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

Abstract

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 order alongx orŷ as attractive renormalization group fixed points of the Kitaev chain. Then in a global phase space composed of the anisotropy λ of the XY interaction and the coupling ∆ of the ∆σ z σ z interaction we find that the above fixed points remain attractive in the two dimesional parameter space. We therefore classify the gapped phases of the XYZ spin chain as: (1) either attracted to the Ising limit of the Kitaev-chain which in turn is characterized by winding number ±1 depending whether the Ising order parameter is alongx orŷ directions; or (2) attracted to the Mott phases of the underlying Jordan-Wigner fermions which is characterized by zero winding number. We therefore establish that the exact phase boundaries of the XYZ model in Baxter's solution indeed correspond to topological phase transitions. The topological nature of the phase transitions of the XYZ model justifies why our analytical solution of the three-site problem which is at the core of the renormalization group treatment is able to produce the exact phase diagram of Baxter's solution. We argue that the distribution of the winding numbers between the three Ising phases is a matter of choice of the coordinate system, and therefore the Mott-Ising phase is entitled to host apprpriate form of zero modes. We further observe that the renormalization group flow can be cast into a geometric progression of a properly identified parameter. We show that this new parameter is actually the size of the (Majorana) zero modes.

Zero-energy quasiparticles in an interacting nanowire containing a topological Josephson junction

“…From the known properties of the XYZ model, 24 it can then be inferred that either attractive or repulsive interactions close the superconducting gap when |U | = W +∆. For interactions stronger than this critical value, the system is again in a gapped phase.…”

Topological phase transition of the anisotropic XY model with Dzyaloshinskii-Moriya interaction

“…As can be seen there are two fixed points. Repulsive fixed point at λ 0 * = 0 and two attractors at λ ± * = ±1 27 . These values do not change by replacing D = 0 with D = 4.…”

“…In this process some coefficients from the ground state wave function will be collected. For the coupling of neighboring coarse-grained spins one needs to collect interactions on the bonds connecting a cluster to its neighbors 27 . The effective Hamiltonian in the n'th step of the above process will be…”

“…We find that there are two global attractors that attract the flow to gapped states which correspond to Ising phases polarized along x or y directions 7 . These two are separated by a gap-closing and hence should correspond to topologically non-trivial phases, similar to its one-dimensional counterpart 27 . We corroborate the topological nature of this quantum phase transition with the calculation of concurrence.…”

“…The whole λ = 0 line in the plane of λ and D will be a gapless repeller which is unstable with respect to smallest anisotropy λ (irrespective of the sign of λ). This is reminiscent of the pairing instability in a gapless system of Jordan-Winger fermions 27 which from the exact solution of the one-dimensional problem can be interpreted as the p-wave pairing interaction. Indeed such a p-wave pairing resulting from the anisotropy λ can be obtained from the study of equivalent fermions coupled to Chern-Simons gauge fields on the honeycomb lattice 28 .…”