Chaotic Systems Exhibit Predictable Randomness in Fundamental Fermion Properties

Researchers are investigating the complex behaviour of the Sachdev-Ye-Kitaev (SYK) model, a theoretical framework used to explore the intersection of quantum chaos, gravity, and information. Valérie Bettaque and Brian Swingle, both from Brandeis University, detail a statistical analysis of fermion string expectation values within the SYK model using path integral techniques. Their work, conducted in collaboration, reveals that chaotic SYK Hamiltonians produce Gaussian random variables, differing significantly from the non-Gaussian behaviour observed in integrable variants. This distinction allows for the quantification of ‘magic’ within thermal states, utilising measures such as robustness and stabilizer Rényi entropy. Significantly, the findings demonstrate quantitative agreement with a dual gravity calculation, linking the variance of operator strings to wormhole geometries stabilised by massive particles, thus providing a novel setting to explore the relationship between randomness, wormholes, and closed universes through a holographic duality.

Scientists have established a precise connection between quantum information, gravity, and the behaviour of complex quantum systems, potentially unlocking new insights into the fundamental nature of spacetime and quantum gravity. This work centres on the Sachdev-Ye-Kitaev (SYK) model, a theoretical framework used to simulate the properties of black holes and other exotic quantum phenomena.

Researchers discovered that, within a chaotic SYK model, the expectation values of strings of fermion operators behave as real Gaussian random variables, a statistically predictable pattern despite the system’s inherent complexity. This finding is significant because it demonstrates a quantifiable relationship between the seemingly disparate worlds of quantum mechanics and gravity.

In contrast, integrable variants of the SYK model display non-Gaussian behaviour, highlighting the importance of chaos in establishing this connection. The study shows that these statistical properties can be accurately reproduced using a dual gravity calculation, linking the behaviour of quantum information to the geometry of wormholes, theoretical tunnels connecting different points in spacetime.

Specifically, the variance of these fermion strings directly corresponds to the geometry of the wormholes, implying a precise mathematical relationship between the two. This holographic duality, where a quantum system is described by a gravitational system in a higher dimension, provides a concrete setting to explore the interplay between randomness, wormholes, and the concept of closed universes.

A detailed examination of Majorana fermion strings underpins this work, employing path integral techniques to characterise their statistical properties within the SYK model. The SYK model, a widely-used tool in many-body physics and holographic quantum gravity, was selected due to its tractable nature at large N via path integrals and the existence of an integrable variant for comparison.

Researchers computed statistical moments of thermal one-point functions by integrating over multiple copies of the system, enabling analysis at larger system sizes and moving beyond simple diagonalisation. To establish a foundation for analysis, the study defined a system of N Majorana fermions, represented by Hermitian operators obeying a specific algebraic relation ensuring paired fermion creation and annihilation.

Complete sets of Hermitian operators, known as Majorana strings, were constructed from these fermions, each uniquely identified by a string of indices and possessing a defined weight. These strings form a basis for describing all possible states within the system, with the fermion parity operator playing a special role due to its commutation with the Hamiltonian.

An ensemble of Hamiltonians, known as the q-body SYK model, was then considered, defined by Gaussian random couplings between the Majorana fermions. The variance of these couplings was carefully calibrated, and the ensemble averaged to provide statistically meaningful results, with the choice of q ≥ 4 ensuring quantum chaotic behaviour. Thermal states were then generated using the Gibbs ensemble, setting the stage for calculating the crucial thermal expectation values of the Majorana strings.

Further analysis extends to measures of ‘magic’ within the SYK thermal state, including the robustness of magic and the stabilizer Rényi entropy, providing insights into the state’s complexity and non-stabilizerness at low temperatures. By quantitatively reproducing these results with a dual gravity calculation, the work provides a microscopic verification of theoretical proposals linking operator randomness to wormhole geometries.

This dual gravity model posits a relationship where the variance of a microscopic operator string directly corresponds to a wormhole geometry stabilised by a massive particle, the dual of the operator string itself. Scientists have demonstrated a remarkable correspondence between the seemingly disparate realms of quantum chaos, gravity, and quantum information.

This isn’t merely a mathematical curiosity; it’s a significant step towards understanding how complex quantum systems behave and, potentially, the very fabric of spacetime itself. For years, physicists have sought a concrete link between the abstract mathematics describing black holes and the behaviour of entangled quantum particles, a connection that could illuminate the elusive nature of quantum gravity.

Researchers found that the expectation values of these fermion strings behave like random variables, and crucially, this behaviour can be accurately predicted by calculations rooted in general relativity and wormhole geometry. However, the model relies on specific conditions, notably, a highly chaotic system and low temperatures, limiting the immediate applicability of these findings. Future research will likely focus on extending this framework to more realistic quantum systems and exploring whether these connections can be harnessed for practical applications, such as improving quantum error correction or developing new quantum technologies.