Moments and SU(N) algebra for embedded unitary ensemble
A talk in the Seminar Zufallsmatrizen series by
V.K.B. Kota
| Abstract: | Embedded random matrix ensembles with k-body interactions, usually called EE(k), introduced 50 years back in the context of nuclear shell model, are now well established to be appropriate for understanding statistical properties of many quantum systems [1]. Say m fermions (or bosons) are in N degenerate single particle states and interacting with k-body interactions. Then, with direct product representation of the many-particle states,
the k and m fermion space dimensions are $\binom{N}{k}$ and $\binom{N}{m}$ respectively. Now, with a GUE representation for the
Hamiltonian (H) matrix in the k particle space, the m-particle H matrix will be EGUE(k) - embedded GUE with k-
body interactions. Similarly, we have EGOE(k) and EGSE(k). Note that for k=m we have the classical GOE, GUE
and GSE. Recently, using the formulas for the moments up to order 8, it is established that the one-point
function, ensemble averaged density of eigenvalues, follows the so called q-normal distribution for EGUE(k)
[also for EGOE(k)] with q defined by the fourth moment [2]. The q-normal generates Gaussian density for k <<
m and semi-circle for k=m. Unlike the one-point function, till today there is no success in deriving the two-
point correlation function for EGUE(k) or EGOE(k) even in the limit of k < [1] V.K.B. Kota, Embedded Random Matrix Ensembles in Quantum Physics (Springer, Heidelberg, 2014); V.K.B. Kota and N.D. Chavda, Int. J. Mod. Phys. E 27, 1830001 (2018). [2] Manan Vyas and V.K.B. Kota, J. Stat. Mech.: Theory and Experiment 2019, 103103 (2019). [3] K. K. Mon and J.B. French, Ann. Phys. (N.Y.) 95, 90 (1975); L. Benet, T. Rupp, and H.A. Weidenmüller, Ann. Phys. (N.Y.) 292, 67 (2001); V.K.B. Kota, J. Math. Phys. 46, 033514 (2005); R.A. Small and S. Müller, Ann. Phys. (N.Y.) 356, 269 (2015); V.K.B. Kota, arXiv:2208.11312 (2022). Please contact Mario Kieburg (m.kieburg@unimelb.edu.au) for details regarding access. Within the CRC this talk is associated to the project(s): C6 |