Complexity transitions in chaotic quantum systems: Nonstabilizerness, entanglement, and fractal dimension in SYK and random matrix models

Abstract

Complex quantum systems -- composed of many, interacting particles -- are intrinsically difficult to model. When a quantum many-body system is subject to disorder, it can undergo transitions to regimes with varying non-ergodic and localized behavior, which can significantly reduce the number of relevant basis states. It remains an open question whether such transitions are also directly related to an abrupt change in the system's complexity. In this work, we study the transition from chaotic to integrable phases in several paradigmatic models, the power-law random banded matrix model, the Rosenzweig--Porter model, and a hybrid SYK+Ising model, comparing three complementary complexity markers -- fractal dimension, von Neumann entanglement entropy, and stabilizer Rényi entropy. For all three markers, finite-size scaling reveals sharp transitions between high- and low-complexity regimes, which, however, can occur at different critical points. As a consequence, while in the ergodic and localized regimes the markers align, they diverge significantly in the presence of an intermediate fractal phase. Additionally, our analysis reveals that the stabilizer Rényi entropy is more sensitive to underlying many-body symmetries, such as fermion parity and time reversal, than the other markers. As our results show, different markers capture complementary facets of complexity, making it necessary to combine them to obtain a comprehensive diagnosis of phase transitions. The divergence between different complexity markers also has significant consequences for the classical simulability of chaotic many-body systems.

Figures (11)

• Figure 1: Complexity markers across a transition from a chaotic (ergodic) to an integrable (localized) phase. (a) As the control parameter $\mu$ drives a random matrix model towards localization, the eigenvalue distribution evolves from the Wigner--Dyson semicircle to a Gaussian profile. (b) Comparison of three complexity markers (data for the RP model). A high (low) fractal dimension $D_2$ indicates delocalization (localization) in the computational basis. High (low) magic $\mathcal{M}_2$ implies support over an extensive set of $4^N$ Pauli strings (concentration on a limited subset). Strong (weak) entanglement $S_\mathrm{vN}$ reflects high (low) Schmidt rank and the presence of many (few) computational basis states in the superposition. While all three markers feature a high-complexity regime, the transition towards intermediate- or low-complexity regimes can happen at distinct points ($\mu_{1,2}^{D_2} \neq \mu_{1,2}^{\mathcal{M}_2} \neq \mu_{1,2}^{S}$). Regimes of different complexity thus overlap but do not coincide. (c) The phase transition points in the RP model, obtained through finite-size scaling analysis, differ across the three complexity markers (orange: high [$\mathcal{O}(1)$] complexity; green: intermediate; blue: low complexity [$\mathcal{O}(1/N)$]). Faded regions indicate numerical uncertainty in determining regime boundaries. Notably, the ground state exhibits a sharp transition directly from high to low complexity, bypassing the intermediate regime. In contrast, bulk eigenstates display a well-defined intermediate phase, with a significant variation in transition points across the considered markers.
• Figure 2: Three model Hamiltonians exhibiting a transition from chaotic (ergodic) to localized phases, controlled by tunable parameters. (a) Rosenzweig--Porter (RP) model: A random matrix ensemble where the off-diagonal elements have decreasing weight as the parameter $\gamma$ increases. (b) Power-Law Random Banded Matrix (PLRBM) model: Characterized by off-diagonal elements that decay with a power-law determined by the exponent $\alpha$. (c) SYK$_4$+Ising hybrid model: The coupling strength $\lambda$ interpolates between the Sachdev--Ye--Kitaev (SYK$_4$) model with all-to-all random four-body interactions drawn from a Gaussian distribution (left, $\lambda=0$) and an Ising chain, expressed in Majorana fermions (right, $\lambda =1$).
• Figure 3: (a1-c1) Complexity markers (normalized with respect to $N$), averaged over the central $20\%$ of the bulk eigenstates of the RP model for different system sizes from $N=4$ up to $N=12$, and averaged over samples from $20000$ for $N=4$ to $500$ for $N=12$. All markers plateau at the maximum value in an extended regime at small $\gamma$, overlapping with the model's ergodic regime. The $N$-dependent value matches with predictions from the GUE ensemble that is exactly recovered at $\gamma=0$. The plateau's extent in terms of $\gamma$ depends on the marker: the plateau is largest for $S_{\rm vN}$ and smallest for $D_2$. At large $\gamma$, the model enters a localized regime, characterized by a low value of all markers. The slope at which the model goes from the ergodic to the localized regime is different for all markers, as becomes clear from the derivatives (a2-c2). The crossings in the derivatives (insets in a2-c2) determine two different transition points $\gamma_{1}$ and $\gamma_2$. There are, therefore, three different regimes of high, intermediate, and low complexity, which do not coincide for the different complexity markers used.
• Figure 4: Complexity markers for the ground state of the RP model. Inset: first derivative with respect to $\gamma$. A direct transition from an ergodic to a localized phase is observed. The transition points detected through fractal dimension $D_2$ (a) and SRE density $\mathcal{M}_2/N$ (c) coincide within error bars. The entanglement entropy $S_\mathrm{vN}$ (b) indicates the transition at a slightly smaller value of $\gamma$, but with a larger error bar due to the even--odd effect.
• Figure 5: (a1-c1) Complexity markers (normalized with respect to $N$), averaged over the central $20\%$ of bulk eigenstates of the PLRBM model for $N=4$ up to $N=9$ ($N=12$ for the fractal dimension) averaged over different sample sizes of $35000$ to $3000$, respectively. Throughout an extended regime at small $\alpha$, all markers plateau at the value matching with predictions from the GUE ensemble (recovered by the model at $\alpha=0$). At large $\alpha$, the model transitions to a localized regime, characterized by a lower value of all the markers. The transition point can be deduced from the derivatives of the complexity markers (a2-c2). The crossings in the derivatives (insets a2-c2) effectively find only one distinct transition point. Therefore, in contrast to the RP model, there are only 2 different regimes of high and low complexity that coincide with the ergodic and localized phases, respectively.