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

• URL: http://arxiv.org/abs/2405.00190v3
• Date: Tue, 29 Oct 2024 17:52:46 GMT
• Title: Distribution of lowest eigenvalue in $k$-body bosonic random matrix ensembles
• Authors: N. D. Chavda, Priyanka Rao, V. K. B. Kota, Manan Vyas,
• Abstract summary: We numerically study the distribution of the lowest eigenvalue of finite many-boson systems with $k$-body interactions. The first four moments of the distribution of lowest eigenvalues have been analyzed as a function of the $q$ parameter.
• Score: 0.8999666725996978
• License:
• 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.

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