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Within the block spin renormalization group we are able to construct the"exact" phase diagram of the XYZ spin chain. First we identify the Ising orderalong $\hat x$ or $\hat y$ as attractive renormalization group fixed points ofthe Kitaev chain. Then in a global phase space composed of the anisotropy$\lambda$ of the XY interaction and the coupling $\Delta$ of the$\Delta\sigma^z\sigma^z$ interaction we find that the above fixed points remainattractive in the two dimesional parameter space. We therefore classify thegapped phases of the XYZ spin chain as: (1) either attracted to the Ising limitof the Kitaev-chain which in turn is characterized by winding number $\pm 1$depending whether the Ising order parameter is along $\hat x$ or $\hat y$directions; or (2) attracted to the Mott phases of the underlying Jordan-Wignerfermions which is characterized by zero winding number. We therefore establishthat the exact phase boundaries of the XYZ model in Baxter's solution indeedcorrespond to topological phase transitions. The topological nature of thephase transitions of the XYZ model justifies why our analytical solution of thethree-site problem which is at the core of the renormalization group treatmentis able to produce the exact phase diagram of Baxter's solution. We argue thatthe distribution of the winding numbers between the three Ising phases is amatter of choice of the coordinate system, and therefore the Mott-Ising phaseis entitled to host apprpriate form of zero modes. We further observe that therenormalization group flow can be cast into a geometric progression of aproperly identified parameter. We show that this new parameter is actually thesize of the (Majorana) zero modes.

Exact phase diagram and topological phase transitions of the XYZ spin chain