Given a simple self-dual quantum Hamiltonian $H=KB+\ensuremath{\Gamma}\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}$, where
K and $\ensuremath{\Gamma}$ are coupling constants, and the condition that $[B,[B,[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]]]=16[B,\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{B}]$, then we construct an infinite set of conserved charges
Q2n;
[H,Q2n]=0. In simple models, like the two-dimensional Ising or Baxter eight-vertex, these charges appear in the associated quantum theories and are equivalent to those which result from the transfer-matrix formulation and exact quantum integrability of the system. The power of our result is that it is an operator statement and does not refer to the number of dimensions or the nature of the space-time manifold: lattice, continuum, or loop space. It is suggested how the establishment of this link between duality and integrability could be used to exploit the Kramers-Wannier-type self-duality of the four-dimensional
SU(N) gauge theory to find hidden symmetry.
We review some recent results on discrete symmetries and topological order in spinor Bose–Einstein condensates (BECs) of 23Na. For spin one bosons with two-body scatterings dominated by a total spin equal to two channel, the BECs are in quantum spin nematic states at a low density limit. We study spin correlations in condensates at different limits and analyze hidden symmetries using a non-perturbative approach developed recently. Furthermore we investigate the influence of hidden Z2 symmetries and U(1) quantum orders in symmetry partially restored states, particularly the effects on topological excitations.