Quantum Chaos: Tunable Transition to Faster Behaviour

Scientists have presented a new solution to the Lévy Sachdev-Ye-Kitaev (LSYK) model, offering insights into chaotic dynamics and non-Fermi liquid behaviour. Budhaditya Bhattacharjee and colleagues at University of Luxembourg present an exact solution in the large- N limit, examining how varying the coupling distribution impacts the model’s chaotic properties and thermodynamic quantities. The collaboration between the University of Luxembourg, National University of Singapore, Centre for Theoretical Physics of Complex Systems, and Centre for Trapped Ions Quantum Science reveals a continuous transition between a free theory and the maximally chaotic Gaussian SYK model, demonstrating non-maximal chaos across an intermediate regime. The research advances understanding of holographic duality and provides a novel connection to the Gaussian SYK model through an alternative decomposition of the Lévy Stable distribution.

Lévy SYK model exhibits continuous chaoticity shift and altered thermodynamic properties

The Lyapunov exponent, a key measure of quantum chaos quantifying the rate of separation of initially close trajectories in phase space, transitions from 2.2 at μ = 0 to approximately 1.2 at μ = 2 within the Lévy Sachdev-Ye-Kitaev (LSYK) model. This represents the first exact solution demonstrating a continuous shift in chaoticity, a crucial development as many condensed matter systems exhibit behaviour between fully ordered and maximally chaotic states. Previously, precise control over the level of chaos necessitated approximations within theoretical frameworks like perturbation theory, which compromised the model’s solvability and hindered detailed analysis of the transition between order and disorder. Utilising a Lévy Stable distribution for interactions, which allows for heavier tails compared to the Gaussian distribution, the new solution allows physicists to explore a regime of non-maximal chaos. In this regime, the system exhibits chaotic behaviour but not at the maximum rate, between μ = 0 and μ = 2. The entropy, free energy, average energy, and specific heat capacity all differ when contrasted with their Gaussian SYK counterparts, providing further validation of the model’s behaviour and highlighting the impact of the altered coupling distribution. A non-trivial connection to Gaussian SYK was established through an alternative representation of the LSYK model, utilising a distinct decomposition of the Lévy Stable distribution and supported by both analytical and numerical results. Analysis of the model’s thermodynamics also considered its holographic dual, a theoretical construct linking quantum gravity in anti-de Sitter space to conformal field theories on the boundary, and potential relevance to non-Fermi liquid theory, expanding the scope of its potential applications in understanding strongly correlated electron systems.

Schwinger-Dyson Equations and Closed-Form Solutions for the Large-N LSYK Model

Schwinger-Dyson equations proved important in unlocking an exact solution to the LSYK model; these equations are a set of integral equations that approximate solutions to complex quantum field theories by relating different Green’s functions, which describe the propagation of particles. Physicists initially employed these equations to map the relationships between different quantum properties within the model, building a self-consistent picture of its behaviour. The equations account for the complex many-body interactions within the system, allowing for a systematic approach to finding approximate solutions. Iteratively refining approximations until a stable, accurate solution emerges enabled scientists to bypass the difficulties of directly solving the intractable many-body problem, which would require exponentially increasing computational resources. Applying these equations within a bosonic oscillator representation of the action, a mathematical technique that transforms the problem into a more manageable form involving harmonic oscillators, scientists derived a closed-form expression, enabling detailed analysis of the model’s properties. Researchers investigated the model, focusing on its behaviour in the large- N limit; N represents the number of Majorana fermions, which are particles that are their own antiparticles, and the model utilises a Lévy Stable distribution, parameterised by a tail exponent, μ, ranging from 0 to 2. The large- N limit simplifies the calculations by effectively averaging over many degrees of freedom, allowing for analytical solutions that would otherwise be inaccessible. This approach allows analysis of chaotic properties and thermodynamic quantities like entropy and free energy, offering a contrast to the standard Gaussian SYK model, which serves as a benchmark for understanding quantum chaos.

Precise control of quantum chaos achieved via an exact solution of the Lévy Sachdev-Ye-Kitaev model

An exact solution for the LSYK model has been unlocked, representing a significant advance in understanding quantum systems exhibiting chaotic behaviour. This breakthrough allows precise control over the degree of chaos within the model, differing from previous approaches reliant on Gaussian randomness, which often leads to maximal chaos. The ability to tune the Lyapunov exponent via the parameter μ provides a powerful tool for investigating the transition from ordered to chaotic behaviour and exploring intermediate regimes. The current solution, however, remains constrained by the large- N limit, a mathematical simplification assuming an infinite number of particles; this limitation raises questions about its applicability to real-world systems with a finite number of interacting components. While the large- N limit provides valuable insights, further research is needed to determine the extent to which these results hold for finite-size systems.

Vital for modelling quantum systems, this level of control has now been achieved. Within the large- N limit, establishing this solution allows physicists to precisely tune the level of chaos present in the system, revealing a continuous shift from predictable behaviour to maximum chaos. This research delivers an exact solution for a complex quantum system where interactions follow a Lévy Stable distribution, opening avenues for exploring novel quantum phenomena and potentially informing the development of new quantum technologies. The ability to manipulate and understand quantum chaos is crucial for advancements in areas such as quantum computing and materials science.

An exact solution to the Lévy Sachdev-Ye-Kitaev model has been found, demonstrating a way to control the degree of chaos within a quantum system. This is important because it moves beyond the standard approach of Gaussian randomness, which typically results in maximal chaos, and allows researchers to explore intermediate levels of chaotic behaviour using the parameter μ. The study computed thermodynamic quantities like entropy and free energy, offering a comparison to the Gaussian SYK model. Researchers established an alternative representation of the model, connecting it to Gaussian SYK and providing further insight into its chaotic properties.