Theory of finite-entanglement scaling at one-dimensional quantum critical points

Studies of entanglement in many-particle systems suggest that most quantum critical ground states have infinitely more entanglement than non-critical states. Standard algorithms for one-dimensional many-particle systems construct model states with limited entanglement, which are a worse approximation to quantum critical states than to others. We give a quantitative theory of previously observed scaling behavior resulting from finite entanglement at quantum criticality: the scaling theory of finite entanglement is only superficially similar to finite-size scaling, and has a different physical origin. We find that finite-entanglement scaling is governed not by the scaling dimension of an operator but by the "central charge" of the critical point, which counts its universal degrees of freedom. An important ingredient is the recently obtained universal distribution of density-matrix eigenvalues at a critical point\cite{calabrese1}. The parameter-free theory is checked against numerical scaling at several quantum critical points.