Quantum phase transition as an interplay of Kitaev and Ising interactions

Abstract

We study the interplay between the Kitaev and Ising interactions on both ladder and two dimensional lattices. We show that the ground state of the Kitaev ladder is a symmetry-protected topological (SPT) phase, which is protected by a Z2 × Z2 symmetry. It is confirmed by the degeneracy of the entanglement spectrum and non-trivial phase factors (inequivalent projective representations of the symmetries), which are obtained within infinite matrix-product representation of numerical density matrix renormalization group. We derive the effective theory to describe the topological phase transition on both ladder and two-dimensional lattices, which is given by the transverse field Ising model with/without next-nearest neighbor coupling based on the primary Ising configurations. The ladder has three phases, namely, the Kitaev SPT, symmetry broken ferro/antiferromagnetic order and classical spin-liquid. The non-zero quantum critical point and its corresponding central charge are provided by the effective theory, which are in full agreement with the numerical results, i.e., the divergence of entanglement entropy at the critical point, change of the entanglement spectrum degeneracy and a drop in the ground-state fidelity. The central charge of the critical points are either c=1 or c=2, with the magnetization and correlation exponents being 1/4 and 1/2, respectively. The transition from the classical spin-liquid phase of the frustrated Ising ladder to the Kitaev SPT phase is mediated by a floating phase, which shows strong finite entanglement scaling. In the absence of frustration, the 2D lattice shows a topological phase transition from the Z2 spin-liquid state to the long-range ordered Ising phase at finite ratio of couplings, while in the presence of frustration, an order-by-disorder transition is induced by the Kitaev term. The 2D classical spin-liquid phase is unstable against the addition of Kitaev term toward an ordered phase before the transition to the Z2 spin-liquid state.

Topological spin liquids in the ruby lattice with anisotropic Kitaev interactions

“…The discovery of such quantum magnets presaged the study of other models with anisotropic interactions on different lattices, including decorated honeycomb [14], triangular [15,16], spin ladder [17,18] and ruby [19,20] lattices. The latter one, the ruby color code (RCC) shown in Fig.1, is central to our work in this paper.…”

Anyonic self-induced disorder in a stabilizer code: Quasi many-body localization in a translational invariant model

“…Here we present a novel mechanism toward quasi-MBL in a family of clean self-correcting memories, in particular the Kitaev toric code [37,38] on ladder geometry, a.k.a. the Kitaev ladder (KL) [39,40]. The elementary excitations of KL are associated with point-like quasi-particles, known as electric (e) and magnetic (m) charges.…”

“…surface code [37]) whose width is one. This model represents Z 2 × Z 2 symmetry-protected topological (SPT) order associated to anyonic parities [40,59]. Now we would like to perturb the KL Hamiltonian such that e (charge) and m (flux), corresponding to A s = −1 and B p = −1, respectively, hop across the ladder and gain kinetic energy.…”

Characterization of topological phases in the compass ladder model

“…[29][30][31][32] A promising platform based on the 'inhomogeneous Kitaev ladder model' has been recently proposed, which can read out Majorana fermion qubit states and also perform non-Abelian braiding. [33][34][35][36][37] Our main objective in this paper is to identify the type of quantum phase transitions and different topological phases of the compass ladder and 1D compass models. The compass ladder includes three phases denoted by A, B, and C-see Fig.…”