We demonstrate that the exact nonequilibrium steady state of the one-dimensional Heisenberg $XXZ$ spin chain driven by boundary Lindblad operators can be constructed explicitly with a matrix product ansatz for the nonequilibrium density matrix. For the isotropic Heisenberg chain, polarized at the boundaries in different directions with a non-zero twist angle, we calculate the exact magnetization profiles and magnetization currents. The in-plane steady-state magnetization profiles are harmonic functions with a frequency proportional to the twist angle. In-plane steady-state magnetization currents are subdiffusive and vanish as the boundary coupling strength increases, while the transverse current is diffusive and saturates as the coupling strength becomes large. The anisotropic chain exhibits spin helix states at special values of the anisotropy where the transverse current is independent of system size, even for non-integrable higher-spin chains.

Recently, the notion of symmetry has been extended from 0-symmetry described by group to higher symmetry described by higher group. In this talk, we show that the notion of symmetry can be generalized even further to "algebraic higher symmetry". Then we will describe an even more general point of view of symmetry, which puts the (generalized) symmetry charges and topological excitation at equal footing: symmetry can be viewed as gravitational anomaly, or symmetry can be viewed as shadow topological order in one higher dimension. This picture allows us to see many duality relations between seeming unrelated symmetries.

The eight-vertex model is an useful description that generalizes several spin systems, as well as the more common six-vertex model, and others. In a special "free-fermion" regime, it is known since the work of Fan, Lin, Wu in the late 60s that the model can be mapped to non-bipartite dimers. However, no general theory is known for dimers in the non-bipartite case, contrary to the extensive rigorous description of Gibbs measures by Kenyon, Okounkov, Sheffield for bipartite dimers. In this talk I will show how to transform these non-bipartite dimers into bipartite ones, on generic planar graphs. I will mention a few consequences: computation of long-range correlations, criticality and critical exponents, and their "exact" application to Z-invariant regimes on isoradial graphs.