Study of quantum phase transitions and topological phases in chains and ladders

Abstract

The ground-state phase diagrams and order parameters of low-dimensional quantum models are analyzed. Those models in their spin representation are the dimerized spin-1/2 XY and XYZ chains, and the two-leg ladders with anisotropy and three different dimerization patterns, in the presence of uniform and staggered transverse fields. The analysis is done by using the effective quadratic fermionic Hamiltonian of models, resulting from the Hatree-Fock mean-field approximation. In the fermionic representation, those models are equivalent to the generic Kitaev -Majorana chains/ladders with the proper parametrizations. An exact solvable model, the XY chain has a rich phase diagram, and its distinct phases are identified by the local and nonlocal (string) order parameters. We have calculated all the local order parameters (spontaneous magnetization) and the nonlocal order parameters within the same systematic framework, along with the winding numbers for all regimes of the phase diagram. By combining the exact and the meanfield methods, the local and string order parameters on the phase diagram of the XYZ chain, are identified and calculated. We found similar qualitative pictures on the phase diagrams of XY and XYZ chains, where the corresponding parameters of the latter model are renormalized by the interaction ∆ = Jz/J. For both models, the topological nontrivial phase is shown to have a peculiar oscillating order with the period of a four lattice spacing, not reported before and awaiting for its experimental confirmation. The detailed analysis of patterns of the string order is given. Moreover, the trivial phases of both the cases are investigated by local order parameters (components of spontaneous magnetization). The special XXZ limit of the model with additional U(1) symmetry is in agreement with the Lieb-Schiltz-Mattis theorem and its extensions, plateaus of magnetization, and some additional conserving quantities. We have shown that within the XYZ chain, where the plateaus are smeared, the robust oscillating string order parameter is continuously connected to its XXZ limit. Also, the nontrivial winding number and zero-energy localized Majorana edge states, as additional attributes of the topological order, are robust in that phase, even off the line of U(1) symmetry. The phase diagram of the isotropic two-leg ladder is investigated by calculating the field-induced magnetization at each point along the external field. In the phase diagram, the two kind of phases, gapped plateau and gapless Lutinger liquid, (LL) are identified. In the applied uniform field, those models are in agreement with the quantization conditions of the magnetization plateaus. The existence of the mid-plateau in the staggered ladder with columnar field, we report for the first time is an indication of a new spin gapped phase in this type of spin structure. For the staggered and columnar ladder, the alternating field only modifies the phase boundaries of the phase diagram. The ladder with the rung dimerization and columnar field exhibits additional quantum phase transition by closing and re-opening the zero-plateau and mid-plateau gapped phases with respect to the alternating field. The Hatree-Fock mean-field Hamiltonian of the ladders with an anisotropy and two dimerization patterns, map onto the sum of two quadratic Majorana Hamiltonians, which are dual to a sum of two (even/odd) XY quantum chains in the alternating transverse fields. The mapping between the effective Hamiltonian of the ladder and the pair of the dual XY chains considerably simplifies calculations of the order parameters, and analyses of the hidden symmetry breaking. The ground state phase diagram of the staggered ladder contains nine phases: four of them are conventional antiferromagnets, while the other five possess the non-local brane orders. Using the dualities and the newly found exact results for the local and string order parameters of the transverse XY chains, we were able to find analytically all the magnetizations and the brane order parameters for the staggered case, as well as the functions of the renormalized couplings of the effective Hamiltonian. The columnar ladder has three ground-state phases, and it does not possess a magnetic long-range order. The brane order parameters for these phases are numerically calculated from the Toeplitz determinants. We expect this study to motivate the search for the real spin-Peierls anisotropic ladder compounds, which can undergo the predicted quantum phase transitions with a gap closure and distinct brane orders.