Holography of K-complexity: Switchbacks and Shockwaves

TL;DR

This work establishes a geometric interpretation of Krylov (K-)complexity in the double-scaled SYK (DSSYK) model by exploiting chord-diagram techniques and a triple-scaling limit that renders the bulk dual as semiclassical JT gravity. It demonstrates a bulk-boundary dictionary in which operator K-complexity maps to geodesic lengths acted upon by shockwaves in JT gravity, and shows, on the boundary, that K-complexity exhibits the switchback effect with universal late-time linear growth. A key methodological advance is the construction of two-sided perturbations of the Lanczos algorithm, which, in the triple-scaled regime, yield perturbed operator complexity that matches emergent ERB lengths with corresponding shockwave insertions in the bulk. The results generalize to multiple insertions and provide a robust holographic framework in which Krylov methods capture the geometric content of complexity, reinforcing K-complexity as a holographic diagnostic of chaotic dynamics.

Abstract

In this paper we study Krylov complexity in the presence of single and multiple operators in the DSSYK model, where we can use the analytical techniques coming from chord diagrammatics. One of the results we obtain is that it showcases the switchback effect, when the appropriate ``triple-scaling limit'' is taken, under which the model becomes dual to semiclassical JT gravity. We build on previous work, where it was shown that, in the continuum limit, Krylov complexity is defined as the sum of expectations value of right and left chord number operators. Here we argue that this property signals the emergence of the geometric nature of this notion of K-complexity. We show that in the regime where DSSYK is dual to semi-classical gravity, the light matter chord corresponds to a shockwave insertion in JT gravity. We identify the geodesic-length dual of the operator complexity and extend the relevant holographic dictionary to describe the details of the matter insertions. Additionally, we define a class of two-sided perturbations of the Lanczos algorithm that allows to analyze the switchback effect. In the appropriate semi-classical limit, this perturbed operator complexity is dual to an ERB length in JT gravity with corresponding shockwave insertions. We thus establish that K-complexity exhibits the expected switchback effect and universal late-time linear growth, consistent with previous findings regarding its geometric nature in the holographic bulk-boundary map.

Figures (8)

• Figure 1: Krylov complexity can be understood as the propagation of a wave packet along the Krylov chain, here labeled by the discrete position $n$ and shown in 1a). In Ambrosini:2024sre it was shown that, in a suitable continuum limit, $\lambda\rightarrow 0$, this propagation becomes ballistic, corresponding to the position of a fully localised wave packet evolved by a Liouville-like Hamiltonian $H_L \pm H_R$ evolving a general class of states, obtained by perturbing the thermofield double state, (see 1b). The choice of sign corresponds to different bulk dual prescriptions, as we show in Section \ref{['sec:bulk_dual']}. In all cases, the bulk geometry is that of a shockwave, with energy $E$ inserted above the black hole mass $M$, where the boundary operator dimension is given by $\Delta = E/M$.
• Figure 2: Krylov complexity can be understood as the propagation of a wave packet along the Krylov chain, here labeled by the discrete position $n$. In Ambrosini:2024sre it was shown that, in a suitable continuum limit, $\lambda\rightarrow 0$, this propagation becomes ballistic, corresponding to the position of a fully localized wave packet whose width goes to zero with $\lambda$, and can be understood as the growth in time of Krylov complexity in the presence of precursor operator insertions. In the so-called triple-scaled limit of DSSYK, which maps to the semiclassical description of JT gravity in an AdS$_2$ bulk, this complexity shows the switchback effect, triggered by the wave packet hitting the critical position $n_s = n(t_s)$ along the, now continuous, Krylov chain which induces the characteristic delay of complexity growth of order of the scrambling time. Indeed, as shown in the complexity profile, $C_K$ lingers around the critical value $C_s$ for a scrambling time. Here we show the case of a single precursor operator, but the picture generalizes to several such insertions.
• Figure 3: Intuitive representation of the DSSYK-JT gravity duality, where the blue dashed lines represent the event horizon of the 2D black hole. The re-normalized length obtained by considering the triple-scaled limit of K-complexity, that is the number of open chords intersected by the black dashed line, is dual to the re-normalized geodesic length anchored at times $t_L=t_R=t$ on the regularized boundary of AdS$_2$. Upon canonical quantization, by the matching of the Hamiltonians with the holographic dictionary \ref{['eq:hol_dict_nomatt']}, this duality is uplifted to an isomorphism between the Hilbert spaces of the quantized theories Rabinovici:2023yex.
• Figure 4: A timefold with six operator insertions, denoted by the red dots, between times $t_L$ and $-t_R$: at $t_1$, $t_2$, $t_4$ and $t_6$ we have switchback insertions, while $t_3$ and $t_5$ are through-going.
• Figure 5: A shockwave (green) insertion amounts to a null shift along its interface when computing the length of a geodesic anchored at points $t_L$, $t_R$ on the regularized boundary.