Abstract
Embedded random matrix ensembles, introduced in 1970 and being explored in considerable detail since 1994, have started occupying an important position in quantum physics in the study of isolated finite quantum many-particle systems. As aptly stated by Weidenmüller : “Although used with increasing frequency in many branches of Physics, random matrix ensembles sometimes are too unspecific to account for important features of the physical system at hand. One refinement which retains the basic stochastic approach but allows for such features consists in the use of embedded ensembles”. In this chapter, given is a general introduction emphasizing the importance of random matrix theory in quantum physics in general and embedded random matrix ensembles in particular. Also given is a short preview of the contents of the next fifteen chapters of the book.
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References
J. Wishart, The generalized product moment distribution in samples from a normal multivariate population. Biometrika 20A, 32–52 (1928)
E.P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548–564 (1955)
E.P. Wigner, Characteristic vectors of bordered matrices with infinite dimensions II. Ann. Math. 65, 203–207 (1957)
E.P. Wigner, Statistical properties of real symmetric matrices with many dimensions, in Can. Math. Congr. Proc. (University of Toronto Press, Toronto, 1957), pp. 174–184
E.P. Wigner, The probability of the existence of a self reproducing unit, reproduced from the logic of personal knowledge: essays in honor of Michael Polanyi, Chap. 19 (Routledge and Kegan Paul, London, 1961), in Symmetries and Reflections: Scientific Essays of E.P. Wigner, ed. by W.J. Moore, M. Scriven (Indiana University Press, Bloomington, 1967), Reprinted by (Ox Bow Press, Woodbridge, 1979), pp. 200–208
J.B. French, Elementary aspects of nuclear quantum chaos, in A Gift of Prophecy—Essays in the Celebration of the Life of R.E. Marshak, ed. by E.C.G. Sudarshan (World Scientific, Singapore, 1994), pp. 156–167
F.J. Dyson, Statistical theory of energy levels of complex systems I. J. Math. Phys. 3, 140–156 (1962)
F.J. Dyson, Statistical theory of energy levels of complex systems II. J. Math. Phys. 3, 157–165 (1962)
F.J. Dyson, Statistical theory of energy levels of complex systems III. J. Math. Phys. 3, 166–175 (1962)
F.J. Dyson, The threefold way: algebraic structures of symmetry groups and ensembles in quantum mechanics. J. Math. Phys. 3, 1199–1215 (1963)
C.E. Porter, Statistical Theories of Spectra: Fluctuations (Academic Press, New York, 1965)
M.L. Mehta, Random Matrices, 3rd edn. (Elsevier, Amsterdam, 2004)
J.B. Garg (ed.), Statistical Properties of Nuclei, Proceedings of the International Conference on Statistical Properties of Nuclei, Held at Albany (N.Y.), August 23–27, 1971 (Plenum, New York, 1972)
T.A. Brody, J. Flores, J.B. French, P.A. Mello, A. Pandey, S.S.M. Wong, Random matrix physics: spectrum and strength fluctuations. Rev. Mod. Phys. 53, 385–479 (1981)
O. Bohigas, M.J. Giannoni, C. Schmit, Characterization of chaotic quantum spectra and universality of level fluctuation laws. Phys. Rev. Lett. 52, 1–4 (1984)
O. Bohigas, Random matrix theories and chaotic dynamics, in Chaos and Quantum Physics, ed. by M.J. Giannoni, A. Voros, J. Zinn-Justin (Elsevier, Amsterdam, 1991), pp. 87–199
S. Heusler, S. Müller, A. Altland, P. Braun, F. Haake, Periodic-orbit theory of level correlations. Phys. Rev. Lett. 98, 044103 (2007)
S. Müller, S. Heusler, A. Altland, P. Braun, F. Haake, Periodic-orbit theory of universal level correlations in quantum chaos. New J. Phys. 11, 103025 (2009)
P. Braun, Beyond the Heisenberg time: semiclassical treatment of spectral correlations in chaotic systems with spin 1/2. J. Phys. A 45, 045102 (2012)
M.V. Berry, M. Tabor, Level clustering in the regular spectrum. Proc. R. Soc. Lond. A 356, 375–394 (1977)
