Fast diagonalization of evolving matrices: application to spin-fermion models

Abstract

Large scale simulation of the colossal magnetoresistance effect in manganitesusing spin-fermion models is often hampered due to the high computational costassociated with computing the eigenvalues of each of the successive Hamiltonianmatrices. Consequently, current spin-fermion model simulations contain no more than63 sites or the equivalent for lower dimensions. This imposes severe limitations on the kinds ofphysical systems that can be studied; for example, the Mn spin concentration in dilutedsemiconductors has to be high enough to be numerically tractable, and the study ofmany band systems becomes computationally difficult. This study presents analgorithm that directly updates the spectrum of a successive Hamiltonian matrix onthe basis of the spectrum of the previous Hamiltonian matrix. This eigenvalueupdating algorithm significantly reduces the computational bottleneck involved inrecomputing the spectrum of the Hamiltonian matrices each time a local configurationalchange is accepted, thereby allowing the simulation of much larger lattice systemsizes. The serial version of the algorithm is an order of magnitude faster thanthe approaches based on direct diagonalization. In addition, this algorithm isamenable to parallel computation and retains excellent accuracy even after manyupdates.