Universality near zero virtuality |
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| Institution: | 1. Department of Mathematics, Faculty of Science, Suez Canal University, Ismailia, Egypt;2. Operations Research and Decision Support Department, Faculty of Computers and Information, Cairo University, Cairo, Egypt;1. Beckman Institute, University of Illinois at Urbana-Champaign, United States;2. Advanced Visualization Laboratory, NCSA, University of Illinois at Urbana-Champaign, United States;3. Department of Molecular Biology and Biotechnology, University of Sheffield, Sheffield, U.K.;4. Department of Chemistry, University of Illinois at Urbana-Champaign, United States;1. Beckman Institute, University of Illinois at Urbana-Champaign, 405 N. Mathews Ave, Urbana, IL 61801, USA;2. Department of Physics, University of Illinois at Urbana-Champaign, 1110 W Green St, Urbana, IL 61801, USA;3. School of Applied and Engineering Physics, Cornell University, 271 Clark Hall, Ithaca, New York 14853, USA;4. Kavli Institute at Cornell for Nanoscale Sciences, 420 Physical Sciences Building, Ithaca, New York 14853, USA;5. Department of Molecular Biology and Biotechnology, University of Sheffield, Western Bank, Sheffield, South Yorkshire S10 2TN, UK |
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| Abstract: | In this paper we study a random matrix model with the chiral and flavor structure of the QCD Dirac operator and a temperature dependence given by the lowest Matsubara frequency. Using the supersymmetric method for random matrix theory, we obtain an exact, analytic expression for the average spectral density. In the large-n limit, the spectral density can be obtained from the solution to a cubic equation. This spectral density is nonzero in the vicinity of eigenvalue zero only for temperatures below the critical temperature of this model. Our main result is the demonstration that the microscopic limit of the spectral density is independent of temperature (apart from a temperature dependent scale factor expressed in terms of the chiral condensate) up to the critical temperature. This is due to a number of remarkable cancellations. This result provides strong support for the conjecture that the microscopic spectral density is universal. In our derivation, we emphasize the symmetries of the partition function and show that this universal behavior is closely related to the existence of an invariant saddle-point manifold. |
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