M.V. Berry, Semiclassical theory of spectral rigidity. Proc. R. Soc. Lond. A 400, 229–251 (1985)
F. Haake, Quantum Signatures of Chaos, 3rd edn. (Springer, Heidelberg, 2010)
H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, New York, 1999)
K.B. Efetov, Supersymmetry in Disorder and Chaos (Cambridge University Press, New York, 1997)
T. Guhr, A. Müller-Groeling, H.A. Weidenmüller, Random-matrix theories in quantum physics: common concepts. Phys. Rep. 299, 189–425 (1998)
A.D. Mirlin, Statistics of energy levels and eigenfunctions in disordered systems. Phys. Rep. 326, 259–382 (2000)
H.A. Weidenmüller, G.E. Mitchell, Random matrices and chaos in nuclear physics: nuclear structure. Rev. Mod. Phys. 81, 539–589 (2009)
G.E. Mitchell, A. Richter, H.A. Weidenmüller, Random matrices and chaos in nuclear physics: nuclear reactions. Rev. Mod. Phys. 82, 2845–2901 (2010)
P.J. Forrester, N.C. Snaith, J.J.M. Verbaarschot (eds.), Special issue: random matrix theory. J. Phys. A 36, R1–R10 and 2859–3646 (2003)
V.A. Marc̆enko, L.A. Pastur, Distribution of eigenvalues for some sets of random matrices. Math. USSR Sb. 1, 457–483 (1967)
L.A. Pastur, On the spectrum of random matrices. Theor. Math. Phys. 10, 67–74 (1972)
N.M. Katz, P. Sarnak, Random Matrices, Frobenius Eigenvalues and Monodromy, Colloquium Publication, vol. 45 (Am. Math. Soc., Providence, 1999)
P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, Courant Lecture Notes, vol. 3 (Am. Math. Soc. and Courant Institute of Mathematical Sciences at New York, Providence, 2000)
A.M. Tulino, S. Verdú, Random Matrix Theory and Wireless Communication (Now Publishers, Hanover, 2004)
J. Baik, T. Kriecherbauer, L. Li, K.D.T.-R. McLaughlin, C. Tomei (eds.), Integrable Systems and Random Matrices, Contemporary Mathematics, vol. 458 (Am. Math. Soc., Providence, 2008)
G. Blower (ed.), Random Matrices: High Dimensional Phenomena, London Mathematical Society Lecture Series, vol. 367 (Cambridge University Press, Cambridge, 2009)
P. Deift, D. Gioev, Random Matrix Theory: Invariant Ensembles and Universality (Am. Math. Soc. and Courant Institute of Mathematical Sciences at New York, Providence, 2009)
G.W. Anderson, A. Guionnet, O. Zeitouni, An Introduction to Random Matrices (Cambridge University Press, New York, 2010)
Z. Bai, J.W. Silverstein, Spectral Analysis of Large Dimensional Random Matrices, 2nd edn. (Springer, New York, 2010)
L. Pastur, M. Shcherbina, Eigenvalue Distribution of Large Random Matrices, Mathematical Surveys and Monographs, vol. 171 (Am. Math. Soc., Providence, 2011)
R. Couillet, M. Debbah, Random Matrix Methods for Wireless Communications (Cambridge University Press, New York, 2012)
M. Wright, R. Weiver, New Directions in Linear Acoustics and Vibrations: Quantum Chaos, Random Matrix Theory and Complexity (Cambridge University Press, New York, 2011)
P.J. Forrester, Log-Gases and Random Matrices (Princeton University Press, Princeton, 2010)
G. Akemann, J. Baik, P. Di Francesco (eds.), The Oxford Handbook of Random Matrix Theory (Oxford University Press, Oxford, 2011)
V.K.B. Kota, Embedded random matrix ensembles for complexity and chaos in finite interacting particle systems. Phys. Rep. 347, 223–288 (2001)
L. Benet, H.A. Weidenmüller, Review of the k-body embedded ensembles of Gaussian random matrices. J. Phys. A 36, 3569–3594 (2003)
T. Papenbrock, H.A. Weidenmüller, Random matrices and chaos in nuclear spectra. Rev. Mod. Phys. 79, 997–1013 (2007)
J.M.G. Gómez, K. Kar, V.K.B. Kota, R.A. Molina, A. Relaño, J. Retamosa, Many-body quantum chaos: recent developments and applications to nuclei. Phys. Rep. 499, 103–226 (2011)
M. Vyas, V.K.B. Kota, Random matrix structure of nuclear shell model Hamiltonian matrices and comparison with an atomic example. Eur. Phys. J. A 45, 111–120 (2010)
J.B. French, S.S.M. Wong, Validity of random matrix theories for many-particle systems. Phys. Lett. B 33, 449–452 (1970)
