  We show that gas disks around the components of an orbiting binary system
(so-called minidisks) may be susceptible to a resonant instability which causes
the minidisks to become significantly eccentric. Eccentricity is injected by,
and also induces, regular impacts between the minidisks at roughly the orbital
period of the binary. Such eccentric minidisks are seen in vertically
integrated, two-dimensional simulations of a circular, equal-mass binary
accreting from a circumbinary gas disk with a $\Gamma$-law equation of state.
Minidisk eccentricity is suppressed by the use of an isothermal equation of
state. However, the instability still operates, and can be revealed in a
minimal disk-binary simulation by removing the circumbinary disk, and feeding
the minidisks from the component positions. Minidisk eccentricity is also
suppressed when the gravitational softening length is large ($\gtrsim 4\%$ of
the binary semi-major axis), suggesting that its absence could be an artifact
of widely adopted numerical approximations; a follow-up study in three
dimensions with well-resolved, geometrically thin minidisks (aspect ratios
$\lesssim 0.02$) may be needed to assess whether eccentric minidisks can occur
in real astrophysical environments. If they can, the electromagnetic signature
may be important for discriminating between binary and single black hole
scenarios for quasi-periodic oscillations in active galactic nuclei; in turn,
this might aid in targeted searches with pulsar timing arrays for individual
supermassive black hole binary sources of low-frequency gravitational waves.
