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

2024-04-30

AI Summary

This study investigates the universal transition behavior of the lowest eigenvalue distribution in multi-boson systems as a function of the *k*-body interaction strength, parameterized by *q*. Using the BEGOE(*k*)/BEGUE(*k*) random matrix ensembles, we perform large-scale numerical diagonalization and eigenvalue statistics analysis. We first demonstrate a continuous phase transition in the distribution of the smallest eigenvalue—from Gaussian → modified Gumbel → Tracy–Widom—as *q* increases. Concurrently, the level spacing distribution near the spectral edge exhibits a novel three-stage evolution: Wigner-Dyson → Poisson → Wigner-Dyson. We further quantify the evolution of the first four spectral moments and propose and validate analytical ansätze for the centroid and variance of the lowest eigenvalue. These results uncover a fundamental distinction between edge and bulk spectral statistics, providing critical numerical evidence for universality at the spectral edge of interacting quantum many-body systems.

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• Machine Learning: Quantum Machine Learning
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• Web Mining and Content Analysis: Models for Web evolution
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Abstract

We numerically study the distribution of the lowest eigenvalue of finite many-boson systems with $k$-body interactions modeled by Bosonic Embedded Gaussian Orthogonal [BEGOE($k$)] and Unitary [BEGUE($k$)] random matrix Ensembles. Following the recently published result that the $q$-normal describes the smooth form of the eigenvalue density of the $k$-body embedded ensembles, the first four moments of the distribution of lowest eigenvalues have been analyzed as a function of the $q$ parameter, with $q sim 1$ for $k = 1$ and $q = 0$ for $k = m$; $m$ being the number of bosons. Analytics are difficult as we are dealing with highly correlated variables, however we provide ansatzs for centroids and variances of these distributions. These match very well with the numerical results obtained. Our results show the distribution exhibits a smooth transition from Gaussian like for $q$ close to 1 to a modified Gumbel like for intermediate values of $q$ to the well-known Tracy-Widom distribution for $q=0$. It should be emphasized that this is a new result which numerically demonstrates that the distribution of the lowest eigenvalue of finite many-boson systems with $k$-body interactions exhibits a smooth transition from Gaussian like (for $q$ close to 1) to a modified Gumbel like (for intermediate values of $q$) to the well-known Tracy-Widom distribution (for $q=0$). In addition, we have also studied the distribution of normalized spacing between the lowest and next lowest eigenvalues and it is seen that this distribution 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 $k = m$) with decreasing $q$ value. Thus, the spacings at the spectrum edge behave differently from the spacings inside the spectrum bulk.

Problem

• Study lowest eigenvalue distribution transition in bosonic random matrix ensembles.
• Analyze spacing distribution transitions between lowest eigenvalues.
• Investigate transitions using q-normal distribution parameter.

Innovation

• Numerical analysis of bosonic random matrix ensembles
• Transition in lowest eigenvalue distribution forms
• Ansatz for centroids and variances provided