Chasing infinity with matrix product states by embracing divergences

Abstract: In this paper, we present a formalism for representing infinite systems in quantum mechanics by employing a strategy that embraces divergences rather than avoiding them. We do this by representing physical quantities such as inner products, expectations, etc., as maps from natural numbers to complex numbers which contain information about how these quantities diverge, and in particular whether they scale linearly, quadratically, exponentially, etc. with the size of the system. We build our formalism on a variant of matrix product states, as this class of states has a structure that naturally provides a way to obtain the scaling function. We show that the states in our formalism form a module over the ring of functions that are made up of sums of exponentials times polynomials and delta functions. We analyze properties of this formalism and show how it works for selected systems. Finally, we discuss how our formalism relates to other work.