J.B. French, S.S.M. Wong, Some random-matrix level and spacing distributions for fixed-particle-rank interactions. Phys. Lett. B 35, 5–7 (1971)
J.B. French, Analysis of distant-neighbour spacing distributions for k-body interaction ensembles. Rev. Mex. Fis. 22, 221–229 (1973)
O. Bohigas, J. Flores, Two-body random Hamiltonian and level density. Phys. Lett. B 34, 261–263 (1971)
O. Bohigas, J. Flores, Spacing and individual eigenvalue distributions of two-body random Hamiltonians. Phys. Lett. B 35, 383–387 (1971)
J.B. French, Special topics in spectral distributions, in Moment Methods in Many Fermion Systems, ed. by B.J. Dalton, S.M. Grimes, J.P. Vary, S.A. Williams (Plenum, New York, 1980), pp. 91–108
K.K. Mon, J.B. French, Statistical properties of many-particle spectra. Ann. Phys. (N.Y.) 95, 90–111 (1975)
J.B. French, V.K.B. Kota, A. Pandey, S. Tomsovic, Statistical properties of many-particle spectra VI. Fluctuation bounds on N-N T-noninvariance. Ann. Phys. (N.Y.) 181, 235–260 (1988)
V.V. Flambaum, A.A. Gribakina, G.F. Gribakin, M.G. Kozlov, Structure of compound states in the chaotic spectrum of the Ce atom: localization properties, matrix elements, and enhancement of weak perturbations. Phys. Rev. A 50, 267–296 (1994)
V.V. Flambaum, G.F. Gribakin, F.M. Izrailev, Correlations within eigenvectors and transition amplitudes in the two-body random interaction model. Phys. Rev. E 53, 5729–5741 (1996)
V.V. Flambaum, F.M. Izrailev, Statistical theory of finite Fermi systems based on the structure of chaotic eigenstates. Phys. Rev. E 56, 5144–5159 (1997)
M. Horoi, V. Zelevinsky, B.A. Brown, Chaos vs thermalization in the nuclear shell model. Phys. Rev. Lett. 74, 5194–5197 (1995)
V. Zelevinsky, B.A. Brown, N. Frazier, M. Horoi, The nuclear shell model as a testing ground for many-body quantum chaos. Phys. Rep. 276, 85–176 (1996)
Ph. Jacquod, D.L. Shepelyansky, Emergence of quantum chaos in finite interacting Fermi systems. Phys. Rev. Lett. 79, 1837–1840 (1997)
B. Georgeot, D.L. Shepelyansky, Breit-Wigner width and inverse participation ratio in finite interacting Fermi systems. Phys. Rev. Lett. 79, 4365–4368 (1997)
V.K.B. Kota, R. Sahu, Information entropy and number of principal components in shell model transition strength distributions. Phys. Lett. B 429, 1–6 (1998)
Y. Alhassid, Statistical theory of quantum dots. Rev. Mod. Phys. 72, 895–968 (2000)
V.K.B. Kota, R. Sahu, Structure of wavefunctions in (1+2)-body random matrix ensembles. Phys. Rev. E 64, 016219 (2001)
W.G. Brown, L.F. Santos, D.J. Starling, L. Viola, Quantum chaos, delocalization and entanglement in disordered Heisenberg models. Phys. Rev. E 77, 021106 (2008)
M. Rigol, V. Dunjko, M. Olshanii, Thermalization and its mechanism for generic isolated quantum systems. Nature (London) 452, 854–858 (2008)
L.F. Santos, M. Rigol, Onset of quantum chaos in one dimensional bosonic and fermionic systems and its relation to thermalization. Phys. Rev. E 81, 036206 (2010)
L.F. Santos, M. Rigol, Localization and the effects of symmetries in the thermalization properties of one-dimensional quantum systems. Phys. Rev. E 82, 031130 (2010)
J.B. French, Elementary principles of spectral distributions, in Moment Methods in Many Fermion Systems, ed. by B.J. Dalton, S.M. Grimes, J.P. Vary, S.A. Williams (Plenum, New York, 1980), pp. 1–16
V.K.B. Kota, R.U. Haq, Spectral Distributions in Nuclei and Statistical Spectroscopy (World Scientific, Singapore, 2010)
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Kota, V.K.B. (2014). Introduction. In: Embedded Random Matrix Ensembles in Quantum Physics. Lecture Notes in Physics, vol 884. Springer, Cham. https://doi.org/10.1007/978-3-319-04567-2_1
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DOI: https://doi.org/10.1007/978-3-319-04567-2_1
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