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arXiv:2311.11684

Emergence of new topological gapless phases in the modified square-lattice Kitaev model

Jihyeon Park, Gun Sang Jeon

Published Nov 20, 2023• 0 citations
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Abstract

We investigate emergent topological gapless phases in the square-lattice Kitaev model with additional hopping terms. In the presence of nearest-neighbor hopping only, the model is known to exhibit gapless phases with two topological gapless points. When the strength of the newly added next-nearest-neighbor hopping is smaller than a certain value, qualitatively the same phase diagram persists. We find that further increase of the extra hopping results in a new topological phase with four gapless points. We construct a phase diagram to clarify the regions of emergent topological gapless phases as well as topologically trivial ones in the space of the chemical potential and the next-nearest-neighbor hopping strength. We examine the evolution of the gapless phases in the energy dispersions of the bulk as the chemical potential varies. The topological properties of the gapless phases are characterized by the winding numbers of the present gapless points. We also consider the ribbon geometry to examine the corresponding topological edge states. It is revealed that Majorana-fermion edge modes exist as flat bands in topological gapless phases. We also perform the analytical calculation as to Majorana-fermion zero-energy modes and discuss its implications on the numerical results.

DOI
10.48550/arxiv.2311.11684

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We investigate emergent topological gapless phases in the square-lattice Kitaev model with additional hopping terms. In the presence of nearest-neighbor hopping only, the model is known to exhibit gapless phases with two topological gapless points. When the strength of the newly added next-nearest-neighbor hopping is smaller than a certain value, qualitatively the same phase diagram persists. We find that further increase of the extra hopping results in a new topological phase with four gapless points. We construct a phase diagram to clarify the regions of emergent topological gapless phases as well as topologically trivial ones in the space of the chemical potential and the next-nearest-neighbor hopping strength. We examine the evolution of the gapless phases in the energy dispersions of the bulk as the chemical potential varies. The topological properties of the gapless phases are characterized by the winding numbers of the present gapless points. We also consider the ribbon geometry to examine the corresponding topological edge states. It is revealed that Majorana-fermion edge modes exist as flat bands in topological gapless phases. We also perform the analytical calculation as to Majorana-fermion zero-energy modes and discuss its implications on the numerical results.
We study the effects of the next-nearest-neighbor hopping and nearest-neighbor interactions on topological phases in a one-dimensional generalized Kitaev model. In the noninteracting case, we define a topological number and calculate exactly the phase diagram of the system. With addition of the next-nearest-neighbor hopping, the change of phase boundary between the topological and trivial regions can be described by an effective shift of the chemical potential. In the interacting case, we obtain the entanglement spectrum, the degeneracies of which correspond to the topological edge modes, by using the infinite time-evolving block decimation method. The results show that the interactions change the phase boundary as adding an effective chemical potential which can be explained by the change of the average number of particles.
We study the topological feature of gapless states in the fermionic Kitaev model on a square lattice. There are two types of gapless states which are topologically trivial and nontrivial. We show that the topological gapless phase lives in a wide two-dimensional parameter region and are characterized by two vertices of an auxiliary vector field de-fined in the two-dimensional momentum space, with opposite winding numbers. The isolated band touching points, as the topological defects of the field, move, emerge, and disappear as the parameters vary. The band gap starts to open only at the merg-ing points, associated with topologically trivial gapless states. The symmetry protect-ing the topological gapless phase and the robustness under perturbations are also discussed.
The topological properties of the Kitaev chain model with extended-range interactions are investigated, focusing on cases where the topological winding number is preserved. We assume that the pairing and hopping terms decay algebraically in space with exponents ααα and βββ, respectively. We show that in the truncated-range scenario, there are as many distinct topological phases as the number of coupled neighboring sites. In addition, an explicit analytical formulation is provided to evaluate the topological invariant and the phase transitions that emerge in these systems. Besides the analytical description, we introduce a new physical insight into the topological excitations of the ground-state by measuring the spatial distribution of the edge modes with the Majorana average position. Taking the next-nearest neighbor Kitaev chains as a probe, various numerical calculations of Majorana edge states were performed in finite-size clusters to determine the sensitivity of the topological zero energy modes to the parameters of interest. The occupation of edge-to-edge non-local fermion states is computed and defined as an effective parity. Such an effective parity exhibits new interesting features beyond the energetic exchange from the ground-state fermion parity switches, which are related to the distribution of the respective edge modes. Our calculations show a direct correlation between the ground-state fermion parity and the edge occupation numbers, which are translated into localization and delocalization of the Majorana average position.
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Abstract We introduce a new type of one-dimensional Kitaev chain with staggered p-wave superconducting pairing. We find three physical regimes in this model by tuning the p-wave pairing and the chemical potential of the system. In the topologically nontrivial phase, there are two Majorana zero modes localized at the opposite ends of the lattice, which are characterized and protected by nonzero topological invariants. More interestingly, we also find a regime where the system can hold four unprotected nonzero-energy edge modes in the trivial phase, which is analogous to a weak topological phase. The third regime is also trivial but holds no edge modes. The emergence of zero- and nonzero-energy edge modes in the system are analyzed by transforming the lattice model into a ladder consisting of Majorana fermions, where the competition between the intra- and inter-leg couplings leads to different phases. We further investigate the properties of edge modes under the influences of dissipation, which is represented by introducing a imaginary part in the chemical potential. Our work unveils the exotic properties induced by the staggered p-wave pairing and provides a new platform for further exploration of Majorana edge modes.
A time-reversal invariant Kitaev-type model is introduced in which spins (Dirac matrices) on the square lattice interact via anisotropic nearest-neighbor and next-nearest-neighbor exchange interactions. The model is exactly solved by mapping it onto a tight-binding model of free Majorana fermions coupled with static Z2{\mathbb{Z}}_{2}Z2​ gauge fields. The Majorana fermion model can be viewed as a model of time-reversal-invariant superconductor and is classified as a member of symmetry class DIII in the Altland-Zirnbauer classification. The ground-state phase diagram has two topologically distinct gapped phases which are distinguished by a Z2{\mathbb{Z}}_{2}Z2​ topological invariant. The topologically nontrivial phase supports both a Kramers' pair of gapless Majorana edge modes at the boundary and a Kramers' pair of zero-energy Majorana states bound to a 0-flux vortex in the $\ensuremath{\pi}$-flux background. Power-law decaying correlation functions of spins along the edge are obtained by taking the gapless Majorana edge modes into account. The model is also defined on the one-dimension ladder, in which case again the ground-state phase diagram has Z2{\mathbb{Z}}_{2}Z2​ trivial and nontrivial phases.