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Singular values of the Dirac operator at nonzero density
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At nonzero density the eigenvalues of the Dirac operator move into the complex plane, while its singular values remain real and nonnegative. In QCD-like theories, the singular-value spectrum carries information on the diquark (or pionic) condensate. We have constructed low-energy effective theories in different density regimes and derived a number of exact results for the Dirac singular values, including Banks-Casher-type relations for the diquark (or pionic) condensate, Smilga-Stern-type relations for the slope of the singular-value density, and Leutwyler-Smilga-type sum rules for the inverse singular values. We also present a rigorous index theorem for non-Hermitian Dirac operators.
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