Phases of matter are splendid, and get more complicated from physical systems to biochemical systems. Simple physics principles are proposed to categorise various phases of matter. The phys- ical principles are often exemplified by simple idealised models. When they are applied to more complicated biochemical systems, some generalisations may be needed.
In this thesis, two idealised models, dipolar dimer liquid and topological metamaterials with odd elasticity, are discussed. The first model, Dipolar Dimer Liquid (DDL), is motivated by water. It is a simplified model consisting of dipolar dimers on lattice with a Coulombic interaction between the dimers. The Coulombic interaction is an idealised model for the hydrogen bond interaction. It is found that there exists a liquid-liquid phase transition in this model. To understand this phase transition, an exact mapping from the DDL to the annealed Ising model on random graphs has been constructed. Consequently, one may identify the order parameter charactering the liquid-liquid phase transition of the DDL with the order parameter of the Ising model, magnetisation. One may further bound the critical temperature with the help of this exact mapping and the exactly solved Ising models in 2D. The estimation of the critical temperature has been confirmed by the numerical simulations. A quantum generalisation of the DDL has also been proposed. Interesting interplay of the Coulombic interaction and the quantum solid-liquid transition is discussed.
The second model is motivated by active matters. In the presence of active energy pumping, a strange stress response, the odd elasticity, is admissible. In our model, the odd elasticity comes from an idealised model of metabeams, and the dynamics of the system becomes non-Hermitian. Interestingly, the topological properties still survive in the non-Hermitian dynamics. But the bulk modes also become exponentially localised, which is called non-Hermitian skin eﬀect. To characterise the topological properties, the definition of the Brillouin zone has been generalised, and the definition of the Zak-Berry phase has been clarified for the non-Hermitian systems. The 1D rotor chain and 2D rotor lattice have been discussed in detail as demonstrating examples.