Counting local integrals of motion in disordered spinless-fermion and Hubbard chains

We develop a procedure which systematically generates all conserved operators in the disordered models of interacting fermions. Among these operators, we identify and count the independent and local integrals of motion (LIOM) which represent the hallmark of the many-body localization (MBL). The method is tested first on the prototype disordered chain of interacting spinless fermions. As expected for full MBL, we find for large enough disorder $N_M=2^M-1$ independent and quasi-local LIOM with support on $M$ consecutive sites. On the other hand, the study of the disordered Hubbard chain reveals that $3^M-1< N_M < 4^M/2$ which is less than required for full MBL but much more than in the case of spinless fermions.