Distribution of lowest eigenvalue in $k$-body bosonic random matrix ensembles

Abstract

We present numerical investigations demonstrating the result that the distribution of the lowest eigenvalue of finite many-boson systems (say we have $m$ number of bosons) with $k$-body interactions, modeled by Bosonic Embedded Gaussian Orthogonal [BEGOE($k$)] and Unitary [BEGUE($k$)] random matrix Ensembles of $k$-body interactions, exhibits a smooth transition from Gaussian like (for $k=1$) to a modified Gumbel like (for intermediate values of $k$) to the well-known Tracy-Widom distribution (for $k=m$) form. We also provide ansatz for centroids and variances of the lowest eigenvalue distributions. In addition, we show that the distribution of normalized spacing between the lowest and the next lowest eigenvalues exhibits a transition from Wigner's surmise (for $k=1$) to Poisson (for intermediate $k$ values with $k\le m/2$) to Wigner's surmise (starting from $k=m/2$ to $k=m$) form. We analyze these transitions as a function of $q$ parameter defining $q$-normal distribution for eigenvalue densities.

25 pages, 11 figures, version accepted for publication in Physica A: Statistical Mechanics and its Applications (2025)

References in corpus (38)

• Random Matrix Theories in Quantum Physics: Common Concepts
• Gapless Spin-Fluid Ground State in a Random Quantum Heisenberg Magnet
• Level-Spacing Distributions and the Airy Kernel
• On Orthogonal and Symplectic Matrix Ensembles
• The Statistical Theory of Quantum Dots
• Quantum Chaos and Thermalization in Isolated Systems of Interacting Particles
• Universality Classes for Extreme Value Statistics
• Spectral and thermodynamic properties of the Sachdev-Ye-Kitaev model
• Level-Spacing Distributions and the Airy Kernel
• Universal Fluctuations in Correlated Systems
• Extreme Value Statistics of Eigenvalues of Gaussian Random Matrices
• Extreme value statistics of correlated random variables: a pedagogical review
• Notes on the complex Sachdev-Ye-Kitaev model
• Analytical Spectral Density of the Sachdev-Ye-Kitaev Model at finite N
• Distribution of the largest eigenvalue for real Wishart and Gaussian random matrices and a simple approximation for the Tracy-Widom distribution
• Quantum chaos transition in a two-site SYK model dual to an eternal traversable wormhole
• Random Matrices and Chaos in Nuclear Spectra
• Non-interacting fermions at finite temperature in a $d$-dimensional trap: universal correlations
• Review of the k-Body Embedded Ensembles of Gaussian Random Matrices
• General Features of the Relaxation Dynamics of Interacting Quantum Systems
• Spectral Properties of the k-Body Embedded Gaussian Ensembles of Random Matrices
• Extreme statistics and spacing distribution in a Brownian gas correlated by resetting
• Regularities of Many-body Systems Interacting by a Two-body Random Ensemble
• Spectral Statistics of the Two-Body Random Ensemble Revisited
• Chaos and Thermalization in a Dynamical Model of Two Interacting Particles
• Near-extreme eigenvalues and the first gap of Hermitian random matrices
• Chaos on the hypercube
• About the Functional Form of the Parisi Overlap Distribution for the Three-Dimensional Edwards-Anderson Ising Spin Glass
• Quantum Chaos on Edge
• Particle Diagrams and Statistics of Many-Body Random Potentials
• Quenched many-body quantum dynamics with $k$-body interactions using $q$-Hermite polynomials
• Random Matrix Ensembles For Many-Body Quantum Systems
• A smooth transition towards a Tracy-Widom distribution for the largest eigenvalue of interacting $k$-body fermionic Embedded Gaussian Ensembles
• Ground-state energy distribution of disordered many-body quantum systems
• Structure of wavefunction for interacting bosons in mean-field with random $k$-body interactions
• Thermalization in many-fermion quantum systems with one- plus random $k$-body interactions
• Two species $k$-body embedded Gaussian unitary ensembles: $q$-normal form of the eigenvalue density
• Bivariate moments of the two-point correlation function for embedded Gaussian unitary ensemble with $k$-body interactions