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

Abstract

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.

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

References in corpus (4)

• Virial expansion for almost diagonal random matrices
• Density of states for almost diagonal random matrices
• T-duality in supersymmetric theory of disordered quantum systems
• Energy level statistics of a critical random matrix ensemble

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