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At nonzero density the eigenvalues of the Dirac operator move into thecomplex plane, while its singular values remain real and nonnegative. InQCD-like theories, the singular-value spectrum carries information on thediquark (or pionic) condensate. We have constructed low-energy effectivetheories in different density regimes and derived a number of exact results forthe Dirac singular values, including Banks-Casher-type relations for thediquark (or pionic) condensate, Smilga-Stern-type relations for the slope ofthe singular-value density, and Leutwyler-Smilga-type sum rules for the inversesingular values. We also present a rigorous index theorem for non-HermitianDirac operators.

Singular values of the Dirac operator at nonzero density