Matrices hold a prominent position in physics, generally serving as fundamental tools for describing relationships between physical quantities governed by linear laws. On the other hand, matrices themselves can be considered as physical entities. Such idea traces back to Wigner in the 1950s when he approximated the Hamiltonians of heavy nuclei using random matrices, which exhibited remarkable agreements with experimental data. This concept has continued to inspire researchers and finds applications in areas such as quantum chaos, quantum chromodynamics, and string theory. The mathematical framework for studying random matrices lies in matrix integrals, involving integrations of functions with matrix variables.

Furthermore, matrices can also be treated as quantum variables, giving rise to significant physical problems known as Matrix Quantum Mechanics (MQM). One important example is the BFSS matrix model, comprising nine matrices with bosonic elements and sixteen matrices with fermionic elements. It was conjectured that at large N, this model is dual to the eleven dimensional M-theory.

Large N multi-matrix systems, including both matrix integrals and MQM, thus play a vital role in theories of interest. However, except for specific cases, they are typically unsolvable due to the high nonlinearity and the rapid growth of degrees of freedom at large N. Therefore, it is desirable to develop a systematic framework for approaching these problems analytically or numerically.

Xianlong Liu, PhD’24 with collaborators, including his advisor Professor Antal Jevicki, has made progress in addressing this challenge by devising and improving an effective numerical scheme. This scheme can be summarized as “loop space optimization with master field variables”.

Large N matrix systems can be described using the collective field formulation proposed by Prof. Jevicki and Sakita in the 1980s. This formulation provides a systematic transformation of the original variables into invariant variables, such that it preserves Hermiticity. In matrix systems, the invariant variables are “loops”, i.e. traces of various matrix products. These loops constitute an (over) complete set, and the matrix systems can be reformulated in terms of a collective field theory whose dynamics is governed by a collective Hamiltonian, whose minimum gives the ground state. The loops meanwhile satisfy Schwarz inequalities, imposing infinite constraints on the system, which should be considered when performing optimization. Since the constraints are inherent properties of matrix traces, the explicit use of matrix elements, also known as master variables, to represent loop variables automatically satisfies the Schwarz inequalities, enabling direct minimization. As such, the scheme efficiently addresses this constrained optimization problem.

This method was tested on two-matrix integrals and also two-MQM. Constrained minimizations with nearly 10,000 master variables were accomplished, and the results exhibited excellent agreement with available analytical solutions.

The research also delved into the 1/N expansions and the spectra of small fluctuations. This involved expanding the Hamiltonian to order 1 and calculating the corresponding eigenvalues. Numerous energy levels for low-lying spectra were obtained, revealing degeneracy lifting with increasing interacting couplings. These findings are conjectured to hold true for generic multi-matrix systems.

Building upon this systematic study, Xianlong Liu aims to investigate more sophisticated matrix models in the future, including the BFSS matrix model at both zero and finite temperatures. These endeavors have the potential to study the BFSS conjecture and shed light on the properties of black holes based on the Gauge/Gravity duality principle." - Xianlong Liu

When we asked Xianlong how he anticipates the award to impact his research, he said he is honored to receive this prestigious fellowship, enabling him to delve deeper into his field and allowing him to engage with leading experts.

Congratulations, Xianlong!