Symmetry fractionalization: Symmetry-protected topological phases of the bond-alternating spin-1/21/2 Heisenberg chain

We study different phases of the one-dimensional bond-alternating spin-$1/2$ Heisenberg model by using the symmetry fractionalization mechanism. We employ the infinite matrix-product state representation of the ground state (through the infinite-size density matrix renormalization group algorithm) to obtain inequivalent projective representations of the (unbroken) symmetry groups of the model, which are used to identify the different phases. We find that the model exhibits trivial as well as symmetry-protected topological phases. The symmetry-protected topological phases are Haldane phases on even/odd bonds, which are protected by the time-reversal (acting on the spin as $\sigma\rightarrow-\sigma$), parity (permutation of the chain about a specific bond), and dihedral ($\pi$-rotations about a pair of orthogonal axes) symmetries. Additionally, we investigate the phases of the most general two-body bond-alternating spin-$1/2$ model, which respects the time-reversal, parity, and dihedral symmetries, and obtain its corresponding twelve different types of the symmetry-protected topological phases.Comment: 9 pages, 5 figure

Quantum phase transition as an interplay of Kitaev and Ising interactions

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 $\mathbb{Z}_2 \times \mathbb{Z}_2$ 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. 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. In the absence of frustration, the 2D lattice shows a topological phase transition from the $\mathbb{Z}_2$ 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 $\mathbb{Z}_2$ spin-liquid state.Comment: 16 pages, 18 figure

Real space renormalization of Majorana fermions in quantum nano-wire superconductors

We have applied the real space quantum renormalization group approach to study the topological quantum phase transition in the one-dimensional chain of a spinless p-wave superconductor. We investigate the behavior of local compressibility and ground-state fidelity of the Kitaev chain. We show that the topological phase transition is signaled by the maximum of local compressibility at the quantum critical point tuned by the chemical potential. Moreover, a sudden drop of the ground-state fidelity and the divergence of fidelity susceptibility at the topological quantum critical point have been used as a proper indicators for the topological quantum phase transition, which signals the appearance of Majorana fermions. We also present the scaling analysis of ground-state fidelity near the critical point that manifests the universal information about the topological phase transition.Comment: 7 pages, 9 figures; to appear in Journal of the Physical Society of Japan (JPSJ

Phase Diagram of the One Dimensional S=1/2S=1/2 XXZXXZ model with Ferromagnetic nearest-neighbor and Antiferromagnetic next-nearest neighbor interactions

We have studied the phase diagram of the one dimensional $S=1/2$ $XXZ$ model with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest neighbor interactions. We have applied the quantum renormalization group (QRG) approach to get the stable fixed points and the running of coupling constants. The second order QRG has been implemented to get the self similar Hamiltonian. This model shows a rich phase diagram which consists of different phases which possess the quantum spin-fluid and dimer phases in addition to the classical N\'{e}el and ferromagnetic ones. The border between different phases has been shown as a projection onto two different planes in the phase space

Second order quantum renormalisation group of XXZ chain with next nearest neighbour interactions

We have extended the application of quantum renormalisation group (QRG) to the anisotropic Heisenberg model with next-nearest neighbour (n-n-n) interaction. The second order correction has to be taken into account to get a self similar renormalized Hamiltonian in the presence of n-n-n-interaction. We have obtained the phase diagram of this model which consists of three different phases, i.e, spin-fluid, dimerised and Ne'el types which merge at the tri-critical point. The anisotropy of the n-n-n-term changes the phase diagram significantly. It has a dominant role in the Ne'el-dimer phase boundary. The staggered magnetisation as an order parameter defines the border between fluid-Ne'el and Ne'el-dimer phases. The improvement of the second order RG corrections on the ground state energy of the Heisenberg model is presented. Moreover, the application of second order QRG on the spin lattice model has been discussed generally. Our scheme shows that higher order corrections lead to an effective Hamiltonian with infinite range of interactions.Comment: 10 pages, 4 figures and 1 tabl