Emergence of different interesting and insightful phenomena in different length scales is the heart of quantum many-body system. We present emergence of quantum phases for the interacting helical liquid of topological quantum matter. We also observe that Luttinger liquid parameter plays a significant role to determine different quantum phases. We use three sets of renormalisation group (RG) equations to solve emergent quantum phases for our model Hamiltonian system. Two of them are the quantum Berezinskii–Kosterlitz–Thouless (BKT) equations. We show explicitly from the study of length scale-dependent emergent physics that there is no evidence of Majorana–Ising transition for the two sets of quantum BKT equations, i.e., the system is either in the topological superconducting phase or in the Ising phase. The whole set of RG equation shows the evidence of length scale-dependent Majorana–Ising transition. Emergence of length scale-dependent quantum phases can be observed in topological materials which exhibit fundamentally new physical phenomena with potential applications for novel devices and quantum information technology.

We study and present the results of curvature for different symmetry classes (BDI, AIII and A) of model Hamiltonians and also present the transformation of model Hamiltonian from one distinct symmetry class to the other based on the curvature property. We observe the mirror symmetric curvature for the Hamiltonian with BDI symmetry class but there is no evidence of such behaviour for Hamiltonians of AIII symmetry class. We show the origin of torsion and its consequences on the parameter space of topological phase of the system. We find the evidence of torsion for the Hamiltonian of A symmetry class. We present Serret–Frenet equations for all model Hamiltonians in R$^3$ space. To the best of our knowledge, this is the first application of curvature theory to the model Hamiltonian of different symmetry classes which belong to the topological state of matter.

An attempt is made to quantum simulate the topological characterisations, such as winding number, geometric phase and symmetry properties for a quantum-simulated Kitaev chain. We find that α (ratio between the spin–orbit coupling and magnetic field) and the range of momentum space of consideration, play crucial roles in topological classification. We show explicitly that, for this quantum-simulated Kitaev chain, the topological quantum phase transition occurs for finite range of momentum. We observe that the quasiparticle mass of thefermion plays a significant role in topological quantum phase transition. We also show that the symmetry properties of the simulated Kitaev chain is the same as that of the original Kitaev chain. The exact solution of the simulated Kitaev chain is given. This work provides a new perspective on new emerging quantum simulator and also for the topological state of matter.