Exact excited states of nonintegrable models

Abstract

We discuss a method of numerically identifying exact energy eigenstates for a finite system, whose form can then be obtained analytically. We demonstrate our method by identifying and deriving exact analytic expressions for several excited states, including an infinite tower, of the one dimensional spin-1 AKLT model, a celebrated non-integrable model. The states thus obtained for the AKLT model can be interpreted as one-to-an extensive number of quasiparticles on the ground state or on the highest excited state when written in terms of dimers. Included in these exact states is a tower of states spanning energies from the ground state to the highest excited state. To our knowledge, this is the first time that exact analytic expressions for a tower of excited states have been found in non-integrable models. Some of the states of the tower appear to be in the bulk of the energy spectrum, allowing us to make conjectures on the strong Eigenstate Thermalization Hypothesis (ETH). We also generalize these exact states including the tower of states to the generalized integer spin AKLT models. Furthermore, we establish a correspondence between some of our states and those of the Majumdar-Ghosh model, yet another non-integrable model, and extend our construction to the generalized integer spin AKLT models. arXiv:1708.05021v2 [cond-mat.str-el]

Quantum many-body scar states with emergent kinetic constraints and finite-entanglement revivals

“…It is intriguing to note that a similar structure might be at work in the tower of scarred eigenstates of the AKLT model derived in Ref. [17], which also feature emergent kinetic constraints like the ones arising in this work. Ref.…”

“…(2.1) relies on showing that H λ |S n = 0 and proceeds along the lines of the analogous calculation in Ref. [17]; we present it in Appendix A. Physically, the state |S n contains n magnons (i.e., spin flips, or 1s in a background of 0s), each carrying momentum k = π. In fact, when n/L is finite it can be viewed as a condensate of such magnons, as it possesses off-diagonal long-range order (ODLRO) [25] with respect to the "order parameter" Q † , similar to Refs.…”

“…Exact finite-energy-density eigenstates with equal energy spacing and sub-volume-law entanglement [i.e., satisfying conditions ii) and iii)] were found in the Affleck-Kennedy-Lieb-Tasaki (AKLT) spin chain in Refs. [17,18]. However, it is not yet known what quench is necessary to observe revivals of the many-body wavefunction-i.e., it is not known whether condition i) is satisfied in this model.…”

“…This is due to the emergence of a kinetic constraint that forbids them from occupying neighboring sites (a similar constraint was found in Refs. [17,18]). This kinetic constraint is precisely the "Fibonacci" constraint that arises in the experimentally relevant Rydberg-blockaded model [20,21], but in this case is emergent in that it is not present in the underlying Hamiltonian and its other eigenstates.…”

Exact eigenstates in the Lesanovsky model, proximity to integrability and the PXP model, and approximate scar states

“…By adopting the approach of Refs. [18,35] and examining the Schmidt numbers of eigenstates obtained from exact diagonalization (ED) of finite systems, we found two new states with finite Schmidt number of 12 and 16, indicating these states might be "simple" and hinting at their exact expressions.…”