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yr, 2000, arxiv , 61 pages: arXiv:hep-ph/0003017v2 26 Jun 2000

Abstract:

Random matrix theory is developed to calculate universal correlations of eigenvalues in complex systems. It can also be utilized as a schematic model quantum systems with disorder. Related applications include also chiral random matrix theory applied to the calculation of the QCD partition function. The authors prove that constraints imposed by the chiral symmetry and its spontaneous breaking determine the structure of low-energy effective partition functions for the Dirac spectrum. Exact results are derived for the low-lying eigenvalues of the QCD Dirac operator. It is also proposed that the statistical properties of such eigenvalues are universal, and thus can be described by a random matrix theory with the global symmetries of the QCD partition function. The total number of such eigenvalues is shown to increase with the square root of the Euclidean four-volume. The spectral density for larger eigenvalues (but still well below a typical hadronic mass scale) follows also from the same low-energy effective partition function. The validity of the random matrix approach is already confirmed by many lattice QCD simulations for a wide range of parameters. The random matrix model is also extended to nonzero temperature and chemical potential that allow one to obtain qualitative results for the QCD phase diagram and the spectrum of the QCD Dirac operator; the nature of the quenched approximation is discussed, and the quenched Dirac spectra at nonzero baryon density are analyzed in terms of an effective partition function. KEYWORDS: random matrix theory, QCD, chiral symmetry, effective low energy theories, finite volume partition function, lattice QCD A concise quote from the paper regarding symmetry breaking is as follows: "QCD and Chiral Symmetry: We illustrate the concept of spontaneous symmetry breaking using the simpler example of a classical spin system with two rotational degrees of freedom. The Hamiltonian of this system has a certain symmetry: It is invariant under rotations. In mathematical language, the symmetry group is G = O(3). However, at low temperatures, the ground state of the system does not exhibit this symmetry. In a small external magnetic field, which breaks the rotational invariance explicitly, the spins will polarize in the direction of the magnetic field. In the thermodynamic limit, the spins remain polarized even if the magnetic field is switched off completely. This phenomenon is known as spontaneous magnetization. The ground state is no longer invariant under O(3) rotations but only under O(2) rotations in the plane perpendicular to the spontaneous magnetization. In mathematical language, the full symmetry group G = O(3) is spontaneously broken to a smaller symmetry group H = O(2). The spontaneously broken phase is characterized by low-energy excitations in the form of spin waves in the plane perpendicular to the spontaneous magnetization.This is a consequence of a general theorem known as Goldstone's theorem [3], which tells us that spontaneous breaking of a continuous symmetry leads to low-lying excitations, the Goldstone modes, with a mass that vanishes in the absence of a symmetry-breaking field. The Goldstone modes are given by the fluctuations in the plane perpendicular to the direction of the spontaneous magnetization. Thus, the spin system has two Goldstone modes. In general, spontaneous symmetry breaking in a system of spins with n components is associated with n -1 Goldstone modes. This number is also equal to the number of generators of the coset G/H [the number of generators of O(n) is n(n-1)/2]. A spontaneously broken symmetry is characterized by an order parameter, which in this case is the spontaneous magnetization. At nonzero temperature, the alignment of the spins is counteracted by their thermal motion, and above a critical temperature (the Curie temperature) the spontaneous magnetization vanishes." On line PDF and PS download at: http://arxiv.org/PS_cache/hep-ph/pdf/0003/0003017v2.pdf

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http://arxiv.org/PS_cache/hep-ph/pdf/0003/0003017v2.pdf Open access

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