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By employing QCD inequalities, we discuss appearance of the pion condensatefor both real and imaginary isospin chemical potentials, taking also intoaccount imaginary quark chemical potential. We show that the charged pion cancondense for real isospin chemical potential, but not for imaginary one.Furthermore, we evaluate the expectation value of the neutral-pion field forimaginary isospin chemical potential by using framework of the twisted mass. Asa result, it is found that the expectation value becomes zero for the finitecurrent-quark mass, whereas the Banks-Casher relation is obtained in themassless limit.

QCD-inequality analyses on pion condensate at real and imaginary isospin chemical potentials under finite imaginary quark chemical potential