of this discussion). The analytical estimate $b_n^2=J^2\, c_{n/2}(n)$, given in section \ref{['subsect:bn_asymptotic_limit']}, is found to agree excellently with the Lanczos coefficients for all $n\leq N$. Top right: Expectation value of total chord number of the Krylov elements $|\psi_n\rangle$. For $n\leq N$ (i.e. before truncation effects kick in) we observe that $\langle \psi_n |\widehat{n}|\psi_n\rangle\approx n$, suggesting that those Krylov elements are peaked on the corresponding chord sector. For $n>N$ the chord number expectation value starts to decrease as a function of $n$, reflecting the fact that the wave function of the subsequent Krylov vectors probes orthogonal directions within the chord sectors that are not included in the span of the Krylov elements with $n\leq N$. Bottom left: Participation ratio of the chord sectors in each Krylov vector. For $n\leq N$ we find that it is equal to one, confirming that each Krylov vector is confined within one chord sector. Putting this together with the plot on the top right, one can conclude that $|\psi_n\rangle \in \mathcal{H}_{1p}^{(n)}$ for $n\leq N$, i.e. the Krylov vectors, which are Krylov complexity eigenstates, are simultaneously total chord number eigenstates. Bottom right: For illustration purposes, this plot compares the profile of the $n=N$ Krylov vector over the sector $\mathcal{H}_{1p}^{(N)}$, i,e, the wave function $\psi^{(N)}_{Nk}$ in \ref{['Krylov_element_coordinates']}, and compares it to the (normalized) binomial Ansatz \ref{['Binomial_Ansatz']}, showing good qualitative agreement. The horizontal axis in this case is the left chord number $k$, labeling the basis elements $|k,N-k\rangle$ in the aforementioned sector. Such basis elements are not orthogonal, and therefore this plot may be taken as a qualitative demonstration (similar to figure \ref{['fig:3dplots_main']}), rather than as a quantitative analysis of norm contributions. Nevertheless, the binomial Ansatz is seen to match very closely the numerically obtained wave function. The overall conclusion emerging from this figure is the following: For $n\leq N$, the Krylov elements $|\psi_n\rangle$ are localized in sectors of total chord number $n$, within which they are satisfactorily described by the binomial Ansatz \ref{['Binomial_Ansatz']}; furthermore, they solve the Lanczos recursion \ref{['Lanczos_recursion_1p']} with Lanczos coefficients that are accurately described by $b_n=J\sqrt{c_{n/2}(n)}$, where $c_k(n)$ is given in \ref{['Ckn_simplified']}, an approximation that will be argued for in section \ref{['subsect:bn_asymptotic_limit']} (always in the regime that is not affected by the truncation protocol).
• Figure 4: Numerical solution of the Lanczos algorithm with PRO for a finite truncation of the one-particle Hilbert space with chord sectors up to $N=50$, for the parameters $\lambda=0.5$ and $\widetilde{q}=0.577$ (i.e. same parameters as in \ref{['fig:PRO_results_lambda0pt05']} but with a bigger value of $\lambda$). In plots whose horizontal axis is $n$, the vertical lines $n=\frac{1}{\lambda},N$ have been included for reference. Top left: Lanczos coefficients. We start to observe small deviations between the numerics and the semiclassical approximation $b_n=J\sqrt{c_{n/2}(n)}$ (cf. section \ref{['subsect:bn_asymptotic_limit']}). Top right: Total chord number expectation value of the Krylov vectors $|\psi_n\rangle$. Even before the truncation, it deviates from a line with unit slope, reflecting the fact that Krylov vectors start to develope a non-negligible tail over sectors of smaller total chord number (cf. figure \ref{['fig:3dplots_main']}). Bottom left: Consistently, the chord sector participation ratios deviate from $1$. Bottom right: As an illustration, the profile of the Krylov element $|\psi_N\rangle$ over the sector $\mathcal{H}_{1p}^{(N)}$, captured by the wave function $\psi^{(N)}_{Nk}$, is seen to deviate slightly from the (normalized) binomial Ansatz \ref{['Binomial_Ansatz']}.
• Figure 5: The Lanczos coefficients of \ref{['eq:lanczos_coeff']} with $J=1$ and $\lambda=0.01$, for $\Tilde{q}=0.5$ and $\Tilde{q}=0.99$ (in the inset: a zoomed-in version for better visualization of the small $n$ regimes). For $\Tilde{q}=0.5$, the thick gray lines show the small and large $n$ approximations in \ref{['eq:op_lanczos_regimes']}. For $\Tilde{q}=0.99$ the gray lines show \ref{['eq:bn_second_order']} for small $n$ and \ref{['eq:bn_qtilde1_largen']} for large $n$; in the inset we show in gray the linear approximation \ref{['eq:bn_linear_approx']}.