Approximating observables on eigenstates of large many-body localized systems

Eigenstates of fully many-body localized (FMBL) systems can be organized into spin algebras based on quasilocal operators called l bits. These spin algebras define quasilocal l-bit measurement (tau(z)(i)) and l-bit flip (tau(x)(i)) operators. For a disordered Heisenberg spin chain in the MBL regime we approximate l-bit flip operators by first calculating them exactly on small windows of systems using an algorithm called operator localization optimization. We then extend the l-bit operators onto the whole system by exploiting their quasilocal nature. We subsequently use these operators to represent approximate eigenstates of the Hamiltonian. Finally, we describe a method to calculate products of local observables on these eigenstates for systems of size L in O(L-2) time. This method is used to calculate the variance of the energy of the approximate eigenstates, yielding an estimate of the error of the approximation.