  We introduce a novel family of translationally-invariant su$(m|n)$
supersymmetric spin chains with long-range interaction not directly associated
to a root system. We study the symmetries of these models, establishing in
particular the existence of a boson-fermion duality characteristic of this type
of systems. Taking advantage of the relation of the new chains with an
associated many-body supersymmetric spin dynamical model, we are able to
compute their partition function in closed form for all values of $m$ and $n$
and for an arbitrary number of spins. When both $m$ and $n$ are even, we show
that the partition function factorizes as the product of the partition
functions of two supersymmetric Haldane-Shastry spin chains, which in turn
leads to a simple expression for the thermodynamic free energy per spin in
terms of the Perron eigenvalue of a suitable transfer matrix. We use this
expression to study the thermodynamics of a large class of these chains,
showing in particular that the specific heat presents a single Schottky peak at
approximately the same temperature as a suitable $k$-level model. We also
analyze the critical behavior of the new chains, and in particular the ground
state degeneracy and the existence of low energy excitations with a linear
energy-momentum dispersion relation. In this way we are able to show that the
only possible critical chains are the ones with $m=0,1,2$. In addition, using
the explicit formula for the partition function we are able to establish the
criticality of the su$(0|n)$ and su$(2|n)$ chains with even $n$, and to
evaluate the central charge of their associated conformal field theory.
