The new localized excitations of some (2+1)-dimensional integrable models obtained by the multi-linear variable separation approach (MLVSA) are reviewed. A universal formula with some arbitrary functions is obtained for some suitable physical quantities for various integrable models. By selecting the arbitrary functions appropriately, one may obtain abundant interesting localized excitations like the multi-dromions, lumps, ring solitons, breathers, instantons, peakons, compactons, foldons, chaotic and fractal patterns and so on. The localized excitations possess quite rich interaction behaviors.

No abstract

Density Matrix Renormalization Group (DMRG) and its extensions in the form of Matrix Product States (MPS) are arguably the choice for the study of one dimensional quantum systems in the last three decades. However, due to the limited entanglement encoded in the wave-function ansatz, to maintain the accuracy of DMRG with the increase of the system size in the study of two dimensional systems, exponentially increased resources are required, which limits the applicability of DMRG to only narrow systems. In this work, we introduce a new ansatz in which DMRG is augmented with disentanglers to encode area-law-like entanglement entropy (entanglement entropy supported in the new ansatz scales as $l$ for a $l \times l$ system). In the new method, the $O(D^3)$ low computational cost of DMRG is kept (with an overhead of $O(d^4)$ and $d$ the dimension of the physical degree of freedom). We perform benchmark calculations with this approach on the two dimensional transverse Ising and Heisenberg models. This new ansatz extends the power of DMRG in the study of two-dimensional quantum systems.