  The higher fusion level logarithmic minimal models LM(P,P';n) have recently
been constructed as the diagonal GKO cosets (A_1^{(1)})_k oplus (A_1^{(1)})_n /
(A_1^{(1)})_{k+n} where n>0 is an integer fusion level and k=nP/(P'-P)-2 is a
fractional level. For n=1, these are the logarithmic minimal models LM(P,P').
For n>1, we argue that these critical theories are realized on the lattice by n
x n fusion of the n=1 models. For n=2, we call them logarithmic superconformal
minimal models LSM(p,p') where P=|2p-p'|, P'=p' and p,p' are coprime, and they
share the central charges of the rational superconformal minimal models
SM(P,P'). Their mathematical description entails the fused planar
Temperley-Lieb algebra which is a spin-1 BMW tangle algebra with loop fugacity
beta_2=x^2+1+x^{-2} and twist omega=x^4 where x=e^{i(p'-p)pi/p'}. Examples are
superconformal dense polymers LSM(2,3) with c=-5/2, beta_2=0 and superconformal
percolation LSM(3,4) with c=0, beta_2=1. We calculate the free energies
analytically. By numerically studying finite-size spectra on the strip with
appropriate boundary conditions in Neveu-Schwarz and Ramond sectors, we argue
that, in the continuum scaling limit, these lattice models are associated with
the logarithmic superconformal models LM(P,P';2). For system size N, we propose
finitized Kac character formulas whose P,P' dependence only enters in the
fractional power of q in a prefactor. These characters involve Motzkin and
Riordan polynomials defined in terms of q-trinomial coefficients. Using the
Hamiltonian limit, we argue that there exist reducible yet indecomposable
representations for which the Virasoro dilatation operator L_0 exhibits rank-2
Jordan blocks confirming that these theories are indeed logarithmic. We relate
these results to the N=1 superconformal representation theory.
