  Interacting one-dimensional electron systems are generally referred to as
"Luttinger liquids", after the effective low-energy theory in which spin and
charge behave as separate degrees of freedom with independent energy scales.
The "spin-incoherent Luttinger liquid" describes a finite-temperature regime
that is realized when the temperature is very small relative to the Fermi
energy, but larger than the characteristic spin energy scale. Similar physics
can take place in the ground-state, when a Luttinger Liquid is coupled to a
spin bath, which effectively introduces a "spin temperature" through its
entanglement with the spin degree of freedom. We show that the spin-incoherent
state can be written as a factorized wave-function, with a spin wave-function
that can be described within a valence bond formalism. This enables us to
calculate exact expressions for the momentum distribution function and the
entanglement entropy. This picture holds not only for two antiferromagnetically
coupled t-J chains, but also for the t-J-Kondo chain with strongly interacting
conduction electrons. We argue that this theory is quite universal and may
describe a family of problems that could be dubbed "spin-incoherent".
