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arXiv:1603.03166 · Physical review. B./Physical review. B

Topological and dynamical properties of a generalized cluster model in one dimension

Takumi Ohta, Shu Tanaka, Ippei Danshita, Keisuke Totsuka

Published Apr 18, 2016• 21 citations
arXivHTMLPDF

Abstract

We study the ground-state phase diagram and dynamics of the one-dimensional cluster model with several competing interactions. Paying particular attention to the relation between the entanglement spectrum (ES) and the bulk topological (winding) number, we first map out the ground-state phases of the model and determine the universality classes of the transitions from the exact solution. We then investigate the dynamical properties during interaction sweeps through the critical points of topological phase transitions. When the sweep speed is slow, the correlation functions and the entanglement entropy exhibit spatially periodic structures. On top of this, the levels in the ES oscillate temporally during the dynamics. By explicitly calculating the above quantities for excited states, we attribute these behaviors to the Bogoliubov quasiparticles generated near the critical points. We also show that the ES reflects the strength of the Majorana correlation even for the excited states.

DOI
10.1103/physrevb.93.165423

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We study the ground-state phase diagram and dynamics of the one-dimensional cluster model with several competing interactions. Paying particular attention to the relation between the entanglement spectrum (ES) and the bulk topological (winding) number, we first map out the ground-state phases of the model and determine the universality classes of the transitions from the exact solution. We then investigate the dynamical properties during interaction sweeps through the critical points of topological phase transitions. When the sweep speed is slow, the correlation functions and the entanglement entropy exhibit spatially periodic structures. On top of this, the levels in the ES oscillate temporally during the dynamics. By explicitly calculating the above quantities for excited states, we attribute these behaviors to the Bogoliubov quasiparticles generated near the critical points. We also show that the ES reflects the strength of the Majorana correlation even for the excited states.
We numerically study quantum phase transitions and dynamical properties in the one-dimensional cluster model with several interactions by using the time-evolving block decimation method for infinite systems and the exact diagonalization. First, boundaries among several quantum phases of the model are determined from energy gap and each phase is characterized by order parameters and the entanglement spectrum (ES). We confirm that in the model with open boundary condition the degeneracy of the lowest levels in the ES corresponds to that of the ground states. Then, using the time-dependent Bogoliubov transformation with open boundary condition, we investigate dynamical properties during an interaction sweep through the critical point which separates two topological phases involving four-fold degeneracy in the ground state. After a slow sweep across the critical point, we observe spatially periodic structures in the string correlation functions and the entanglement entropy. It is shown that the periodicities stem from the Bogoliubov quasiparticles generated near the critical point.
Within the framework of a one-dimensional model of interacting electrons, the ground state of an electron liquid is studied. Using the exact solution of the model, the ground state phase diagram and zero-energy Majorana edge functions in a finite chain are calculated. The winding number invariant reflects the topological nature of the electron liquid. The phase diagram includes two topological phases with different winding number invariants, the topologically trivial Mott insulator phase, and three critical phase transition points. Numerical calculations confirm and illustrate the analytical results.
We numerically study the quantum phase transitions and the stability of Majorana zero modes in a generalized Kitaev model in one dimension when the chemical potential is periodically modulating in space. By using the exact diagonalization method for open boundary condition, we investigate the ground-state phases in terms of the non-local properties such as the entanglement spectrum (ES) and the string correlation functions. When we vary the phase of the modulation, the number of the Majorana zero modes changes, which manifests itself in the degeneracy of the lowest level of the ES. Next, we study the quantum phase transitions driven by the change in the amplitude of the modulation. In particular, for certain values of the wave number and the phase of the modulation, we observe a quantum phase transition from one topological phase into another where the string correlation function oscillates in space. We also show a case where the degeneracy of the ES does not change even for large enough amplitude of the modulation. Finally, we characterize the phases of the system with periodic boundary condition by the topological invariant, which reflects the number of the zero-energy excitations.
We study the phases of an exactly solvable one dimensional model with 4−4-4−dimensional Γ−Γ-Γ−matrix degrees of freedom on each site. The Γ−Γ-Γ−matrix model has a large set of competing interactions and displays a rich phase diagram with critical lines and multi-critical points. We work with the model with certain Z2Z_2Z2​ symmetries and identify the allowed symmetry protected topological phases using the winding number as the topological invariant. The model belongs to the CII-class of the 10−10-10−fold classification and allows for integer values of the winding number. We confirm that the system also hosts localized zero energy Majorana edge modes, consistent with the integer value of the winding number of the corresponding phase. We further study scaling and universality behaviour of the various topological phase transitions.
Within the framework of an one-dimension model of interacting electrons, the ground state of an electron liquid is studied. Using the exact solution of the model, the ground state phase diagram and zero-energy Majorana edge functions in a finite chain are calculated. The winding number invariant reflects the topological nature of the electron liquid. The phase diagram includes two topological phases with different winding number invariants, the topologically trivial Mott insulator phase, and three critical phase transition points. Numerical calculations confirm and illustrate the analytical results.
We investigate the one-dimensional spin-1/2 XX chain with anisotropic three-spin interactions, which hosts four distinct topological phases separated by critical lines. Using exact fermionization, we analyze the spin squeezing parameter, entanglement entropy, and bulk entanglement spectrum to characterize the ground state. Our results show that the system remains unsqueezed across the entire phase diagram, with maximal unsqueezing at criticality. The entanglement entropy exhibits cusp-like singularities at the critical lines, while the extracted central charges reveal distinct critical behaviors. Furthermore, the lowest level of the bulk entanglement spectrum is four-fold degenerate in two phases and two-fold in the others, reflecting the topological nature of each phase. These findings offer deeper insight into the interplay between topology, criticality, and quantum correlations in low-dimensional spin systems.