Non-adiabatic dynamics of the entanglement entropy in a symmetry-breaking Haldane insulator

We study the non-adiabatic dynamics of a typical symmetry-protected topological phase-the Haldane insulator phase with broken bond-centered inversion. By continuously breaking the middle chain, we find the gap closes at a critical point in the deep Haldane insulator regime with a change of particle number partition of the left or right system. The adiabatic evolution fails at this critical point and we show how to predict the dynamics of the entanglement entropy near this point using a two-level model. These results show that one can find a critical regime where the entanglement measurement is relatively robust against perturbation that breaks the protecting symmetries in the Haldane insulator. This is in contrast to the common belief that the symmetry-protected topological phases are fragile without the protecting symmetries.

PACKING SHELVES WITH ITEMS THAT DIVIDE THE SHELVES' LENGTH: A CASE OF A UNIVERSAL NUMBER PARTITION PROBLEM

This note examines the likelihood of packing two identical one dimensional shelves of integer length L by items whose individual lengths are divisors of L, given that their combined length sums-up to 2L. We compute the number of packing failures and packing successes for integer shelve lengths L, 1 ≤ L ≤ 1000, by implementing a dynamic programming scheme using a problem specific "boundedness property". The computational results indicate that the likelihood of a packing failure is very rare. We observe that the existence of packing failures is tied to the number of divisors of L and prove that the number of divisors has to be at least 8 for a packing failure to exist.