Magic and Wormholes in the Sachdev-Ye-Kitaev Model

TL;DR

This work analyzes the ensemble of thermal Majorana-string observables in the SYK model, showing that chaotic $q\geq 4$ SYK yields Gaussian fluctuations for the thermal 1-point functions with variance set by operator weight, while the integrable $q=2$ case remains non-Gaussian. Using a multi-replica path integral, the authors derive Schwinger-Dyson equations and implement an anticirculant replica symmetry to solve for the moments in the large-$N$ limit, revealing cross-replica correlations essential for nonzero higher moments. They connect these statistical properties to quantum magic measures, deriving bounds on robustness of magic and analyzing stabilizer Rényi entropy, with the results consistent with a holographic interpretation via wormholes in JT gravity at low temperature. The holographic mapping identifies the variance of operator strings with a wormhole geometry stabilized by a bulk heavy particle, providing a concrete link between randomness, non-stabilizerness, and closed universes in a quantum-gravity setting. Overall, the paper delivers a quantitative framework for operator randomness in SYK, demonstrates a holographic dual for magic, and paves the way for future explorations of wormholes, randomness, and quantum information in many-body systems.

Abstract

Any quantum state is fully specified by the expectation values of a complete set of Hermitian operators. For a system of Majorana fermions, such as the Sachdev-Ye-Kitaev (SYK) model, this set of observables can be taken to be all possible strings of Majorana fermion operators. The expectation values of these fermion strings in a thermal state depend erratically on the microscopic couplings that specify the SYK Hamiltonian, and we study their statistical properties directly in the thermodynamic limit using path integral techniques. When the underlying SYK Hamiltonian is chaotic, we find that these expectation values are well-modeled as real Gaussian random variables with zero mean and a variance that we compute. In contrast, for the integrable variant of SYK, we find that the expectation values are actually non-Gaussian. We then use these results to study measures of magic in the SYK thermal state, including the robustness of magic and the stabilizer Rényi entropy. We also show that our results can be quantitatively reproduced with a dual gravity calculation in the chaotic case at sufficiently low temperature. In this dual gravity model the variance of a given microscopic operator string is related to a wormhole geometry stabilized by a massive particle which is dual to the operator string. Our results thus provide a concrete and quantitative setting in which to study the relationship between randomness, wormholes, and closed universes as well as a holographic dual of quantum magic.

Figures (21)

• Figure 1: (A) The basic entry in the holographic dictionary states that the ensemble averaged SYK partition function is equal to a gravitational path integral that includes a sum over geometries and different topologies. The bulk theory entering this path integral reduces to JT gravity for simple observables but in general includes a rich and not well understood collection of bulk degrees of freedom. Below, we will model this bulk theory as JT gravity coupled to $N$ free fermions. (B) Boundary correlations of the fermion operators are related to correlations of the bulk fermion fields propagating in the gravitational geometry. For ease of analysis, we will treat these bulk correlations within a geodesic approximation. (C) For some observables, the sum over topologies is important in that there is no analog of the disk contribution and the first contribution instead comes a geometry with a non-trivial topology. In our case, a wormhole geometry serves to capture the variance of the erratic thermal 1-point functions.
• Figure 2: $L$ dependence of the single-replica determinant $\det(\mathcal{D}^+ - (\Delta \tau)^2 \, \Sigma) \approx (\Delta\tau)^2 \mathop{\mathrm{tr}}\nolimits(\mathbb{1} \, \Sigma)$ evaluated at $w = 0$ and for $q = 4$. In all cases a fit of the values (offset rounded to five decimals) indicates that both the determinant and its zero mode approximation decrease with about the same inverse power of $L$ towards 0, at least approximately. The only outlier for this trend is the case of $\beta = 5$, where no good fit can be made because the determinant is initially negative.
• Figure 3: $L$ dependence of the two-replica determinant $\det(I_2 \otimes \mathcal{D}^+ - (\Delta \tau)^2 \, \Sigma) \approx (\Delta\tau)^4 |\mathop{\mathrm{tr}}\nolimits(\mathbb{1} \, (\Sigma_0 + i \, \Sigma_1))|^2$ evaluated at $w = 0.5$ and for $q = 4$. In all cases a fit of the values (rounded to two decimals) indicates that both the determinant and its zero mode approximation -- while not matching at larger $\beta J$ -- are either essentially independent of $L$ (up to small corrections) or converge towards a non-zero value at large $L$.
• Figure 4: One potential choice for initial values of $G$ whose replica structures are not equivalent under signed permutations, up to $R = 8$. We also comment on if a given initial value can be block-diagonalized, has been observed to converge towards a different equivalence class of replica structures, or does not seem to converge at all under certain circumstances.
• Figure 5: Matrix heatmaps of all distinct (non-diagonal) replica structures ($R = 2, 4, 6, 8$) which are left invariant by the Schwinger-Dyson equations and result in "well-behaved" saddle points for all $q$ and $\beta$. Orange represents a value of $+1/2$, blue a value of $-1/2$ and gray a value of $0$. Arrows indicate self-similarity relationships between replicas of different sizes.