Level compressibility in a critical random matrix ensemble: The second virial coefficient

Details Level compressibility in a critical random matrix ensemble: The second virial coefficient Level compressibility in a critical random matrix ensemble: The second virial coefficient

2005-10-14

arxiv.org/abs/cond-mat/0510378v3

J. Phys. A: Math. Gen. 39 (2006) 2021-2034

Physics

Condensed Matter

Disordered Systems and Neural Networks

several typos and the unfolding factor are corrected, Erratum has been added

Scientific paper

10.1088/0305-4470/39/9/003

We study spectral statistics of a Gaussian unitary critical ensemble of almost diagonal Hermitian random matrices with off-diagonal entries $<|H_{ij}|^{2} > \sim b^{2} |i-j|^{-2}$ small compared to diagonal ones $<|H_{ii}|^{2} > \sim 1$. Using the recently suggested method of {\it virial expansion} in the number of interacting energy levels (J.Phys.A {\bf 36},8265 (2003)), we calculate a coefficient $\propto b^{2}\ll 1$ in the level compressibility $\chi(b)$. We demonstrate that only the leading terms in $\chi(b)$ coincide for this model and for an exactly solvable model suggested by Moshe, Neuberger and Shapiro (Phys.Rev.Lett. {\bf 73}, 1497 (1994)), the sub-leading terms $\sim b^{2}$ being different. Numerical data confirms our analytical calculation.

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