Real-space quantum renormalization groups

Finite automata for caching in matrix product algorithms

“…Thus, if we start by minimizing the energy with respect to site 1, and then sweep to the right (i.e., site 2, site 3, up to site N), then although it took us O(N ) time to compute R 1 , we get the R i matrices for all of the rest of the sites up through N for free. Once we hit site N, we start moving left back through N − 1, N − 2, etc., and at each step it only takes us one additional matrix 7 By "amortized" here we mean that although it takes O(N ) time to initialize L 1 and R 1 at the first step, it takes O(1) for all remaining steps, and there are typically at least N steps, so on average the operation takes O(1) time per step.…”

Renormalization group and quantum information

“…White and Noack [7] realized that the problem lies in the choice of block eigenstates as the states to be kept: these states belong to a small system, namely, the block, in which the boundary conditions are very important. In other words, isolating a block from its neighbours destroys the quantum correlations between them, which are very important for the low-energy spectrum of the total system (with N sites).…”

“…These correlations are somehow recovered by the renormalization of the couplings, but in a small amount. A partial remedy is the "combination of boundary conditions" approach [7], namely, to consider block states corresponding to various boundary conditions. This approach is effective in some cases only.…”

“…So White and Noack [7] proposed to diagonalize a larger block, the "superblock", which includes the basic block. Then the problem is to project the "superblock" state (or states) onto block states: we need a criterium to select which ones to keep.…”

Study of the spectral properties of spin ladders in different representations via a renormalization procedure

“…During the last decades a considerable amount of procedures relying on the renormalization group concept introduced by Wilson [4] have been proposed and tested. Some of them are specifically devised for quantum spin systems, like the Real Space Renormalization Group (RSRG) [5,6] and the Density Matrix Renormalization Group (DMRG) [7,8].…”