Matrix Model Calculations Simplify with New Analytical Technique

Dibakar Roychowdhury, from the Indian Institute of Technology Roorkee, and colleagues have determined Lanczos coefficients across varying deformation limits of the BMN matrix model using a systematic reduction to the pulsating fuzzy sphere model. The analytical framework advances understanding of complexity in gauge/gravity duality, offering a new approach to calculating key parameters relevant to quantum chaos and black hole information retrieval.

Lanczos coefficient scaling reveals linear behaviour in strongly deformed systems

Lanczos coefficients are fundamental to quantifying Krylov complexity, a measure of how quickly information spreads within a quantum system. These coefficients arise in the Lanczos algorithm, an iterative method used to find the eigenvalues and eigenvectors of a large, sparse matrix, and in this context, describe the expansion of a Krylov basis. The researchers have now demonstrated that these coefficients scale as an n ∼μ and b n ∼μ in the large deformation limit (μ ≫1), representing a significant departure from previous approximations which became inaccurate beyond leading order. This linear scaling, observed when the mass parameter μ exceeds 1, enables a level of detailed analysis previously hindered by computational limitations. Earlier methods relied on approximations that failed when the system was strongly deformed, meaning the parameters describing the system’s deviation from equilibrium were large. A systematic reduction of the BMN matrix model to the pulsating fuzzy sphere model facilitated this analytical calculation of the coefficients, unlocking exploration of Krylov state complexity in regimes inaccessible before. The pulsating fuzzy sphere model simplifies the original BMN model while retaining key features relevant to the calculation of Krylov complexity, allowing for tractable analytical solutions. Corrections to the Lanczos coefficients at leading order appear as order μ² in the small deformation limit, providing a complete picture of the scaling behaviour across all scales of deformation. The coefficient a₀, representing the diagonal element of the Hamiltonian, equals 1.15625μ + 1/4μ² for large μ, while the off-diagonal coefficient b₁, measuring the interaction between adjacent Krylov states, is approximately 0.848942μ. Furthermore, b₂, another off-diagonal element representing interactions between states further apart in the Krylov basis, scales as 10.8206μ^−10.7106 in the same limit; these precise values were obtained through iterative construction of the Krylov basis and subsequent orthogonalization procedures, ensuring numerical stability and accuracy. The ability to accurately determine these coefficients is crucial because they directly influence the rate of information propagation within the quantum system being modelled.

Krylov complexity calculations illuminate information scrambling in simplified quantum models

These coefficient calculations provide a novel perspective for examining strongly interacting systems, offering analytical control that is particularly valuable given the limitations of purely numerical approaches when dealing with extreme deformations. Numerical simulations, while powerful, often struggle with the computational demands of strongly interacting systems and can be prone to errors. A systematic analytical method for quantifying Krylov complexity, a measure of information spread within a quantum system, has been established within the BMN matrix model. The BMN matrix model itself is a simplification of more complex string theory models, designed to exhibit properties relevant to the gauge/gravity duality, a conjectured equivalence between quantum gravity in certain spacetime geometries and quantum field theories on their boundaries. Reducing the complex model to the pulsating fuzzy sphere allows calculation of these parameters across a broad range of deformations, offering a key testing ground for ideas about information scrambling in quantum systems, and potentially black holes. Information scrambling refers to the process by which information about a quantum system becomes distributed throughout the system, making it difficult to determine the initial state. This process is thought to be crucial for understanding the behaviour of black holes, where information that falls into the black hole appears to be lost, violating fundamental principles of quantum mechanics. The gauge/gravity duality suggests that this information is not actually lost, but rather scrambled and encoded on the boundary of the black hole. The current framework remains rooted within the BMN matrix model and its simplification, which raises an important question regarding the broader applicability of these findings. While the pulsating fuzzy sphere model captures essential features of the BMN model, it is still an approximation, and it is important to investigate whether the observed scaling behaviour of the Lanczos coefficients holds in more general settings. Future work will likely extend these calculations beyond the current model, beginning to explore broader theoretical fields and potentially informing our understanding of the connection between quantum chaos, gravity, and black hole information. Specifically, researchers may investigate whether similar scaling behaviour emerges in other matrix models or in more realistic models of quantum gravity, potentially bridging the gap between theoretical calculations and observations of black holes.

The research successfully calculated Lanczos coefficients within a simplified BMN matrix model, known as the pulsating fuzzy sphere model, across various deformations. This calculation provides a key analytical framework for studying information scrambling, a process central to understanding how information behaves in quantum systems. Information scrambling is particularly relevant to theoretical investigations of black holes, where it may explain how information is preserved despite appearing to be lost. Researchers intend to extend these calculations to other matrix models and quantum gravity theories to assess the broader relevance of these findings.