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Superbosonization of invariant random matrix ensembles. (English) Zbl 1156.82008

Summary: ‘Superbosonization’ is a new variant of the method of commuting and anti-commuting variables as used in studying random matrix models of disordered and chaotic quantum systems. We here give a concise mathematical exposition of the key formulas of superbosonization. Conceived by analogy with the bosonization technique for Dirac fermions, the new method differs from the traditional one in that the superbosonization field is dual to the usual Hubbard-Stratonovich field. The present paper addresses invariant random matrix ensembles with symmetry group \(\mathrm{U}_n\), \(\mathrm{O}_n\), or \(\mathrm{USp}_n\), giving precise definitions and conditions of validity in each case. The method is illustrated at the example of Wegner’s \(n\)-orbital model. Superbosonization promises to become a powerful tool for investigating the universality of spectral correlation functions for a broad class of random matrix ensembles of non-Gaussian and/or non-invariant type.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
60B20 Random matrices (probabilistic aspects)
60H25 Random operators and equations (aspects of stochastic analysis)
81Q50 Quantum chaos
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[1] Berezin, F. A., Introduction to Superanalysis (1987), Dordrecht: Reidel Publishing Co., Dordrecht · Zbl 0659.58001
[2] Berruto, F.; Brower, R. C.; Svetitsky, B., Effective Lagrangian for strongly coupled domain wall fermions, Phys. Rev. D, 64, 114504 (2001) · doi:10.1103/PhysRevD.64.114504
[3] Bunder, J. E.; Efetov, K. B.; Kravtsov, V. E.; Yevtushenko, O. M.; Zirnbauer, M. R., Superbosonization formula and its application to random matrix theory, J. Stat. Phys., 129, 809-832 (2007) · Zbl 1136.82005 · doi:10.1007/s10955-007-9405-y
[4] Disertori, M.; Pinson, H.; Spencer, T., Density of states of random band matrices, Commun. Math. Phys., 232, 83-124 (2002) · Zbl 1019.15014 · doi:10.1007/s00220-002-0733-0
[5] Dyson, F. J., The threefold way: algebraic structure of symmetry groups and ensembles in quantum mechanics, J. Math. Phys., 3, 1199-1215 (1962) · Zbl 0134.45703 · doi:10.1063/1.1703863
[6] Efetov, K. B., Supersymmetry in disorder and chaos (1999), Cambridge: Cambridge University Press, Cambridge
[7] Efetov, K. B.; Schwiete, G.; Takahashi, K., Bosonization for disordered and chaotic systems, Phys. Rev. Lett., 92, 026807 (2004) · doi:10.1103/PhysRevLett.92.026807
[8] Fyodorov, Y. V., Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation, Nucl. Phys. B, 621, 643-674 (2002) · Zbl 1024.82013 · doi:10.1016/S0550-3213(01)00508-9
[9] Guhr, T., Arbitrary unitarily invariant random matrix ensembles and supersymmetry, J. Phys. A, 39, 13191-13223 (2006) · Zbl 1206.82051 · doi:10.1088/0305-4470/39/42/002
[10] Hackenbroich, G.; Weidenmüller, H. A., Universality of Random-Matrix Results for Non-Gaussian Ensembles, Phys. Rev. Lett., 74, 4118-4121 (1995) · doi:10.1103/PhysRevLett.74.4118
[11] Howe, R., Remarks on classical invariant theory, TAMS, 313, 539-570 (1989) · Zbl 0674.15021 · doi:10.2307/2001418
[12] Howe, R.: Perspectives on invariant theory: Schur duality, multiplicity free actions and beyond. In the Schur Lectures, Providence, RI: Amer. Math. Soc., 1995 · Zbl 0844.20027
[13] Kawamoto, N.; Smit, J., Effective Lagrangian and dynamical symmetry breaking in strongly coupled lattice QCD, Nucl. Phys. B, 192, 100-124 (1981) · doi:10.1016/0550-3213(81)90196-6
[14] Kostant, B., Graded Lie theory and prequantization, Lecture Notes in Math., 570, 177-306 (1977) · Zbl 0358.53024 · doi:10.1007/BFb0087788
[15] Lehmann, N.; Saher, D.; Sokolov, V. V.; Sommers, H.-J., Chaotic scattering – the supersymmetry method for large number of channels, Nucl. Phys. A, 582, 223-256 (1995) · doi:10.1016/0375-9474(94)00460-5
[16] Luna, D., Fonctions differentiables invariantes sous l’operation d’un groupe reductif, Ann. Inst. Fourier, 26, 33-49 (1976) · Zbl 0315.20039
[17] Schäfer, L.; Wegner, F., Disordered system with n orbitals per site: Lagrange formulation, hyperbolic symmetry, and Goldstone modes, Z. Phys. B, 38, 113-126 (1980) · doi:10.1007/BF01598751
[18] Schwarz, G., Lifting smooth homotopy of orbit spaces, Pub. Math. IHES, 51, 37 (1980) · Zbl 0449.57009
[19] Schwarz, A.; Zaboronsky, O., Supersymmetry and localization, Commun. Math. Phys., 183, 463-476 (1997) · Zbl 0873.58003 · doi:10.1007/BF02506415
[20] Wegner, F. J., Disordered system with N orbitals per site: N = ∞ limit, Phys. Rev. B, 19, 783-792 (1979) · doi:10.1103/PhysRevB.19.783
[21] Zirnbauer, M.R.: Supersymmetry methods in random matrix theory. In: Encyclopedia of Mathematical Physics, Vol. 5, Amsterdam: Elsevier, 2006 pp. 151-160
[22] Zirnbauer, M. R., Riemannian symmetric superspaces and their origin in random matrix theory, J. Math. Phys., 37, 4986-5018 (1996) · Zbl 0871.58005 · doi:10.1063/1.531675
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