Finite Size Scaling Around One Dimensional Topological Quantum Phase Transitions

Abstract

The critical point of a topological phase transition is described by a conformal field theory. We first investigate the finite-size scaling away from criticality of the ground state energy and find a scaling function, which discriminates between phases with different topological indexes. This function appears to be universal for all five Altland-Zirnbauer symmetry classes with non-trivial topology in one spatial dimension. We obtain an analytic form of the scaling function and compare it with numerical results. Then we verify the universality of the scaling function for the topological transition between dimerized and Haldane phases in bilinear-biquadratic spin-1 chain. To this end we perform high-accuracy variational matrix product state simulations. We show that the scaling function, expressed in terms of $L/\xi$, where $L$ is the chain length and $\xi$ is the correlation length, coincides with that of three species of non-interacting massive Majorana fermions. The latter is known to be a proper description of the conformal critical theory with central charge $c=3/2$. We have shown that it still holds away from the conformal point, including the finite size corrections. Finally we consider scaling of the entanglement entropy across a topological quantum phase transition for the Kitaev chain model. The change of the topology manifests itself in a sub-leading term, which scales as $L^{-1/\alpha}$ with the size of the subsystem $L$, here $\alpha$ is the R\'{e}nyi index. This term reveals the scaling function $h_\alpha(L/\xi)$, where $\xi$ is the correlation length, which is sensitive to the topological index. The scaling function $h_\alpha(L/\xi)$ is independent of model parameters, suggesting some degree of its universality.