  We analyze a two-body nonhermitian two-site Sachdev-Ye-Kitaev model with the
couplings of one site complex conjugated to the other site. This model, with no
explicit coupling between the sites, shows an infinite number of phase
transitions which is a consequence of the partition function factorizing into a
product over Matsubara frequencies. We calculate the quenched free energy in
two different ways, first in terms of the single-particle energies, and second
by solving the Schwinger-Dyson equations. The first calculation can be done
entirely in terms of a one-site model. The conjugate replica enters due to
non-analyticities when Matsubara frequencies enter the spectral support of the
coupling matrix. The second calculation is based on the replica trick of the
two-site partition function. Both methods give the same result. The
free-fermion partition function can be rephrased as a matrix model for the
coupling matrix. Up to minor details, this model is the random matrix model
that describes the chiral phase transition of QCD, and the order parameter of
the two-body model corresponds to the chiral condensate of QCD. Comparing to
the corresponding four-body model, we are able to determine which features of
the free energy are due to chaotic nature of the four-body model. The
high-temperature phase of both models is entropy dominated, and in both cases
is determined by the spectral density. The four-body SYK model has a
low-temperature phase whose free energy is almost temperature-independent,
signaling an effective gap of the theory even though the actual spectrum does
not exhibit a gap. However the low-temperature free energy of the two-body SYK
model is not flat, in fact it oscillates to arbitrarily low temperature. This
indicates a less desirable feature that the entropy of the two-body model is
not always positive, which most likely is a consequence of the nonhermiticity.
