Density of states for almost-diagonal random matrices

Oleg Yevtushenko; Vladimir E. Kravtsov

doi:10.1103/PhysRevE.69.026104

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Density of states for almost-diagonal random matrices

Oleg Yevtushenko^*

Vladimir E. Kravtsov^†

The Abdus Salam ICTP, Strada Costiera 11, 34100, Trieste, Italy

The Abdus Salam ICTP, Strada Costiera 11, 34100, Trieste, Italy and
Landau Institute for Theoretical Physics, 2 Kosygina Street, 117940 Moscow, Russia

^*Electronic address: bom@ictp.trieste.it
^†Electronic address: kravtsov@ictp.trieste.it

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Phys. Rev. E 69, 026104 – Published 17 February, 2004

DOI: https://doi.org/10.1103/PhysRevE.69.026104

Abstract

We study the density of states (DOS) for disordered systems whose spectral statistics can be described by a Gaussian ensemble of almost-diagonal Hermitian random matrices. The matrices have independent random entries $H_{i >~ j}$ with small off-diagonal elements: $〈 | H_{i \neq j} |^{2} 〉 ≪ 〈 | H_{ii} |^{2} 〉 \sim 1.$ Using the recently suggested method of a virial expansion in the number of interacting energy levels [J. Phys. A 36, 8265 (2003)], we calculate the leading correction to the Poissonian DOS in the cases of the Gaussian orthogonal and unitary ensembles. We apply the general formula to the critical power-law banded random matrices and the unitary Moshe-Neuberger-Shapiro model and compare the DOS’s of these models.

References (20)

A.D. Mirlin, Y.V. Fyodorov, F.M. Dittes, J. Quezada, and T.H. Seligman, Phys. Rev. E 54, 3221 (1996).
V.E. Kravtsov and K.A. Muttalib, Phys. Rev. Lett. 79, 1913 (1997).
F. Evers and A.D. Mirlin, Phys. Rev. Lett. 84, 3690 (2000); Phys. Rev. B 62, 7920 (2000).
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L.S. Levitov, Phys. Rev. Lett. 64, 547 (1990); Ann. Phys. (Leipzig) 8, 697 (1999).
V.E. Kravtsov and A.M. Tsvelik, Phys. Rev. B 62, 9888 (2000).
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B.B. Hu, B.W. Li, J. Liu, and Y. Gu, Phys. Rev. Lett. 82, 4224 (1999).
M. Moshe, H. Neuberger, and B. Shapiro, Phys. Rev. Lett. 73, 1497 (1994).
A.M. García-García and J.J.M. Verbaarschot, Phys. Rev. E 67, 046104 (2003).
M. Gaudin, Nucl. Phys. 85, 545 (1966).
M.L. Ndwana and V.E. Kravtsov, J. Phys. A 36, 3639 (2003).
Y.V. Fyodorov and A.D. Mirlin, Phys. Rev. Lett. 67, 2405 (1991).
K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997).
O. Yevtushenko and V.E. Kravtsov, J. Phys. A 36, 8265 (2003).
N. Rosenzweig and C.E. Porter, Phys. Rev. 120, 1698 (1960).
M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic Press, New York, 1980).
The condition α=1 is not necessary for the application of the virial expansion to DOS and it will be removed in the forthcoming paper.
If the sum in the real space diverges the virial expansion fails and we cannot consider a finite number of interacting energy levels to derive DOS.
This is similar to the first correction to the Poissonian level compressibility calculated in Ref. [15]. This correction is governed by behavior of the correlation function F at the large distances and, therefore, is the same in PLRBM and MNS.

[c1] A.D. Mirlin, Y.V. Fyodorov, F.M. Dittes, J. Quezada, and T.H. Seligman, Phys. Rev. E 54, 3221 (1996).

[c2] V.E. Kravtsov and K.A. Muttalib, Phys. Rev. Lett. 79, 1913 (1997).

[c3] F. Evers and A.D. Mirlin, Phys. Rev. Lett. 84, 3690 (2000); Phys. Rev. B 62, 7920 (2000).

[c4] B.L. Altshuler and L.S. Levitov, Phys. Rep. 288, 487 (1997).

[c5] L.S. Levitov, Phys. Rev. Lett. 64, 547 (1990); Ann. Phys. (Leipzig) 8, 697 (1999).

[c6] V.E. Kravtsov and A.M. Tsvelik, Phys. Rev. B 62, 9888 (2000).

[c7] F. Calogero, J. Math. Phys. 10, 2191 (1969); ibid.10, 2197 (1969); ibid.12, 419 (1971); ibid.B. Sutherland, 12, 246 (1971); ibid.12, 251 (1971).

[c8] B.B. Hu, B.W. Li, J. Liu, and Y. Gu, Phys. Rev. Lett. 82, 4224 (1999).

[c9] M. Moshe, H. Neuberger, and B. Shapiro, Phys. Rev. Lett. 73, 1497 (1994).

[c10] A.M. García-García and J.J.M. Verbaarschot, Phys. Rev. E 67, 046104 (2003).

[c11] M. Gaudin, Nucl. Phys. 85, 545 (1966).

[c12] M.L. Ndwana and V.E. Kravtsov, J. Phys. A 36, 3639 (2003).

[c13] Y.V. Fyodorov and A.D. Mirlin, Phys. Rev. Lett. 67, 2405 (1991).

[c14] K. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, Cambridge, 1997).

[c15] O. Yevtushenko and V.E. Kravtsov, J. Phys. A 36, 8265 (2003).

[c16] N. Rosenzweig and C.E. Porter, Phys. Rev. 120, 1698 (1960).

[c17] M. Reed and B. Simon, Methods of Modern Mathematical Physics (Academic Press, New York, 1980).

[c18] The condition α=1 is not necessary for the application of the virial expansion to DOS and it will be removed in the forthcoming paper.

[c19] If the sum in the real space diverges the virial expansion fails and we cannot consider a finite number of interacting energy levels to derive DOS.

[c20] This is similar to the first correction to the Poissonian level compressibility calculated in Ref. [15]. This correction is governed by behavior of the correlation function F at the large distances and, therefore, is the same in PLRBM and MNS.

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Density of states for almost-diagonal random matrices

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